diff --git "a/random/test_formal_informal.json" "b/random/test_formal_informal.json" new file mode 100644--- /dev/null +++ "b/random/test_formal_informal.json" @@ -0,0 +1,3942 @@ +[ + { + "formal": "exists_open_nhds_one_split ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : Monoid M inst\u271d : ContinuousMul M s : Set M hs : s \u2208 \ud835\udcdd 1 \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 ** 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) ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : Monoid M inst\u271d : ContinuousMul M s : Set M hs : s \u2208 \ud835\udcdd 1 this : (fun a => a.1 * a.2) \u207b\u00b9' s \u2208 \ud835\udcdd (1, 1) \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 ** simpa only [prod_subset_iff] using exists_nhds_square this ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : Monoid M inst\u271d : ContinuousMul M s : Set M hs : s \u2208 \ud835\udcdd 1 \u22a2 s \u2208 \ud835\udcdd (1 * 1) ** simpa only [one_mul] using hs ** Qed", + "informal": "" + }, + { + "formal": "ContMDiffAt.comp_of_eq ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s s\u2081 t : Set M x\u271d : M m n : \u2115\u221e g : M' \u2192 M'' x : M y : M' hg : ContMDiffAt I' I'' n g y hf : ContMDiffAt I I' n f x hx : f x = y \u22a2 ContMDiffAt I I'' n (g \u2218 f) x ** subst hx ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s s\u2081 t : Set M x\u271d : M m n : \u2115\u221e g : M' \u2192 M'' x : M hf : ContMDiffAt I I' n f x hg : ContMDiffAt I' I'' n g (f x) \u22a2 ContMDiffAt I I'' n (g \u2218 f) x ** exact hg.comp x hf ** Qed", + "informal": "" + }, + { + "formal": "midpoint_vsub ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2077 : Ring R inst\u271d\u2076 : Invertible 2 inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' x y z p\u2081 p\u2082 p : P \u22a2 midpoint R p\u2081 p\u2082 -\u1d65 p = \u215f2 \u2022 (p\u2081 -\u1d65 p) + \u215f2 \u2022 (p\u2082 -\u1d65 p) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Language.Theory.models_iff_finset_models ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L \u22a2 T \u22a8\u1d47 \u03c6 \u2194 \u2203 T0, \u2191T0 \u2286 T \u2227 \u2191T0 \u22a8\u1d47 \u03c6 ** simp only [models_iff_not_satisfiable] ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L \u22a2 \u00acIsSatisfiable (T \u222a {Formula.not \u03c6}) \u2194 \u2203 T0, \u2191T0 \u2286 T \u2227 \u00acIsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) ** rw [\u2190 not_iff_not, not_not, isSatisfiable_iff_isFinitelySatisfiable, IsFinitelySatisfiable] ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L \u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194 \u00ac\u2203 T0, \u2191T0 \u2286 T \u2227 \u00acIsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) ** push_neg ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L \u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194 \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) ** letI := Classical.decEq (Sentence L) ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) \u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194 \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) ** constructor ** case mp L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) \u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2192 \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) ** intro h T0 hT0 ** case mp L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) h : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0 T0 : Finset (Sentence L) hT0 : \u2191T0 \u2286 T \u22a2 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) ** 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 _)) ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) h : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0 T0 : Finset (Sentence L) hT0 : \u2191T0 \u2286 T \u22a2 \u2191(T0 \u222a {Formula.not \u03c6}) \u2286 T \u222a {Formula.not \u03c6} ** simp only [Finset.coe_union, Finset.coe_singleton] ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) h : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0 T0 : Finset (Sentence L) hT0 : \u2191T0 \u2286 T \u22a2 \u2191T0 \u222a {Formula.not \u03c6} \u2286 T \u222a {Formula.not \u03c6} ** exact Set.union_subset_union hT0 (Set.Subset.refl _) ** case mpr L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) \u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})) \u2192 \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0 ** intro h T0 hT0 ** case mpr L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) h : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) T0 : Finset (Sentence L) hT0 : \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u22a2 IsSatisfiable \u2191T0 ** exact IsSatisfiable.mono (h (T0.erase (Formula.not \u03c6))\n (by simpa using hT0)) (by simp) ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) h : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) T0 : Finset (Sentence L) hT0 : \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u22a2 \u2191(Finset.erase T0 (Formula.not \u03c6)) \u2286 T ** simpa using hT0 ** L : Language T : Theory L \u03b1 : Type w n : \u2115 \u03c6 : Sentence L this : DecidableEq (Sentence L) := Classical.decEq (Sentence L) h : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6}) T0 : Finset (Sentence L) hT0 : \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u22a2 \u2191T0 \u2286 \u2191(Finset.erase T0 (Formula.not \u03c6)) \u222a {Formula.not \u03c6} ** simp ** Qed", + "informal": "" + }, + { + "formal": "Matrix.blockDiagonal'_diagonal ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 p : Type u_5 q : Type u_6 m' : o \u2192 Type u_7 n' : o \u2192 Type u_8 p' : o \u2192 Type u_9 R : Type u_10 S : Type u_11 \u03b1 : Type u_12 \u03b2 : Type u_13 inst\u271d\u00b3 : DecidableEq o inst\u271d\u00b2 : Zero \u03b1 inst\u271d\u00b9 : Zero \u03b2 inst\u271d : (i : o) \u2192 DecidableEq (m' i) d : (i : o) \u2192 m' i \u2192 \u03b1 \u22a2 (blockDiagonal' fun k => diagonal (d k)) = diagonal fun ik => d ik.fst ik.snd ** ext \u27e8i, k\u27e9 \u27e8j, k'\u27e9 ** case a.mk.h.mk l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 p : Type u_5 q : Type u_6 m' : o \u2192 Type u_7 n' : o \u2192 Type u_8 p' : o \u2192 Type u_9 R : Type u_10 S : Type u_11 \u03b1 : Type u_12 \u03b2 : Type u_13 inst\u271d\u00b3 : DecidableEq o inst\u271d\u00b2 : Zero \u03b1 inst\u271d\u00b9 : Zero \u03b2 inst\u271d : (i : o) \u2192 DecidableEq (m' i) d : (i : o) \u2192 m' i \u2192 \u03b1 i : o k : m' i j : o k' : m' j \u22a2 blockDiagonal' (fun k => diagonal (d k)) { fst := i, snd := k } { fst := j, snd := k' } = diagonal (fun ik => d ik.fst ik.snd) { fst := i, snd := k } { fst := j, snd := k' } ** simp only [blockDiagonal'_apply, diagonal] ** case a.mk.h.mk l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 p : Type u_5 q : Type u_6 m' : o \u2192 Type u_7 n' : o \u2192 Type u_8 p' : o \u2192 Type u_9 R : Type u_10 S : Type u_11 \u03b1 : Type u_12 \u03b2 : Type u_13 inst\u271d\u00b3 : DecidableEq o inst\u271d\u00b2 : Zero \u03b1 inst\u271d\u00b9 : Zero \u03b2 inst\u271d : (i : o) \u2192 DecidableEq (m' i) d : (i : o) \u2192 m' i \u2192 \u03b1 i : o k : m' i j : o k' : m' j \u22a2 (if h : i = j then \u2191of (fun i_1 j => if i_1 = j then d i i_1 else 0) k (cast (_ : m' { fst := j, snd := k' }.fst = m' { fst := i, snd := k }.fst) k') else 0) = \u2191of (fun i j => if i = j then d i.fst i.snd else 0) { fst := i, snd := k } { fst := j, snd := k' } ** obtain rfl | hij := Decidable.eq_or_ne i j ** case a.mk.h.mk.inl l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 p : Type u_5 q : Type u_6 m' : o \u2192 Type u_7 n' : o \u2192 Type u_8 p' : o \u2192 Type u_9 R : Type u_10 S : Type u_11 \u03b1 : Type u_12 \u03b2 : Type u_13 inst\u271d\u00b3 : DecidableEq o inst\u271d\u00b2 : Zero \u03b1 inst\u271d\u00b9 : Zero \u03b2 inst\u271d : (i : o) \u2192 DecidableEq (m' i) d : (i : o) \u2192 m' i \u2192 \u03b1 i : o k k' : m' i \u22a2 (if h : i = i then \u2191of (fun i_1 j => if i_1 = j then d i i_1 else 0) k (cast (_ : m' { fst := i, snd := k' }.fst = m' { fst := i, snd := k }.fst) k') else 0) = \u2191of (fun i j => if i = j then d i.fst i.snd else 0) { fst := i, snd := k } { fst := i, snd := k' } ** simp ** case a.mk.h.mk.inr l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 p : Type u_5 q : Type u_6 m' : o \u2192 Type u_7 n' : o \u2192 Type u_8 p' : o \u2192 Type u_9 R : Type u_10 S : Type u_11 \u03b1 : Type u_12 \u03b2 : Type u_13 inst\u271d\u00b3 : DecidableEq o inst\u271d\u00b2 : Zero \u03b1 inst\u271d\u00b9 : Zero \u03b2 inst\u271d : (i : o) \u2192 DecidableEq (m' i) d : (i : o) \u2192 m' i \u2192 \u03b1 i : o k : m' i j : o k' : m' j hij : i \u2260 j \u22a2 (if h : i = j then \u2191of (fun i_1 j => if i_1 = j then d i i_1 else 0) k (cast (_ : m' { fst := j, snd := k' }.fst = m' { fst := i, snd := k }.fst) k') else 0) = \u2191of (fun i j => if i = j then d i.fst i.snd else 0) { fst := i, snd := k } { fst := j, snd := k' } ** simp [hij] ** Qed", + "informal": "" + }, + { + "formal": "Real.exp_zero ** x y : \u211d \u22a2 rexp 0 = 1 ** simp [Real.exp] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.Pivot.isTwoBlockDiagonal_listTransvecCol_mul_mul_listTransvecRow ** n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 \u22a2 IsTwoBlockDiagonal (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) ** constructor ** case left n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 \u22a2 toBlocks\u2081\u2082 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) = 0 ** ext i j ** case left.a.h n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 i : Fin r j : Unit \u22a2 toBlocks\u2081\u2082 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j ** have : j = unit := by simp only [eq_iff_true_of_subsingleton] ** case left.a.h n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 i : Fin r j : Unit this : j = () \u22a2 toBlocks\u2081\u2082 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j ** simp [toBlocks\u2081\u2082, this, listTransvecCol_mul_mul_listTransvecRow_last_row M hM] ** n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 i : Fin r j : Unit \u22a2 j = () ** simp only [eq_iff_true_of_subsingleton] ** case right n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 \u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) = 0 ** ext i j ** case right.a.h n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 i : Unit j : Fin r \u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j ** have : i = unit := by simp only [eq_iff_true_of_subsingleton] ** case right.a.h n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 i : Unit j : Fin r this : i = () \u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j ** simp [toBlocks\u2082\u2081, this, listTransvecCol_mul_mul_listTransvecRow_last_col M hM] ** n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u00b3 : Field \ud835\udd5c inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : DecidableEq p inst\u271d : CommRing R r : \u2115 M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c hM : M (inr ()) (inr ()) \u2260 0 i : Unit j : Fin r \u22a2 i = () ** simp only [eq_iff_true_of_subsingleton] ** Qed", + "informal": "" + }, + { + "formal": "VitaliFamily.ae_tendsto_measure_inter_div ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 \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) ** let t := toMeasurable \u03bc s ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s \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) ** 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 _ _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) \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) ** 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] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) B : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1 \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) ** filter_upwards [A, B] with x hx h'x ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) B : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1 x : \u03b1 hx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x)) h'x : indicator t 1 x = 1 \u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1) ** rw [h'x] at hx ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) B : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1 x : \u03b1 hx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1) h'x : indicator t 1 x = 1 \u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1) ** apply hx.congr' _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) B : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1 x : \u03b1 hx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1) h'x : indicator t 1 x = 1 \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 ** filter_upwards [v.eventually_filterAt_measurableSet x] with _ ha ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) B : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1 x : \u03b1 hx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1) h'x : indicator t 1 x = 1 a\u271d : Set \u03b1 ha : MeasurableSet a\u271d \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 ** congr 1 ** case h.e_a \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) B : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1 x : \u03b1 hx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1) h'x : indicator t 1 x = 1 a\u271d : Set \u03b1 ha : MeasurableSet a\u271d \u22a2 \u2191\u2191\u03bc (t \u2229 a\u271d) = \u2191\u2191\u03bc (s \u2229 a\u271d) ** exact measure_toMeasurable_inter_of_sigmaFinite ha _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s \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)) ** apply ae_mono restrict_le_self ** case a \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s \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 ** apply ae_tendsto_measure_inter_div_of_measurableSet ** case a.hs \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s \u22a2 MeasurableSet t ** exact measurableSet_toMeasurable _ _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1 ** refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable \u03bc s) _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (toMeasurable \u03bc s), indicator t 1 x = 1 ** filter_upwards [ae_restrict_mem (measurableSet_toMeasurable \u03bc s)] with _ hx ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 t : Set \u03b1 := toMeasurable \u03bc s A : \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)) a\u271d : \u03b1 hx : a\u271d \u2208 toMeasurable \u03bc s \u22a2 indicator t 1 a\u271d = 1 ** simp only [hx, Pi.one_apply, indicator_of_mem] ** Qed", + "informal": "" + }, + { + "formal": "NormedRing.summable_geometric_of_norm_lt_1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 R : Type u_4 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R h : \u2016x\u2016 < 1 \u22a2 Summable fun n => x ^ n ** have h1 : Summable fun n : \u2115 \u21a6 \u2016x\u2016 ^ n := summable_geometric_of_lt_1 (norm_nonneg _) h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 R : Type u_4 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R h : \u2016x\u2016 < 1 h1 : Summable fun n => \u2016x\u2016 ^ n \u22a2 Summable fun n => x ^ n ** refine' summable_of_norm_bounded_eventually _ h1 _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 R : Type u_4 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R h : \u2016x\u2016 < 1 h1 : Summable fun n => \u2016x\u2016 ^ n \u22a2 \u2200\u1da0 (i : \u2115) in cofinite, \u2016x ^ i\u2016 \u2264 \u2016x\u2016 ^ i ** rw [Nat.cofinite_eq_atTop] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 R : Type u_4 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R h : \u2016x\u2016 < 1 h1 : Summable fun n => \u2016x\u2016 ^ n \u22a2 \u2200\u1da0 (i : \u2115) in atTop, \u2016x ^ i\u2016 \u2264 \u2016x\u2016 ^ i ** exact eventually_norm_pow_le x ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.restrict_singleton ** \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 \u22a2 restrict \u03bc {a} = \u2191\u2191\u03bc {a} \u2022 dirac a ** ext1 s hs ** case h \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** by_cases ha : a \u2208 s ** case pos \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : a \u2208 s \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** have : s \u2229 {a} = {a} := by simpa ** case pos \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : a \u2208 s this : s \u2229 {a} = {a} \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** simp [*] ** \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : a \u2208 s \u22a2 s \u2229 {a} = {a} ** simpa ** case neg \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : \u00aca \u2208 s \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** have : s \u2229 {a} = \u2205 := inter_singleton_eq_empty.2 ha ** case neg \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : \u00aca \u2208 s this : s \u2229 {a} = \u2205 \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** simp [*] ** Qed", + "informal": "" + }, + { + "formal": "IsCyclotomicExtension.discr_prime_pow ** p : \u2115+ k : \u2115 K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hcycl : IsCyclotomicExtension {p ^ k} K L hp : Fact (Nat.Prime \u2191p) h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k) hirr : Irreducible (cyclotomic (\u2191(p ^ k)) K) \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1))) ** cases' k with k k ** case zero p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p ^ Nat.zero} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero) hirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K) \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ Nat.zero) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1))) ** 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] ** case zero p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p ^ Nat.zero} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero) hirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K) \u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u03b6 ^ \u2191i) (\u03b6 ^ \u2191j)) = \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1))) ** have h\u03b6one : \u03b6 = 1 := by simpa using h\u03b6 ** case zero p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p ^ Nat.zero} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero) hirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K) h\u03b6one : \u03b6 = 1 \u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u03b6 ^ \u2191i) (\u03b6 ^ \u2191j)) = \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1))) ** 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] ** case zero p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p ^ Nat.zero} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero) hirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K) h\u03b6one : \u03b6 = 1 \u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u2191(algebraMap K L) 1 ^ \u2191i) (\u2191(algebraMap K L) 1 ^ \u2191j)) = \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1))) ** simp only [traceMatrix, map_one, one_pow, Matrix.det_unique, traceForm_apply, mul_one] ** case zero p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p ^ Nat.zero} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero) hirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K) h\u03b6one : \u03b6 = 1 \u22a2 \u2191Matrix.of (fun i j => \u2191(trace K L) 1) default default = \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1))) ** rw [\u2190 (algebraMap K L).map_one, trace_algebraMap, finrank _ hirr] ** case zero p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p ^ Nat.zero} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero) hirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K) h\u03b6one : \u03b6 = 1 \u22a2 \u2191Matrix.of (fun i j => \u03c6 \u2191(p ^ Nat.zero) \u2022 1) default default = \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1))) ** norm_num ** p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p ^ Nat.zero} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero) hirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K) \u22a2 1 / 2 = 0 ** rfl ** p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p ^ Nat.zero} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero) hirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K) \u22a2 \u03b6 = 1 ** simpa using h\u03b6 ** case succ p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** by_cases hk : p ^ (k + 1) = 2 ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk : p ^ (k + 1) = 2 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** have coe_two : 2 = ((2 : \u2115+) : \u2115) := rfl ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk : p ^ (k + 1) = 2 coe_two : 2 = \u21912 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** 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 ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk : p ^ (k + 1) = 2 coe_two : 2 = \u21912 hp : p = 2 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** rw [hp, \u2190 PNat.coe_inj, PNat.pow_coe] at hk ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** nth_rw 2 [\u2190 pow_one 2] at hk ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 hk : \u21912 ^ (k + 1) = \u2191(2 ^ 1) coe_two : 2 = \u21912 hp : p = 2 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** replace hk := Nat.pow_right_injective rfl.le hk ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k + 1 = 1 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** rw [add_left_eq_self] at hk ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** rw [hp, hk] at h\u03b6 ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ succ 0) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** norm_num at h\u03b6 ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 \u21912 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** rw [\u2190 coe_two] at h\u03b6 ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** rw [coe_basis, powerBasis_gen] ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \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))) ** simp only [hp, hk] ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \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))) ** norm_num ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \u22a2 (discr K fun i => \u03b6 ^ \u2191i) = 1 ** simp_rw [h\u03b6.eq_neg_one_of_two_right, show (-1 : L) = algebraMap K L (-1) by simp] ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \u22a2 (discr K fun i => \u2191(algebraMap K L) (-1) ^ \u2191i) = 1 ** simp only [discr, traceMatrix_apply, Matrix.det_unique, Fin.default_eq_zero, Fin.val_zero,\n _root_.pow_zero, traceForm_apply, mul_one] ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \u22a2 \u2191(trace K L) 1 = 1 ** rw [\u2190 (algebraMap K L).map_one, trace_algebraMap, finrank _ hirr, hp, hk] ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \u22a2 \u03c6 \u2191(2 ^ succ 0) \u2022 1 = 1 ** norm_num ** case pos p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \u22a2 \u2191(\u03c6 \u21912) = 1 ** simp [\u2190 coe_two] ** p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk : p ^ (k + 1) = 2 coe_two : 2 = \u21912 \u22a2 p = 2 ** rw [\u2190 PNat.coe_inj, PNat.pow_coe, \u2190 pow_one 2] at hk ** p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u2191p ^ (k + 1) = \u21912 hk : \u2191p ^ (k + 1) = \u2191(2 ^ 1) coe_two : 2 = \u21912 \u22a2 p = 2 ** replace hk :=\n eq_of_prime_pow_eq (prime_iff.1 hp.out) (prime_iff.1 Nat.prime_two) (succ_pos _) hk ** p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u2191p ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hk : \u2191p = 2 \u22a2 p = 2 ** rwa [coe_two, PNat.coe_inj] at hk ** p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp\u271d : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk\u271d : \u21912 ^ (k + 1) = \u21912 coe_two : 2 = \u21912 hp : p = 2 hk : k = 0 h\u03b6 : IsPrimitiveRoot \u03b6 2 \u22a2 -1 = \u2191(algebraMap K L) (-1) ** simp ** case neg p : \u2115+ K : Type u L : Type v \u03b6 : L inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L hp : Fact (Nat.Prime \u2191p) k : \u2115 hcycl : IsCyclotomicExtension {p ^ succ k} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k) hirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K) hk : \u00acp ^ (k + 1) = 2 \u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1))) ** exact discr_prime_pow_ne_two h\u03b6 hirr hk ** Qed", + "informal": "" + }, + { + "formal": "isOpen_pi_iff ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) \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 ** rw [isOpen_iff_nhds] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) \u22a2 (\u2200 (a : (a : \u03b9) \u2192 \u03c0 a), a \u2208 s \u2192 \ud835\udcdd a \u2264 \ud835\udcdf 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 ** simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) \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 \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 ** refine ball_congr fun a _ => \u27e8?_, ?_\u27e9 ** case refine_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s \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 \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 ** rintro \u27e8I, t, \u27e8h1, h2\u27e9\u27e9 ** case refine_1.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (i : \u03b9) \u2192 Set (\u03c0 i) h1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1 h2 : Set.pi (\u2191I) t \u2286 s \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 ** refine \u27e8I, fun a => eval a '' (I : Set \u03b9).pi fun a => (h1 a).choose, fun i hi => ?_, ?_\u27e9 ** case refine_1.intro.intro.intro.refine_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (i : \u03b9) \u2192 Set (\u03c0 i) h1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 hi : i \u2208 I \u22a2 IsOpen ((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) \u2227 a i \u2208 (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 ** 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)] ** case refine_1.intro.intro.intro.refine_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (i : \u03b9) \u2192 Set (\u03c0 i) h1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 hi : i \u2208 I \u22a2 IsOpen (Exists.choose (_ : \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1)) \u2227 a i \u2208 Exists.choose (_ : \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) ** exact (h1 i).choose_spec.2 ** case refine_1.intro.intro.intro.refine_2 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (i : \u03b9) \u2192 Set (\u03c0 i) h1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1 h2 : Set.pi (\u2191I) t \u2286 s \u22a2 (Set.pi \u2191I 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)) \u2286 s ** exact Subset.trans\n (Set.pi_mono fun i hi => (Set.eval_image_pi_subset hi).trans (h1 i).choose_spec.1) h2 ** case refine_2 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s \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 \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 ** rintro \u27e8I, t, \u27e8h1, h2\u27e9\u27e9 ** case refine_2.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s \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 ** refine \u27e8I, fun a => ite (a \u2208 I) (t a) Set.univ, fun i => ?_, ?_\u27e9 ** case refine_2.intro.intro.intro.refine_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 \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 ** by_cases hi : i \u2208 I ** case pos \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 hi : i \u2208 I \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 ** use t i ** case h \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 hi : i \u2208 I \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 ** simp_rw [if_pos hi] ** case h \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 hi : i \u2208 I \u22a2 t i \u2286 t i \u2227 IsOpen (t i) \u2227 a i \u2208 t i ** exact \u27e8Subset.rfl, (h1 i) hi\u27e9 ** case neg \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 hi : \u00aci \u2208 I \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 ** use Set.univ ** case h \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 hi : \u00aci \u2208 I \u22a2 univ \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen univ \u2227 a i \u2208 univ ** simp_rw [if_neg hi] ** case h \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s i : \u03b9 hi : \u00aci \u2208 I \u22a2 univ \u2286 univ \u2227 IsOpen univ \u2227 a i \u2208 univ ** exact \u27e8Subset.rfl, isOpen_univ, mem_univ _\u27e9 ** case refine_2.intro.intro.intro.refine_2 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s \u22a2 (Set.pi \u2191I fun a => if a \u2208 I then t a else univ) \u2286 s ** rw [\u2190 Set.univ_pi_ite] ** case refine_2.intro.intro.intro.refine_2 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 \u03ba : Type u_7 inst\u271d : TopologicalSpace \u03b1 T : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) f : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i s : Set ((a : \u03b9) \u2192 \u03c0 a) a : (a : \u03b9) \u2192 \u03c0 a x\u271d : a \u2208 s I : Finset \u03b9 t : (a : \u03b9) \u2192 Set (\u03c0 a) h1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1 h2 : Set.pi (\u2191I) t \u2286 s \u22a2 (Set.pi univ fun i => if i \u2208 \u2191I then if i \u2208 I then t i else univ else univ) \u2286 s ** simp only [\u2190 ite_and, \u2190 Finset.mem_coe, and_self_iff, Set.univ_pi_ite, h2] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.eqOn_open_of_ae_eq ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X f g : X \u2192 Y h : f =\u1da0[ae (restrict \u03bc U)] g hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U \u22a2 EqOn f g U ** replace h := ae_imp_of_ae_restrict h ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X f g : X \u2192 Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : \u2200\u1d50 (x : X) \u2202\u03bc, x \u2208 U \u2192 f x = g x \u22a2 EqOn f g U ** simp only [EventuallyEq, ae_iff, not_imp] at h ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X f g : X \u2192 Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0 \u22a2 EqOn f g U ** 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') ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X f g : X \u2192 Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0 this : IsOpen (U \u2229 {a | f a \u2260 g a}) \u22a2 EqOn f g U ** replace := (this.eq_empty_of_measure_zero h).le ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X f g : X \u2192 Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0 this : U \u2229 {a | f a \u2260 g a} \u2264 \u2205 \u22a2 EqOn f g U ** exact fun x hx => Classical.not_not.1 fun h => this \u27e8hx, h\u27e9 ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X f g : X \u2192 Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0 \u22a2 IsOpen (U \u2229 {a | f a \u2260 g a}) ** refine' isOpen_iff_mem_nhds.mpr fun a ha => inter_mem (hU.mem_nhds ha.1) _ ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X f g : X \u2192 Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0 a : X ha : a \u2208 U \u2229 {a | f a \u2260 g a} \u22a2 {a | f a \u2260 g a} \u2208 \ud835\udcdd a ** rcases ha with \u27e8ha : a \u2208 U, ha' : (f a, g a) \u2208 (diagonal Y)\u1d9c\u27e9 ** case intro X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X f g : X \u2192 Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0 a : X ha : a \u2208 U ha' : (f a, g a) \u2208 (diagonal Y)\u1d9c \u22a2 {a | f a \u2260 g a} \u2208 \ud835\udcdd a ** 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') ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.toMatrix'_apply ** R : Type u_1 inst\u271d\u00b2 : CommSemiring R k : Type u_2 l : Type u_3 m : Type u_4 n : Type u_5 inst\u271d\u00b9 : Fintype n inst\u271d : DecidableEq n f : (n \u2192 R) \u2192\u2097[R] m \u2192 R i : m j : n \u22a2 \u2191toMatrix' f i j = \u2191f (fun j' => if j' = j then 1 else 0) i ** simp only [LinearMap.toMatrix', LinearEquiv.coe_mk, of_apply] ** R : Type u_1 inst\u271d\u00b2 : CommSemiring R k : Type u_2 l : Type u_3 m : Type u_4 n : Type u_5 inst\u271d\u00b9 : Fintype n inst\u271d : DecidableEq n f : (n \u2192 R) \u2192\u2097[R] m \u2192 R i : m j : n \u22a2 \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i = \u2191f (fun j' => if j' = j then 1 else 0) i ** refine congr_fun ?_ _ ** R : Type u_1 inst\u271d\u00b2 : CommSemiring R k : Type u_2 l : Type u_3 m : Type u_4 n : Type u_5 inst\u271d\u00b9 : Fintype n inst\u271d : DecidableEq n f : (n \u2192 R) \u2192\u2097[R] m \u2192 R i : m j : n \u22a2 \u2191f (\u2191(stdBasis R (fun x => R) j) 1) = \u2191f fun j' => if j' = j then 1 else 0 ** congr ** case h.e_6.h R : Type u_1 inst\u271d\u00b2 : CommSemiring R k : Type u_2 l : Type u_3 m : Type u_4 n : Type u_5 inst\u271d\u00b9 : Fintype n inst\u271d : DecidableEq n f : (n \u2192 R) \u2192\u2097[R] m \u2192 R i : m j : n \u22a2 \u2191(stdBasis R (fun x => R) j) 1 = fun j' => if j' = j then 1 else 0 ** ext j' ** case h.e_6.h.h R : Type u_1 inst\u271d\u00b2 : CommSemiring R k : Type u_2 l : Type u_3 m : Type u_4 n : Type u_5 inst\u271d\u00b9 : Fintype n inst\u271d : DecidableEq n f : (n \u2192 R) \u2192\u2097[R] m \u2192 R i : m j j' : n \u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = if j' = j then 1 else 0 ** split_ifs with h ** case neg R : Type u_1 inst\u271d\u00b2 : CommSemiring R k : Type u_2 l : Type u_3 m : Type u_4 n : Type u_5 inst\u271d\u00b9 : Fintype n inst\u271d : DecidableEq n f : (n \u2192 R) \u2192\u2097[R] m \u2192 R i : m j j' : n h : \u00acj' = j \u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = 0 ** apply stdBasis_ne _ _ _ _ h ** case pos R : Type u_1 inst\u271d\u00b2 : CommSemiring R k : Type u_2 l : Type u_3 m : Type u_4 n : Type u_5 inst\u271d\u00b9 : Fintype n inst\u271d : DecidableEq n f : (n \u2192 R) \u2192\u2097[R] m \u2192 R i : m j j' : n h : j' = j \u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = 1 ** rw [h, stdBasis_same] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.inv_zpow' ** n' : Type u_1 inst\u271d\u00b2 : DecidableEq n' inst\u271d\u00b9 : Fintype n' R : Type u_2 inst\u271d : CommRing R A : M h : IsUnit (det A) n : \u2124 \u22a2 A\u207b\u00b9 ^ n = A ^ (-n) ** rw [zpow_neg h, inv_zpow] ** Qed", + "informal": "" + }, + { + "formal": "Sym2.other_spec ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 z : Sym2 \u03b1 h : a \u2208 z \u22a2 Quotient.mk (Rel.setoid \u03b1) (a, Mem.other h) = z ** erw [\u2190 Classical.choose_spec h] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.FiniteMeasure.coeFn_smul ** \u03a9 : Type u_1 inst\u271d\u2075 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2074 : SMul R \u211d\u22650 inst\u271d\u00b3 : SMul R \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650 c : R \u03bc : FiniteMeasure \u03a9 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) s)) = c \u2022 fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s) ** funext ** case h \u03a9 : Type u_1 inst\u271d\u2075 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2074 : SMul R \u211d\u22650 inst\u271d\u00b3 : SMul R \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650 c : R \u03bc : FiniteMeasure \u03a9 x\u271d : Set \u03a9 \u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) x\u271d) = (c \u2022 fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) x\u271d ** simp only [Pi.smul_apply, \u2190 ENNReal.coe_eq_coe, ne_eq, ennreal_coeFn_eq_coeFn_toMeasure,\n ENNReal.coe_smul] ** case h \u03a9 : Type u_1 inst\u271d\u2075 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2074 : SMul R \u211d\u22650 inst\u271d\u00b3 : SMul R \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650 c : R \u03bc : FiniteMeasure \u03a9 x\u271d : Set \u03a9 \u22a2 \u2191\u2191\u2191(c \u2022 \u03bc) x\u271d = c \u2022 \u2191\u2191\u2191\u03bc x\u271d ** norm_cast ** Qed", + "informal": "" + }, + { + "formal": "abs_sub_sq ** \u03b1 : Type u_1 inst\u271d : LinearOrderedCommRing \u03b1 a\u271d b\u271d c d a b : \u03b1 \u22a2 |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b ** rw [abs_mul_abs_self] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedCommRing \u03b1 a\u271d b\u271d c d a b : \u03b1 \u22a2 (a - b) * (a - b) = a * a + b * b - (1 + 1) * a * b ** 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] ** Qed", + "informal": "" + }, + { + "formal": "AddSubmonoid.mul_mem_mul ** \u03b1 : Type u_1 G : Type u_2 M\u271d : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u00b2 : Monoid M\u271d inst\u271d\u00b9 : AddMonoid A inst\u271d : NonUnitalNonAssocSemiring R M N : AddSubmonoid R m n : R hm : m \u2208 M hn : n \u2208 N \u22a2 \u2191(\u2191AddMonoidHom.mul \u2191{ val := m, property := hm }) n = m * n ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Wbtw.trans_sbtw_right ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2077 : OrderedRing R inst\u271d\u2076 : AddCommGroup V inst\u271d\u2075 : Module R V inst\u271d\u2074 : AddTorsor V P inst\u271d\u00b3 : AddCommGroup V' inst\u271d\u00b2 : Module R V' inst\u271d\u00b9 : AddTorsor V' P' inst\u271d : NoZeroSMulDivisors R V w x y z : P h\u2081 : Wbtw R w x z h\u2082 : Sbtw R x y z \u22a2 Sbtw R w y z ** rw [wbtw_comm] at * ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2077 : OrderedRing R inst\u271d\u2076 : AddCommGroup V inst\u271d\u2075 : Module R V inst\u271d\u2074 : AddTorsor V P inst\u271d\u00b3 : AddCommGroup V' inst\u271d\u00b2 : Module R V' inst\u271d\u00b9 : AddTorsor V' P' inst\u271d : NoZeroSMulDivisors R V w x y z : P h\u2081 : Wbtw R z x w h\u2082 : Sbtw R x y z \u22a2 Sbtw R w y z ** rw [sbtw_comm] at * ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2077 : OrderedRing R inst\u271d\u2076 : AddCommGroup V inst\u271d\u2075 : Module R V inst\u271d\u2074 : AddTorsor V P inst\u271d\u00b3 : AddCommGroup V' inst\u271d\u00b2 : Module R V' inst\u271d\u00b9 : AddTorsor V' P' inst\u271d : NoZeroSMulDivisors R V w x y z : P h\u2081 : Wbtw R z x w h\u2082 : Sbtw R z y x \u22a2 Sbtw R z y w ** exact h\u2081.trans_sbtw_left h\u2082 ** Qed", + "informal": "" + }, + { + "formal": "Finset.image_comm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : DecidableEq \u03b2 f\u271d g\u271d : \u03b1 \u2192 \u03b2 s : Finset \u03b1 t : Finset \u03b2 a : \u03b1 b c : \u03b2 \u03b2' : Type u_4 inst\u271d\u00b9 : DecidableEq \u03b2' inst\u271d : DecidableEq \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b1 \u2192 \u03b2 f' : \u03b1 \u2192 \u03b2' g' : \u03b2' \u2192 \u03b3 h_comm : \u2200 (a : \u03b1), f (g a) = g' (f' a) \u22a2 image f (image g s) = image g' (image f' s) ** simp_rw [image_image, comp, h_comm] ** Qed", + "informal": "" + }, + { + "formal": "IsCyclotomicExtension.subsingleton_iff ** n : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B \u22a2 IsCyclotomicExtension S A B \u2194 S = \u2205 \u2228 S = {1} ** constructor ** case mp n : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B \u22a2 IsCyclotomicExtension S A B \u2192 S = \u2205 \u2228 S = {1} ** rintro \u27e8hprim, -\u27e9 ** case mp.mk n : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B hprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n \u22a2 S = \u2205 \u2228 S = {1} ** rw [\u2190 subset_singleton_iff_eq] ** case mp.mk n : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B hprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n \u22a2 S \u2286 {1} ** intro t ht ** case mp.mk n : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B hprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n t : \u2115+ ht : t \u2208 S \u22a2 t \u2208 {1} ** obtain \u27e8\u03b6, h\u03b6\u27e9 := hprim ht ** case mp.mk.intro n : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B hprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n t : \u2115+ ht : t \u2208 S \u03b6 : B h\u03b6 : IsPrimitiveRoot \u03b6 \u2191t \u22a2 t \u2208 {1} ** rw [mem_singleton_iff, \u2190 PNat.coe_eq_one_iff] ** case mp.mk.intro n : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B hprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n t : \u2115+ ht : t \u2208 S \u03b6 : B h\u03b6 : IsPrimitiveRoot \u03b6 \u2191t \u22a2 \u2191t = 1 ** exact_mod_cast h\u03b6.unique (IsPrimitiveRoot.of_subsingleton \u03b6) ** case mpr n : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B \u22a2 S = \u2205 \u2228 S = {1} \u2192 IsCyclotomicExtension S A B ** rintro (rfl | rfl) ** case mpr.inl n : \u2115+ T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B \u22a2 IsCyclotomicExtension \u2205 A B ** exact \u27e8fun h => h.elim, fun x => by convert (mem_top (R := A) : x \u2208 \u22a4)\u27e9 ** n : \u2115+ T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B x : B \u22a2 x \u2208 adjoin A {b | \u2203 n, n \u2208 \u2205 \u2227 b ^ \u2191n = 1} ** convert (mem_top (R := A) : x \u2208 \u22a4) ** case mpr.inr n : \u2115+ T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B \u22a2 IsCyclotomicExtension {1} A B ** rw [iff_singleton] ** case mpr.inr n : \u2115+ T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B \u22a2 (\u2203 r, IsPrimitiveRoot r \u21911) \u2227 \u2200 (x : B), x \u2208 adjoin A {b | b ^ \u21911 = 1} ** exact \u27e8\u27e80, IsPrimitiveRoot.of_subsingleton 0\u27e9,\n fun x => by convert (mem_top (R := A) : x \u2208 \u22a4)\u27e9 ** n : \u2115+ T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : Subsingleton B x : B \u22a2 x \u2208 adjoin A {b | b ^ \u21911 = 1} ** convert (mem_top (R := A) : x \u2208 \u22a4) ** Qed", + "informal": "" + }, + { + "formal": "Orientation.abs_areaForm_of_orthogonal ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E h : inner x y = 0 \u22a2 |\u2191(\u2191(areaForm o) x) y| = \u2016x\u2016 * \u2016y\u2016 ** rw [o.areaForm_to_volumeForm, o.abs_volumeForm_apply_of_pairwise_orthogonal] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E h : inner x y = 0 \u22a2 Pairwise fun i j => inner (Matrix.vecCons x ![y] i) (Matrix.vecCons x ![y] j) = 0 ** intro i j hij ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E h : inner x y = 0 i j : Fin 2 hij : i \u2260 j \u22a2 inner (Matrix.vecCons x ![y] i) (Matrix.vecCons x ![y] j) = 0 ** fin_cases i <;> fin_cases j ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E h : inner x y = 0 \u22a2 (Finset.prod Finset.univ fun i => \u2016Matrix.vecCons x ![y] i\u2016) = \u2016x\u2016 * \u2016y\u2016 ** simp [Fin.prod_univ_succ] ** case head.head E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E h : inner x y = 0 hij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 0, isLt := (_ : 0 < 2) } \u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) = 0 ** simp_all ** case head.tail.head E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E h : inner x y = 0 hij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) = 0 ** simpa using h ** case tail.head.head E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E h : inner x y = 0 hij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 0, isLt := (_ : 0 < 2) } \u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) = 0 ** simpa [real_inner_comm] using h ** case tail.head.tail.head E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E h : inner x y = 0 hij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) = 0 ** simp_all ** Qed", + "informal": "" + }, + { + "formal": "integral_sin_pow ** a b : \u211d n : \u2115 \u22a2 \u222b (x : \u211d) in a..b, sin x ^ (n + 2) = (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 ** have : n + 2 \u2260 0 := by linarith ** a b : \u211d n : \u2115 this : n + 2 \u2260 0 \u22a2 \u222b (x : \u211d) in a..b, sin x ^ (n + 2) = (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 ** have : (n : \u211d) + 2 \u2260 0 := by norm_cast ** a b : \u211d n : \u2115 this\u271d : n + 2 \u2260 0 this : \u2191n + 2 \u2260 0 \u22a2 \u222b (x : \u211d) in a..b, sin x ^ (n + 2) = (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 ** field_simp ** a b : \u211d n : \u2115 this\u271d : n + 2 \u2260 0 this : \u2191n + 2 \u2260 0 \u22a2 (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) * (\u2191n + 2) = sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (\u2191n + 1) * \u222b (x : \u211d) in a..b, sin x ^ n ** convert eq_sub_iff_add_eq.mp (integral_sin_pow_aux n) using 1 ** case h.e'_2 a b : \u211d n : \u2115 this\u271d : n + 2 \u2260 0 this : \u2191n + 2 \u2260 0 \u22a2 (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) * (\u2191n + 2) = (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) + (\u2191n + 1) * \u222b (x : \u211d) in a..b, sin x ^ (n + 2) ** ring ** a b : \u211d n : \u2115 \u22a2 n + 2 \u2260 0 ** linarith ** a b : \u211d n : \u2115 this : n + 2 \u2260 0 \u22a2 \u2191n + 2 \u2260 0 ** norm_cast ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.mapRange_neLocus_eq ** \u03b1 : Type u_1 N : \u03b1 \u2192 Type u_2 inst\u271d\u2075 : DecidableEq \u03b1 M : \u03b1 \u2192 Type u_3 P : \u03b1 \u2192 Type u_4 inst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a) inst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a) inst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a) inst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a) inst\u271d : (a : \u03b1) \u2192 DecidableEq (M a) f g : \u03a0\u2080 (a : \u03b1), N a F : (a : \u03b1) \u2192 N a \u2192 M a F0 : \u2200 (a : \u03b1), F a 0 = 0 hF : \u2200 (a : \u03b1), Function.Injective (F a) \u22a2 neLocus (mapRange F F0 f) (mapRange F F0 g) = neLocus f g ** ext a ** case a \u03b1 : Type u_1 N : \u03b1 \u2192 Type u_2 inst\u271d\u2075 : DecidableEq \u03b1 M : \u03b1 \u2192 Type u_3 P : \u03b1 \u2192 Type u_4 inst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a) inst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a) inst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a) inst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a) inst\u271d : (a : \u03b1) \u2192 DecidableEq (M a) f g : \u03a0\u2080 (a : \u03b1), N a F : (a : \u03b1) \u2192 N a \u2192 M a F0 : \u2200 (a : \u03b1), F a 0 = 0 hF : \u2200 (a : \u03b1), Function.Injective (F a) a : \u03b1 \u22a2 a \u2208 neLocus (mapRange F F0 f) (mapRange F F0 g) \u2194 a \u2208 neLocus f g ** simpa only [mem_neLocus] using (hF a).ne_iff ** Qed", + "informal": "" + }, + { + "formal": "Turing.ToPartrec.Code.id_eval ** v : List \u2115 \u22a2 eval id v = pure v ** simp [id] ** Qed", + "informal": "" + }, + { + "formal": "LipschitzWith.iterate ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b2 : PseudoEMetricSpace \u03b1 inst\u271d\u00b9 : PseudoEMetricSpace \u03b2 inst\u271d : PseudoEMetricSpace \u03b3 K : \u211d\u22650 f\u271d : \u03b1 \u2192 \u03b2 x y : \u03b1 r : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b1 hf : LipschitzWith K f \u22a2 LipschitzWith (K ^ 0) f^[0] ** simpa only [pow_zero] using LipschitzWith.id ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b2 : PseudoEMetricSpace \u03b1 inst\u271d\u00b9 : PseudoEMetricSpace \u03b2 inst\u271d : PseudoEMetricSpace \u03b3 K : \u211d\u22650 f\u271d : \u03b1 \u2192 \u03b2 x y : \u03b1 r : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b1 hf : LipschitzWith K f n : \u2115 \u22a2 LipschitzWith (K ^ (n + 1)) f^[n + 1] ** rw [pow_succ'] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b2 : PseudoEMetricSpace \u03b1 inst\u271d\u00b9 : PseudoEMetricSpace \u03b2 inst\u271d : PseudoEMetricSpace \u03b3 K : \u211d\u22650 f\u271d : \u03b1 \u2192 \u03b2 x y : \u03b1 r : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b1 hf : LipschitzWith K f n : \u2115 \u22a2 LipschitzWith (K ^ n * K) f^[n + 1] ** exact (LipschitzWith.iterate hf n).comp hf ** Qed", + "informal": "" + }, + { + "formal": "Sigma.ext ** \u03b1 : Type u_1 \u03b1\u2081 : Type u_2 \u03b1\u2082 : Type u_3 \u03b2 : \u03b1 \u2192 Type u_4 \u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5 \u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6 x\u2080 x\u2081 : Sigma \u03b2 h\u2080 : x\u2080.fst = x\u2081.fst h\u2081 : HEq x\u2080.snd x\u2081.snd \u22a2 x\u2080 = x\u2081 ** cases x\u2080 ** case mk \u03b1 : Type u_1 \u03b1\u2081 : Type u_2 \u03b1\u2082 : Type u_3 \u03b2 : \u03b1 \u2192 Type u_4 \u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5 \u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6 x\u2081 : Sigma \u03b2 fst\u271d : \u03b1 snd\u271d : \u03b2 fst\u271d h\u2080 : { fst := fst\u271d, snd := snd\u271d }.fst = x\u2081.fst h\u2081 : HEq { fst := fst\u271d, snd := snd\u271d }.snd x\u2081.snd \u22a2 { fst := fst\u271d, snd := snd\u271d } = x\u2081 ** cases x\u2081 ** case mk.mk \u03b1 : Type u_1 \u03b1\u2081 : Type u_2 \u03b1\u2082 : Type u_3 \u03b2 : \u03b1 \u2192 Type u_4 \u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5 \u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6 fst\u271d\u00b9 : \u03b1 snd\u271d\u00b9 : \u03b2 fst\u271d\u00b9 fst\u271d : \u03b1 snd\u271d : \u03b2 fst\u271d h\u2080 : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst = { fst := fst\u271d, snd := snd\u271d }.fst h\u2081 : HEq { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd \u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d } ** cases h\u2080 ** case mk.mk.refl \u03b1 : Type u_1 \u03b1\u2081 : Type u_2 \u03b1\u2082 : Type u_3 \u03b2 : \u03b1 \u2192 Type u_4 \u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5 \u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6 fst\u271d : \u03b1 snd\u271d\u00b9 snd\u271d : \u03b2 fst\u271d h\u2081 : HEq { fst := fst\u271d, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd \u22a2 { fst := fst\u271d, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d } ** cases h\u2081 ** case mk.mk.refl.refl \u03b1 : Type u_1 \u03b1\u2081 : Type u_2 \u03b1\u2082 : Type u_3 \u03b2 : \u03b1 \u2192 Type u_4 \u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5 \u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6 fst\u271d : \u03b1 snd\u271d : \u03b2 fst\u271d \u22a2 { fst := fst\u271d, snd := snd\u271d } = { fst := fst\u271d, snd := snd\u271d } ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Ideal.comap_of_equiv ** R : Type u S : Type v F : Type u_1 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : Ring S inst\u271d : RingHomClass F R S f\u271d : F I\u271d I : Ideal R f : R \u2243+* S \u22a2 comap (\u2191f) (comap (\u2191(RingEquiv.symm f)) I) = I ** 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] ** Qed", + "informal": "" + }, + { + "formal": "ClassGroup.norm_lt ** R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y \u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** obtain \u27e8i\u27e9 := bS.index_nonempty ** case intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 \u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** 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 ** case intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) \u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** set y' : \u2124 := Finset.max' _ him with y'_def ** case intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him \u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** 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) ** case intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' \u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** 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 ** case intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' this : \u2191y' < y \u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** have y'_nonneg : 0 \u2264 y' := le_trans (abv.nonneg _) (hy' i) ** case intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' this : \u2191y' < y y'_nonneg : 0 \u2264 y' \u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** apply (Int.cast_le.mpr (norm_le abv bS a hy')).trans_lt ** case intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' this : \u2191y' < y y'_nonneg : 0 \u2264 y' \u22a2 \u2191(normBound abv bS * y' ^ Fintype.card \u03b9) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** simp only [Int.cast_mul, Int.cast_pow] ** case intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' this : \u2191y' < y y'_nonneg : 0 \u2264 y' \u22a2 \u2191(normBound abv bS) * \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9 < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9 ** apply mul_lt_mul' le_rfl ** R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him \u22a2 \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' ** intro k ** R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him k : \u03b9 \u22a2 \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' ** exact @Finset.le_max' \u2124 _ _ _ (Finset.mem_image.mpr \u27e8k, Finset.mem_univ _, rfl\u27e9) ** R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' \u22a2 \u2191y' < y ** rw [y'_def, \u2190\n Finset.max'_image (show Monotone (_ : \u2124 \u2192 T) from fun x y h => Int.cast_le.mpr h)] ** R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' \u22a2 Finset.max' (Finset.image (fun x => \u2191x) (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)) ?h < y case h R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' \u22a2 Finset.Nonempty (Finset.image (fun x => \u2191x) (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)) ** apply (Finset.max'_lt_iff _ (him.image _)).mpr ** R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' \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 ** simp only [Finset.mem_image, exists_prop] ** R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' \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 ** rintro _ \u27e8x, \u27e8k, -, rfl\u27e9, rfl\u27e9 ** case intro.intro.intro.intro R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' k : \u03b9 \u22a2 \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y ** exact hy k ** case intro.hbd R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' this : \u2191y' < y y'_nonneg : 0 \u2264 y' \u22a2 \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9 < y ^ Fintype.card \u03b9 ** exact pow_lt_pow_of_lt_left this (Int.cast_nonneg.mpr y'_nonneg) (Fintype.card_pos_iff.mpr \u27e8i\u27e9) ** case intro.hb R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' this : \u2191y' < y y'_nonneg : 0 \u2264 y' \u22a2 0 \u2264 \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9 ** exact pow_nonneg (Int.cast_nonneg.mpr y'_nonneg) _ ** case intro.hc R : Type u_1 S : Type u_2 K : Type u_3 L : Type u_4 inst\u271d\u00b9\u2075 : EuclideanDomain R inst\u271d\u00b9\u2074 : CommRing S inst\u271d\u00b9\u00b3 : IsDomain S inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : Algebra R K inst\u271d\u2079 : IsFractionRing R K inst\u271d\u2078 : Algebra K L inst\u271d\u2077 : FiniteDimensional K L inst\u271d\u2076 : IsSeparable K L algRL : Algebra R L inst\u271d\u2075 : IsScalarTower R K L inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : Algebra S L ist : IsScalarTower R S L iic : IsIntegralClosure S R L abv : AbsoluteValue R \u2124 \u03b9 : Type u_5 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : Fintype \u03b9 bS : Basis \u03b9 R S T : Type u_6 inst\u271d : LinearOrderedRing T a : S y : T hy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y i : \u03b9 him : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) y' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him y'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him hy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y' this : \u2191y' < y y'_nonneg : 0 \u2264 y' \u22a2 0 < \u2191(normBound abv bS) ** exact Int.cast_pos.mpr (normBound_pos abv bS) ** Qed", + "informal": "" + }, + { + "formal": "ZNum.div_to_int ** \u22a2 \u2191(0 / 0) = \u21910 / \u21910 ** simp [Int.ediv_zero] ** n d : PosNum \u22a2 \u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(pos d) ** rw [\u2190 Num.to_nat_to_int] ** n d : PosNum \u22a2 \u2191\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(pos d) ** simp ** n d : PosNum \u22a2 -\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(neg d) ** rw [\u2190 Num.to_nat_to_int] ** n d : PosNum \u22a2 -\u2191\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(neg d) ** simp ** n d : PosNum \u22a2 -\u2191(Num.succ' (PosNum.pred' n / Num.pos d)) = -\u2191n / \u2191d ** 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] ** n d : PosNum \u22a2 -\u2191(\u2191(PosNum.pred' n) / \u2191(Num.pos d) + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) / (\u2191\u2191(PosNum.pred' d) + 1) ** change -[n.pred' / \u2191d+1] = -[n.pred' / (d.pred' + 1)+1] ** n d : PosNum \u22a2 -[\u2191(PosNum.pred' n) / \u2191d+1] = -[\u2191(PosNum.pred' n) / (\u2191(PosNum.pred' d) + 1)+1] ** rw [d.to_nat_eq_succ_pred] ** n d : PosNum \u22a2 \u2191(Num.succ' (PosNum.pred' n / Num.pos d)) = -\u2191n / -\u2191d ** 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] ** n d : PosNum \u22a2 \u2191(\u2191(PosNum.pred' n) / \u2191(Num.pos d) + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) / -(\u2191\u2191(PosNum.pred' d) + 1) ** change (Nat.succ (_ / d) : \u2124) = Nat.succ (n.pred' / (d.pred' + 1)) ** n d : PosNum \u22a2 \u2191(Nat.succ (\u2191(PosNum.pred' n) / \u2191d)) = \u2191(Nat.succ (\u2191(PosNum.pred' n) / (\u2191(PosNum.pred' d) + 1))) ** rw [d.to_nat_eq_succ_pred] ** Qed", + "informal": "" + }, + { + "formal": "AddCommGroupCat.asHom_injective ** G : AddCommGroupCat h k : \u2191G w : asHom h = asHom k \u22a2 h = k ** convert congr_arg (fun k : AddCommGroupCat.of \u2124 \u27f6 G => (k : \u2124 \u2192 G) (1 : \u2124)) w <;> simp ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.Monic.prime_of_degree_eq_one ** R : Type u S : Type v T : Type w a b : R n : \u2115 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R p q : R[X] hp1 : degree p = 1 hm : Monic p \u22a2 p = X - \u2191C (-coeff p 0) ** simpa [hm.leadingCoeff] using eq_X_add_C_of_degree_eq_one hp1 ** Qed", + "informal": "" + }, + { + "formal": "Filter.IsBasis.mem_filter_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 p : \u03b9 \u2192 Prop s : \u03b9 \u2192 Set \u03b1 h : IsBasis p s U : Set \u03b1 \u22a2 U \u2208 IsBasis.filter h \u2194 \u2203 i, p i \u2227 s i \u2286 U ** simp only [IsBasis.filter, FilterBasis.mem_filter_iff, mem_filterBasis_iff,\n exists_exists_and_eq_and] ** Qed", + "informal": "" + }, + { + "formal": "MonoidHomClass.isometry_iff_norm ** \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b3 : SeminormedGroup E inst\u271d\u00b2 : SeminormedGroup F inst\u271d\u00b9 : SeminormedGroup G s : Set E a a\u2081 a\u2082 b b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d inst\u271d : MonoidHomClass \ud835\udcd5 E F f : \ud835\udcd5 \u22a2 Isometry \u2191f \u2194 \u2200 (x : E), \u2016\u2191f x\u2016 = \u2016x\u2016 ** simp only [isometry_iff_dist_eq, dist_eq_norm_div, \u2190 map_div] ** \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b3 : SeminormedGroup E inst\u271d\u00b2 : SeminormedGroup F inst\u271d\u00b9 : SeminormedGroup G s : Set E a a\u2081 a\u2082 b b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d inst\u271d : MonoidHomClass \ud835\udcd5 E F f : \ud835\udcd5 \u22a2 (\u2200 (x y : E), \u2016\u2191f (x / y)\u2016 = \u2016x / y\u2016) \u2194 \u2200 (x : E), \u2016\u2191f x\u2016 = \u2016x\u2016 ** refine' \u27e8fun h x => _, fun h x y => h _\u27e9 ** \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b3 : SeminormedGroup E inst\u271d\u00b2 : SeminormedGroup F inst\u271d\u00b9 : SeminormedGroup G s : Set E a a\u2081 a\u2082 b b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d inst\u271d : MonoidHomClass \ud835\udcd5 E F f : \ud835\udcd5 h : \u2200 (x y : E), \u2016\u2191f (x / y)\u2016 = \u2016x / y\u2016 x : E \u22a2 \u2016\u2191f x\u2016 = \u2016x\u2016 ** simpa using h x 1 ** Qed", + "informal": "" + }, + { + "formal": "Finset.induction_on_max_value ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop s : Finset \u03b9 h0 : p \u2205 step : \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) \u22a2 p s ** induction' s using Finset.strongInductionOn with s ihs ** case a F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t \u22a2 p s ** rcases (s.image f).eq_empty_or_nonempty with (hne | hne) ** case a.inl F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : image f s = \u2205 \u22a2 p s ** simp only [image_eq_empty] at hne ** case a.inl F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : s = \u2205 \u22a2 p s ** simp only [hne, h0] ** case a.inr F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : Finset.Nonempty (image f s) \u22a2 p s ** have H : (s.image f).max' hne \u2208 s.image f := max'_mem (s.image f) hne ** case a.inr F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : Finset.Nonempty (image f s) H : max' (image f s) hne \u2208 image f s \u22a2 p s ** simp only [mem_image, exists_prop] at H ** case a.inr F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : Finset.Nonempty (image f s) H : \u2203 a, a \u2208 s \u2227 f a = max' (image f s) hne \u22a2 p s ** rcases H with \u27e8a, has, hfa\u27e9 ** case a.inr.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : Finset.Nonempty (image f s) a : \u03b9 has : a \u2208 s hfa : f a = max' (image f s) hne \u22a2 p s ** rw [\u2190 insert_erase has] ** case a.inr.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : Finset.Nonempty (image f s) a : \u03b9 has : a \u2208 s hfa : f a = max' (image f s) hne \u22a2 p (insert a (erase s a)) ** refine' step _ _ (not_mem_erase a s) (fun x hx => _) (ihs _ <| erase_ssubset has) ** case a.inr.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : Finset.Nonempty (image f s) a : \u03b9 has : a \u2208 s hfa : f a = max' (image f s) hne x : \u03b9 hx : x \u2208 erase s a \u22a2 f x \u2264 f a ** rw [hfa] ** case a.inr.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 inst\u271d : DecidableEq \u03b9 f : \u03b9 \u2192 \u03b1 p : Finset \u03b9 \u2192 Prop h0 : p \u2205 step : \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) s : Finset \u03b9 ihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t hne : Finset.Nonempty (image f s) a : \u03b9 has : a \u2208 s hfa : f a = max' (image f s) hne x : \u03b9 hx : x \u2208 erase s a \u22a2 f x \u2264 max' (image f s) hne ** exact le_max' _ _ (mem_image_of_mem _ <| mem_of_mem_erase hx) ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.multiset_sum_sum_index ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 A : Type u_4 B : Type u_5 C : Type u_6 inst\u271d\u2074 : AddCommMonoid A inst\u271d\u00b3 : AddCommMonoid B inst\u271d\u00b2 : AddCommMonoid C t : \u03b9 \u2192 A \u2192 C h0 : \u2200 (i : \u03b9), t i 0 = 0 h1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y s\u271d : Finset \u03b1 f\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A i : \u03b9 g : \u03b9 \u2192\u2080 A k : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B x : \u03b3 \u03b2 : Type u_7 M : Type u_8 M' : Type u_9 N : Type u_10 P : Type u_11 G : Type u_12 H : Type u_13 R : Type u_14 S : Type u_15 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : AddCommMonoid N f : Multiset (\u03b1 \u2192\u2080 M) h : \u03b1 \u2192 M \u2192 N h\u2080 : \u2200 (a : \u03b1), h a 0 = 0 h\u2081 : \u2200 (a : \u03b1) (b\u2081 b\u2082 : M), h a (b\u2081 + b\u2082) = h a b\u2081 + h a b\u2082 a : \u03b1 \u2192\u2080 M s : Multiset (\u03b1 \u2192\u2080 M) ih : sum (Multiset.sum s) h = Multiset.sum (Multiset.map (fun g => sum g h) s) \u22a2 sum (Multiset.sum (a ::\u2098 s)) h = Multiset.sum (Multiset.map (fun g => sum g h) (a ::\u2098 s)) ** rw [Multiset.sum_cons, Multiset.map_cons, Multiset.sum_cons, sum_add_index' h\u2080 h\u2081, ih] ** Qed", + "informal": "" + }, + { + "formal": "RatFunc.num_mul_eq_mul_denom_iff ** K : Type u inst\u271d : Field K x : RatFunc K p q : K[X] hq : q \u2260 0 \u22a2 num x * q = p * denom x \u2194 x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q ** rw [\u2190 (algebraMap_injective K).eq_iff, eq_div_iff (algebraMap_ne_zero hq)] ** K : Type u inst\u271d : Field K x : RatFunc K p q : K[X] hq : q \u2260 0 \u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x * q) = \u2191(algebraMap K[X] (RatFunc K)) (p * denom x) \u2194 x * \u2191(algebraMap K[X] (RatFunc K)) q = \u2191(algebraMap K[X] (RatFunc K)) p ** conv_rhs => rw [\u2190 num_div_denom x] ** K : Type u inst\u271d : Field K x : RatFunc K p q : K[X] hq : q \u2260 0 \u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x * q) = \u2191(algebraMap K[X] (RatFunc K)) (p * denom x) \u2194 \u2191(algebraMap K[X] (RatFunc K)) (num x) / \u2191(algebraMap K[X] (RatFunc K)) (denom x) * \u2191(algebraMap K[X] (RatFunc K)) q = \u2191(algebraMap K[X] (RatFunc K)) p ** 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] ** K : Type u inst\u271d : Field K x : RatFunc K p q : K[X] hq : q \u2260 0 \u22a2 \u2191(algebraMap K[X] (RatFunc K)) (denom x) \u2260 0 ** exact algebraMap_ne_zero (denom_ne_zero x) ** Qed", + "informal": "" + }, + { + "formal": "Filter.Tendsto.atBot_add ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1 inst\u271d : OrderTopology \u03b1 l : Filter \u03b2 f g : \u03b2 \u2192 \u03b1 C : \u03b1 hf : Tendsto f l atBot hg : Tendsto g l (\ud835\udcdd C) \u22a2 Tendsto (fun x => f x + g x) l atBot ** conv in _ + _ => rw [add_comm] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1 inst\u271d : OrderTopology \u03b1 l : Filter \u03b2 f g : \u03b2 \u2192 \u03b1 C : \u03b1 hf : Tendsto f l atBot hg : Tendsto g l (\ud835\udcdd C) \u22a2 Tendsto (fun x => g x + f x) l atBot ** exact hg.add_atBot hf ** Qed", + "informal": "" + }, + { + "formal": "LieIdeal.derivedSeries_map_eq ** R : Type u L : Type v M : Type w L' : Type w\u2081 inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I J : LieIdeal R L f : L' \u2192\u2097\u2045R\u2046 L k : \u2115 h : Function.Surjective \u2191f \u22a2 map f (derivedSeries R L' k) = derivedSeries R L k ** induction' k with k ih ** case zero R : Type u L : Type v M : Type w L' : Type w\u2081 inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I J : LieIdeal R L f : L' \u2192\u2097\u2045R\u2046 L h : Function.Surjective \u2191f \u22a2 map f (derivedSeries R L' Nat.zero) = derivedSeries R L Nat.zero ** change (\u22a4 : LieIdeal R L').map f = \u22a4 ** case zero R : Type u L : Type v M : Type w L' : Type w\u2081 inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I J : LieIdeal R L f : L' \u2192\u2097\u2045R\u2046 L h : Function.Surjective \u2191f \u22a2 map f \u22a4 = \u22a4 ** rw [\u2190 f.idealRange_eq_map] ** case zero R : Type u L : Type v M : Type w L' : Type w\u2081 inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I J : LieIdeal R L f : L' \u2192\u2097\u2045R\u2046 L h : Function.Surjective \u2191f \u22a2 LieHom.idealRange f = \u22a4 ** exact f.idealRange_eq_top_of_surjective h ** case succ R : Type u L : Type v M : Type w L' : Type w\u2081 inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I J : LieIdeal R L f : L' \u2192\u2097\u2045R\u2046 L h : Function.Surjective \u2191f k : \u2115 ih : map f (derivedSeries R L' k) = derivedSeries R L k \u22a2 map f (derivedSeries R L' (Nat.succ k)) = derivedSeries R L (Nat.succ k) ** simp only [derivedSeries_def, map_bracket_eq f h, ih, derivedSeriesOfIdeal_succ] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.natTrailingDegree_neg ** R : Type u S : Type v a b : R n m : \u2115 inst\u271d : Ring R p : R[X] \u22a2 natTrailingDegree (-p) = natTrailingDegree p ** simp [natTrailingDegree] ** Qed", + "informal": "" + }, + { + "formal": "wellFounded_iff_wellFounded_subrel ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop \u03b2 : Type u_4 s : \u03b2 \u2192 \u03b2 \u2192 Prop inst\u271d : IsTrans \u03b2 s \u22a2 WellFounded s \u2194 \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b}) ** 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 ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop \u03b2 : Type u_4 s : \u03b2 \u2192 \u03b2 \u2192 Prop inst\u271d : IsTrans \u03b2 s wf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b}) b b' : \u03b2 hb' : s b' b \u22a2 Acc s b' ** let f := PrincipalSeg.ofElement s b ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop \u03b2 : Type u_4 s : \u03b2 \u2192 \u03b2 \u2192 Prop inst\u271d : IsTrans \u03b2 s wf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b}) b b' : \u03b2 hb' : s b' b f : Subrel s {b_1 | s b_1 b} \u227ai s := PrincipalSeg.ofElement s b \u22a2 Acc s b' ** obtain \u27e8b', rfl\u27e9 := f.down.mp ((PrincipalSeg.ofElement_top s b).symm \u25b8 hb' : s b' f.top) ** case intro \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop \u03b2 : Type u_4 s : \u03b2 \u2192 \u03b2 \u2192 Prop inst\u271d : IsTrans \u03b2 s wf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b}) b : \u03b2 f : Subrel s {b_1 | s b_1 b} \u227ai s := PrincipalSeg.ofElement s b b' : \u2191{b_1 | s b_1 b} hb' : s (\u2191f.toRelEmbedding b') b \u22a2 Acc s (\u2191f.toRelEmbedding b') ** exact (f.acc b').mp ((wf b).apply b') ** Qed", + "informal": "" + }, + { + "formal": "TypeVec.append1_drop_last ** n : \u2115 \u03b1 : TypeVec.{u_1} (n + 1) i : Fin2 (n + 1) \u22a2 (drop \u03b1 ::: last \u03b1) i = \u03b1 i ** cases i <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "IsLowerSet.memberSubfamily_subset_nonMemberSubfamily ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 \ud835\udc9c \u212c : Finset (Finset \u03b1) s\u271d : Finset \u03b1 a : \u03b1 h : IsLowerSet \u2191\ud835\udc9c s : Finset \u03b1 \u22a2 s \u2208 Finset.memberSubfamily a \ud835\udc9c \u2192 s \u2208 Finset.nonMemberSubfamily a \ud835\udc9c ** rw [mem_memberSubfamily, mem_nonMemberSubfamily] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 \ud835\udc9c \u212c : Finset (Finset \u03b1) s\u271d : Finset \u03b1 a : \u03b1 h : IsLowerSet \u2191\ud835\udc9c s : Finset \u03b1 \u22a2 insert a s \u2208 \ud835\udc9c \u2227 \u00aca \u2208 s \u2192 s \u2208 \ud835\udc9c \u2227 \u00aca \u2208 s ** exact And.imp_left (h <| subset_insert _ _) ** Qed", + "informal": "" + }, + { + "formal": "String.next_of_valid' ** cs cs' : List Char \u22a2 next { data := cs ++ cs' } { byteIdx := utf8Len cs } = { byteIdx := utf8Len cs + csize (List.headD cs' default) } ** simp only [next, get_of_valid] ** cs cs' : List Char \u22a2 { byteIdx := utf8Len cs } + List.headD cs' default = { byteIdx := utf8Len cs + csize (List.headD cs' default) } ** rfl ** Qed", + "informal": "" + }, + { + "formal": "spectrum.units_smul_resolvent_self ** R : Type u A : Type v inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : Ring A inst\u271d : Algebra R A r : R\u02e3 a : A \u22a2 r \u2022 resolvent a \u2191r = resolvent (r\u207b\u00b9 \u2022 a) 1 ** simpa only [Units.smul_def, Algebra.id.smul_eq_mul, Units.inv_mul] using\n @units_smul_resolvent _ _ _ _ _ r r a ** Qed", + "informal": "" + }, + { + "formal": "List.prev_ne_cons_cons ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x y z : \u03b1 h : x \u2208 y :: z :: l hy : x \u2260 y hz : x \u2260 z \u22a2 x \u2208 z :: l ** simpa [hy] using h ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x y z : \u03b1 h : x \u2208 y :: z :: l hy : x \u2260 y hz : x \u2260 z \u22a2 prev (y :: z :: l) x h = prev (z :: l) x (_ : x \u2208 z :: l) ** cases l ** case nil \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 x y z : \u03b1 hy : x \u2260 y hz : x \u2260 z h : x \u2208 [y, z] \u22a2 prev [y, z] x h = prev [z] x (_ : x \u2208 [z]) ** simp [hy, hz] at h ** case cons \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 x y z : \u03b1 hy : x \u2260 y hz : x \u2260 z head\u271d : \u03b1 tail\u271d : List \u03b1 h : x \u2208 y :: z :: head\u271d :: tail\u271d \u22a2 prev (y :: z :: head\u271d :: tail\u271d) x h = prev (z :: head\u271d :: tail\u271d) x (_ : x \u2208 z :: head\u271d :: tail\u271d) ** rw [prev, dif_neg hy, if_neg hz] ** Qed", + "informal": "" + }, + { + "formal": "mem_span_finset ** R : Type u_1 M : Type u_2 N : Type u_3 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : Module R N s : Finset M x : M hx : x \u2208 span R \u2191s \u22a2 x \u2208 span R (_root_.id '' \u2191s) ** rwa [Set.image_id] ** Qed", + "informal": "" + }, + { + "formal": "List.Nat.mem_antidiagonalTuple ** case h0 k n : \u2115 \u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 n \u2194 \u2211 i : Fin 0, Fin.elim0 i = n ** cases n ** case h0.zero k : \u2115 \u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 Nat.zero \u2194 \u2211 i : Fin 0, Fin.elim0 i = Nat.zero ** simp ** case h0.succ k n\u271d : \u2115 \u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 (Nat.succ n\u271d) \u2194 \u2211 i : Fin 0, Fin.elim0 i = Nat.succ n\u271d ** simp [eq_comm] ** case h k n\u271d x\u2080 : \u2115 x : Fin n\u271d \u2192 \u2115 ih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n n : \u2115 \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 ** simp_rw [Fin.sum_cons] ** case h k n\u271d x\u2080 : \u2115 x : Fin n\u271d \u2192 \u2115 ih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n n : \u2115 \u22a2 Fin.cons x\u2080 x \u2208 antidiagonalTuple (n\u271d + 1) n \u2194 x\u2080 + \u2211 i : Fin n\u271d, x i = n ** rw [antidiagonalTuple] ** case h k n\u271d x\u2080 : \u2115 x : Fin n\u271d \u2192 \u2115 ih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n n : \u2115 \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 x\u2080 + \u2211 i : Fin n\u271d, x i = n ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Rack.self_distrib_inv ** R : Type u_1 inst\u271d : Rack R x y z : R \u22a2 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = (x \u25c3\u207b\u00b9 y) \u25c3\u207b\u00b9 x \u25c3\u207b\u00b9 z ** rw [\u2190 left_cancel (x \u25c3\u207b\u00b9 y), right_inv, \u2190 left_cancel x, right_inv, self_distrib] ** R : Type u_1 inst\u271d : Rack R x y z : R \u22a2 (x \u25c3 x \u25c3\u207b\u00b9 y) \u25c3 x \u25c3 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = z ** repeat' rw [right_inv] ** R : Type u_1 inst\u271d : Rack R x y z : R \u22a2 y \u25c3 y \u25c3\u207b\u00b9 z = z ** rw [right_inv] ** Qed", + "informal": "" + }, + { + "formal": "BoxIntegral.BoxAdditiveMap.volume_apply ** \u03b9 : Type u_1 inst\u271d\u00b2 : Fintype \u03b9 E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E I : Box \u03b9 x : E \u22a2 \u2191(\u2191BoxAdditiveMap.volume I) x = (\u220f j : \u03b9, (upper I j - lower I j)) \u2022 x ** rw [BoxAdditiveMap.volume, toSMul_apply] ** Qed", + "informal": "" + }, + { + "formal": "zsmul_eq_mul' ** \u03b1 : Type u_1 M : Type u N : Type v G : Type w H : Type x A : Type y B : Type z R : Type u\u2081 S : Type u\u2082 inst\u271d : Ring R a : R n : \u2124 \u22a2 n \u2022 a = a * \u2191n ** rw [zsmul_eq_mul, (n.cast_commute a).eq] ** Qed", + "informal": "" + }, + { + "formal": "Stream'.get_odd ** \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w n : \u2115 s : Stream' \u03b1 \u22a2 get (odd s) n = get s (2 * n + 1) ** rw [odd_eq, get_even] ** \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w n : \u2115 s : Stream' \u03b1 \u22a2 get (tail s) (2 * n) = get s (2 * n + 1) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Nat.modEq_digits_sum ** n\u271d b b' : \u2115 h : b' % b = 1 n : \u2115 \u22a2 n \u2261 List.sum (digits b' n) [MOD b] ** rw [\u2190 ofDigits_one] ** n\u271d b b' : \u2115 h : b' % b = 1 n : \u2115 \u22a2 ofDigits b' (digits b' n) \u2261 ofDigits 1 (digits b' n) [MOD b] ** convert ofDigits_modEq b' b (digits b' n) ** case h.e'_3.h.e'_3 n\u271d b b' : \u2115 h : b' % b = 1 n : \u2115 \u22a2 1 = b' % b ** exact h.symm ** Qed", + "informal": "" + }, + { + "formal": "IsPrimitiveRoot.sub_one_norm_prime ** p n : \u2115+ A : Type w B : Type z K : Type u L : Type v C : Type w inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : IsCyclotomicExtension {n} A B inst\u271d\u00b2 : Field L \u03b6 : L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n inst\u271d\u00b9 : Field K inst\u271d : Algebra K L hpri : Fact (Nat.Prime \u2191p) hcyc : IsCyclotomicExtension {p} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191p hirr : Irreducible (cyclotomic (\u2191p) K) h : p \u2260 2 \u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p ** replace hirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1)) : \u2115) K) := by simp [hirr] ** p n : \u2115+ A : Type w B : Type z K : Type u L : Type v C : Type w inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : IsCyclotomicExtension {n} A B inst\u271d\u00b2 : Field L \u03b6 : L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n inst\u271d\u00b9 : Field K inst\u271d : Algebra K L hpri : Fact (Nat.Prime \u2191p) hcyc : IsCyclotomicExtension {p} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191p h : p \u2260 2 hirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K) \u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p ** replace h\u03b6 : IsPrimitiveRoot \u03b6 (\u2191(p ^ (0 + 1)) : \u2115) := by simp [h\u03b6] ** p n : \u2115+ A : Type w B : Type z K : Type u L : Type v C : Type w inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : IsCyclotomicExtension {n} A B inst\u271d\u00b2 : Field L \u03b6 : L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n inst\u271d\u00b9 : Field K inst\u271d : Algebra K L hpri : Fact (Nat.Prime \u2191p) hcyc : IsCyclotomicExtension {p} K L h : p \u2260 2 hirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K) h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1)) \u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p ** haveI : IsCyclotomicExtension {p ^ (0 + 1)} K L := by simp [hcyc] ** p n : \u2115+ A : Type w B : Type z K : Type u L : Type v C : Type w inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : IsCyclotomicExtension {n} A B inst\u271d\u00b2 : Field L \u03b6 : L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n inst\u271d\u00b9 : Field K inst\u271d : Algebra K L hpri : Fact (Nat.Prime \u2191p) hcyc : IsCyclotomicExtension {p} K L h : p \u2260 2 hirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K) h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1)) this : IsCyclotomicExtension {p ^ (0 + 1)} K L \u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p ** simpa using sub_one_norm_prime_ne_two h\u03b6 hirr h ** p n : \u2115+ A : Type w B : Type z K : Type u L : Type v C : Type w inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : IsCyclotomicExtension {n} A B inst\u271d\u00b2 : Field L \u03b6 : L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n inst\u271d\u00b9 : Field K inst\u271d : Algebra K L hpri : Fact (Nat.Prime \u2191p) hcyc : IsCyclotomicExtension {p} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191p hirr : Irreducible (cyclotomic (\u2191p) K) h : p \u2260 2 \u22a2 Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K) ** simp [hirr] ** p n : \u2115+ A : Type w B : Type z K : Type u L : Type v C : Type w inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : IsCyclotomicExtension {n} A B inst\u271d\u00b2 : Field L \u03b6 : L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n inst\u271d\u00b9 : Field K inst\u271d : Algebra K L hpri : Fact (Nat.Prime \u2191p) hcyc : IsCyclotomicExtension {p} K L h\u03b6 : IsPrimitiveRoot \u03b6 \u2191p h : p \u2260 2 hirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K) \u22a2 IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1)) ** simp [h\u03b6] ** p n : \u2115+ A : Type w B : Type z K : Type u L : Type v C : Type w inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : IsCyclotomicExtension {n} A B inst\u271d\u00b2 : Field L \u03b6 : L h\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n inst\u271d\u00b9 : Field K inst\u271d : Algebra K L hpri : Fact (Nat.Prime \u2191p) hcyc : IsCyclotomicExtension {p} K L h : p \u2260 2 hirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K) h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1)) \u22a2 IsCyclotomicExtension {p ^ (0 + 1)} K L ** simp [hcyc] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.Nodup.disjSum ** \u03b1 : Type u_1 \u03b2 : Type u_2 s : Multiset \u03b1 t : Multiset \u03b2 s\u2081 s\u2082 : Multiset \u03b1 t\u2081 t\u2082 : Multiset \u03b2 a : \u03b1 b : \u03b2 x : \u03b1 \u2295 \u03b2 hs : Nodup s ht : Nodup t \u22a2 Nodup (disjSum s t) ** refine' ((hs.map inl_injective).add_iff <| ht.map inr_injective).2 fun x hs ht => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 s : Multiset \u03b1 t : Multiset \u03b2 s\u2081 s\u2082 : Multiset \u03b1 t\u2081 t\u2082 : Multiset \u03b2 a : \u03b1 b : \u03b2 x\u271d : \u03b1 \u2295 \u03b2 hs\u271d : Nodup s ht\u271d : Nodup t x : \u03b1 \u2295 \u03b2 hs : x \u2208 Multiset.map inl s ht : x \u2208 Multiset.map inr t \u22a2 False ** rw [Multiset.mem_map] at hs ht ** \u03b1 : Type u_1 \u03b2 : Type u_2 s : Multiset \u03b1 t : Multiset \u03b2 s\u2081 s\u2082 : Multiset \u03b1 t\u2081 t\u2082 : Multiset \u03b2 a : \u03b1 b : \u03b2 x\u271d : \u03b1 \u2295 \u03b2 hs\u271d : Nodup s ht\u271d : Nodup t x : \u03b1 \u2295 \u03b2 hs : \u2203 a, a \u2208 s \u2227 inl a = x ht : \u2203 a, a \u2208 t \u2227 inr a = x \u22a2 False ** obtain \u27e8a, _, rfl\u27e9 := hs ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 s : Multiset \u03b1 t : Multiset \u03b2 s\u2081 s\u2082 : Multiset \u03b1 t\u2081 t\u2082 : Multiset \u03b2 a\u271d : \u03b1 b : \u03b2 x : \u03b1 \u2295 \u03b2 hs : Nodup s ht\u271d : Nodup t a : \u03b1 left\u271d : a \u2208 s ht : \u2203 a_1, a_1 \u2208 t \u2227 inr a_1 = inl a \u22a2 False ** obtain \u27e8b, _, h\u27e9 := ht ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 s : Multiset \u03b1 t : Multiset \u03b2 s\u2081 s\u2082 : Multiset \u03b1 t\u2081 t\u2082 : Multiset \u03b2 a\u271d : \u03b1 b\u271d : \u03b2 x : \u03b1 \u2295 \u03b2 hs : Nodup s ht : Nodup t a : \u03b1 left\u271d\u00b9 : a \u2208 s b : \u03b2 left\u271d : b \u2208 t h : inr b = inl a \u22a2 False ** exact inr_ne_inl h ** Qed", + "informal": "" + }, + { + "formal": "lt_mul_left ** \u03b1 : Type u \u03b2 : Type u_1 inst\u271d : StrictOrderedSemiring \u03b1 a b c d : \u03b1 hn : 0 < a hm : 1 < b \u22a2 a < b * a ** convert mul_lt_mul_of_pos_right hm hn ** case h.e'_3 \u03b1 : Type u \u03b2 : Type u_1 inst\u271d : StrictOrderedSemiring \u03b1 a b c d : \u03b1 hn : 0 < a hm : 1 < b \u22a2 a = 1 * a ** rw [one_mul] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearEquiv.comp_contDiffAt_iff ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F e : F \u2243L[\ud835\udd5c] G \u22a2 ContDiffAt \ud835\udd5c n (\u2191e \u2218 f) x \u2194 ContDiffAt \ud835\udd5c n f x ** simp only [\u2190 contDiffWithinAt_univ, e.comp_contDiffWithinAt_iff] ** Qed", + "informal": "" + }, + { + "formal": "AddSubgroup.cyclic_of_min ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G ha : IsLeast {g | g \u2208 H \u2227 0 < g} a \u22a2 H = closure {a} ** obtain \u27e8\u27e8a_in, a_pos\u27e9, a_min\u27e9 := ha ** case intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a \u22a2 H = closure {a} ** refine' le_antisymm _ (H.closure_le.mpr <| by simp [a_in]) ** case intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a \u22a2 H \u2264 closure {a} ** intro g g_in ** case intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a g : G g_in : g \u2208 H \u22a2 g \u2208 closure {a} ** obtain \u27e8k, \u27e8nonneg, lt\u27e9, _\u27e9 := existsUnique_zsmul_near_of_pos' a_pos g ** case intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a g : G g_in : g \u2208 H k : \u2124 right\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k nonneg : 0 \u2264 g - k \u2022 a lt : g - k \u2022 a < a h_zero : g - k \u2022 a = 0 \u22a2 g \u2208 closure {a} ** simp [sub_eq_zero.mp h_zero, AddSubgroup.mem_closure_singleton] ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a \u22a2 {a} \u2286 \u2191H ** simp [a_in] ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a g : G g_in : g \u2208 H k : \u2124 right\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k nonneg : 0 \u2264 g - k \u2022 a lt : g - k \u2022 a < a \u22a2 g - k \u2022 a = 0 ** by_contra h ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a g : G g_in : g \u2208 H k : \u2124 right\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k nonneg : 0 \u2264 g - k \u2022 a lt : g - k \u2022 a < a h\u271d : \u00acg - k \u2022 a = 0 h : a \u2264 g - k \u2022 a \u22a2 False ** have h' : \u00aca \u2264 g - k \u2022 a := not_le.mpr lt ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a g : G g_in : g \u2208 H k : \u2124 right\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k nonneg : 0 \u2264 g - k \u2022 a lt : g - k \u2022 a < a h\u271d : \u00acg - k \u2022 a = 0 h : a \u2264 g - k \u2022 a h' : \u00aca \u2264 g - k \u2022 a \u22a2 False ** contradiction ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a g : G g_in : g \u2208 H k : \u2124 right\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k nonneg : 0 \u2264 g - k \u2022 a lt : g - k \u2022 a < a h : \u00acg - k \u2022 a = 0 \u22a2 a \u2264 g - k \u2022 a ** refine' a_min \u27e8_, _\u27e9 ** case refine'_1 G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a g : G g_in : g \u2208 H k : \u2124 right\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k nonneg : 0 \u2264 g - k \u2022 a lt : g - k \u2022 a < a h : \u00acg - k \u2022 a = 0 \u22a2 g - k \u2022 a \u2208 H ** exact AddSubgroup.sub_mem H g_in (AddSubgroup.zsmul_mem H a_in k) ** case refine'_2 G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G a : G a_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g} a_in : a \u2208 H a_pos : 0 < a g : G g_in : g \u2208 H k : \u2124 right\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k nonneg : 0 \u2264 g - k \u2022 a lt : g - k \u2022 a < a h : \u00acg - k \u2022 a = 0 \u22a2 0 < g - k \u2022 a ** exact lt_of_le_of_ne nonneg (Ne.symm h) ** Qed", + "informal": "" + }, + { + "formal": "Monotone.map_iInf_of_continuousAt' ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : OrderTopology \u03b1 inst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : OrderClosedTopology \u03b2 inst\u271d\u00b9 : Nonempty \u03b3 \u03b9 : Sort u_1 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 g : \u03b9 \u2192 \u03b1 Cf : ContinuousAt f (iInf g) Mf : Monotone f bdd : autoParam (BddBelow (range g)) _auto\u271d \u22a2 f (\u2a05 i, g i) = \u2a05 i, f (g i) ** rw [iInf, Monotone.map_sInf_of_continuousAt' Cf Mf (range_nonempty g) bdd, \u2190 range_comp, iInf] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : OrderTopology \u03b1 inst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : OrderClosedTopology \u03b2 inst\u271d\u00b9 : Nonempty \u03b3 \u03b9 : Sort u_1 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 g : \u03b9 \u2192 \u03b1 Cf : ContinuousAt f (iInf g) Mf : Monotone f bdd : autoParam (BddBelow (range g)) _auto\u271d \u22a2 sInf (range (f \u2218 g)) = sInf (range fun i => f (g i)) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.center_eq_infi' ** G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N s : Set G g : G S : Set G hS : closure S = \u22a4 \u22a2 center G = \u2a05 g, centralizer \u2191(zpowers \u2191g) ** rw [center_eq_iInf S hS, \u2190 iInf_subtype''] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.rel_zero_right ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 \u03b4 : Type u_3 r : \u03b1 \u2192 \u03b2 \u2192 Prop p : \u03b3 \u2192 \u03b4 \u2192 Prop a : Multiset \u03b1 \u22a2 Rel r a 0 \u2194 a = 0 ** rw [Rel_iff] ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 \u03b4 : Type u_3 r : \u03b1 \u2192 \u03b2 \u2192 Prop p : \u03b3 \u2192 \u03b4 \u2192 Prop a : Multiset \u03b1 \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 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Perm.disjoint_conj ** \u03b1 : Type u_1 f g h\u271d h : Perm \u03b1 x\u271d : \u03b1 \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 ** simp only [mul_apply, eq_inv_iff_eq] ** Qed", + "informal": "" + }, + { + "formal": "isClosedMap_smul\u2080 ** M\u271d : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 G\u2080 : Type u_4 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b1 inst\u271d\u2079 : GroupWithZero G\u2080 inst\u271d\u2078 : MulAction G\u2080 \u03b1 inst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b2 f : \u03b2 \u2192 \u03b1 b : \u03b2 c\u271d : G\u2080 s : Set \u03b2 \ud835\udd5c : Type u_5 M : Type u_6 inst\u271d\u2075 : DivisionRing \ud835\udd5c inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : T1Space M inst\u271d\u00b9 : Module \ud835\udd5c M inst\u271d : ContinuousConstSMul \ud835\udd5c M c : \ud835\udd5c \u22a2 IsClosedMap fun x => c \u2022 x ** rcases eq_or_ne c 0 with (rfl | hne) ** case inl M\u271d : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 G\u2080 : Type u_4 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b1 inst\u271d\u2079 : GroupWithZero G\u2080 inst\u271d\u2078 : MulAction G\u2080 \u03b1 inst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b2 f : \u03b2 \u2192 \u03b1 b : \u03b2 c : G\u2080 s : Set \u03b2 \ud835\udd5c : Type u_5 M : Type u_6 inst\u271d\u2075 : DivisionRing \ud835\udd5c inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : T1Space M inst\u271d\u00b9 : Module \ud835\udd5c M inst\u271d : ContinuousConstSMul \ud835\udd5c M \u22a2 IsClosedMap fun x => 0 \u2022 x ** simp only [zero_smul] ** case inl M\u271d : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 G\u2080 : Type u_4 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b1 inst\u271d\u2079 : GroupWithZero G\u2080 inst\u271d\u2078 : MulAction G\u2080 \u03b1 inst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b2 f : \u03b2 \u2192 \u03b1 b : \u03b2 c : G\u2080 s : Set \u03b2 \ud835\udd5c : Type u_5 M : Type u_6 inst\u271d\u2075 : DivisionRing \ud835\udd5c inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : T1Space M inst\u271d\u00b9 : Module \ud835\udd5c M inst\u271d : ContinuousConstSMul \ud835\udd5c M \u22a2 IsClosedMap fun x => 0 ** exact isClosedMap_const ** case inr M\u271d : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 G\u2080 : Type u_4 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b1 inst\u271d\u2079 : GroupWithZero G\u2080 inst\u271d\u2078 : MulAction G\u2080 \u03b1 inst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b2 f : \u03b2 \u2192 \u03b1 b : \u03b2 c\u271d : G\u2080 s : Set \u03b2 \ud835\udd5c : Type u_5 M : Type u_6 inst\u271d\u2075 : DivisionRing \ud835\udd5c inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : T1Space M inst\u271d\u00b9 : Module \ud835\udd5c M inst\u271d : ContinuousConstSMul \ud835\udd5c M c : \ud835\udd5c hne : c \u2260 0 \u22a2 IsClosedMap fun x => c \u2022 x ** exact (Homeomorph.smulOfNeZero c hne).isClosedMap ** Qed", + "informal": "" + }, + { + "formal": "differentiable_const_add_iff ** \ud835\udd5c : Type u_1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G G' : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' f f\u2080 f\u2081 g : E \u2192 F f' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F x : E s t : Set E L L\u2081 L\u2082 : Filter E c : F h : Differentiable \ud835\udd5c fun y => c + f y \u22a2 Differentiable \ud835\udd5c f ** simpa using h.const_add (-c) ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.monic_of_natDegree_le_of_coeff_eq_one ** R : Type u S : Type v a b c d : R n\u271d m : \u2115 inst\u271d : Semiring R p q : R[X] \u03b9 : Type u_1 n : \u2115 pn : natDegree p \u2264 n p1 : coeff p n = 1 \u22a2 Monic p ** unfold Monic ** R : Type u S : Type v a b c d : R n\u271d m : \u2115 inst\u271d : Semiring R p q : R[X] \u03b9 : Type u_1 n : \u2115 pn : natDegree p \u2264 n p1 : coeff p n = 1 \u22a2 leadingCoeff p = 1 ** nontriviality ** R : Type u S : Type v a b c d : R n\u271d m : \u2115 inst\u271d : Semiring R p q : R[X] \u03b9 : Type u_1 n : \u2115 pn : natDegree p \u2264 n p1 : coeff p n = 1 \u271d : Nontrivial R \u22a2 leadingCoeff p = 1 ** refine' (congr_arg _ <| natDegree_eq_of_le_of_coeff_ne_zero pn _).trans p1 ** R : Type u S : Type v a b c d : R n\u271d m : \u2115 inst\u271d : Semiring R p q : R[X] \u03b9 : Type u_1 n : \u2115 pn : natDegree p \u2264 n p1 : coeff p n = 1 \u271d : Nontrivial R \u22a2 coeff p n \u2260 0 ** exact ne_of_eq_of_ne p1 one_ne_zero ** Qed", + "informal": "" + }, + { + "formal": "HeytingHom.cancel_left ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 inst\u271d\u00b3 : HeytingAlgebra \u03b1 inst\u271d\u00b2 : HeytingAlgebra \u03b2 inst\u271d\u00b9 : HeytingAlgebra \u03b3 inst\u271d : HeytingAlgebra \u03b4 f f\u2081 f\u2082 : HeytingHom \u03b1 \u03b2 g g\u2081 g\u2082 : HeytingHom \u03b2 \u03b3 hg : Injective \u2191g h : comp g f\u2081 = comp g f\u2082 a : \u03b1 \u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a) ** rw [\u2190 comp_apply, h, comp_apply] ** Qed", + "informal": "" + }, + { + "formal": "RingHom.cancel_left ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 x\u271d\u00b2 : NonAssocSemiring \u03b1 x\u271d\u00b9 : NonAssocSemiring \u03b2 x\u271d : NonAssocSemiring \u03b3 g : \u03b2 \u2192+* \u03b3 f\u2081 f\u2082 : \u03b1 \u2192+* \u03b2 hg : Injective \u2191g h : comp g f\u2081 = comp g f\u2082 x : \u03b1 \u22a2 \u2191g (\u2191f\u2081 x) = \u2191g (\u2191f\u2082 x) ** rw [\u2190 comp_apply, h, comp_apply] ** Qed", + "informal": "" + }, + { + "formal": "Units.isUnit_units_mul ** \u03b1 : Type u M\u271d : Type u_1 N : Type u_2 M : Type u_3 inst\u271d : Monoid M u : M\u02e3 a : M x\u271d : IsUnit (\u2191u * a) v : M\u02e3 hv : \u2191v = \u2191u * a \u22a2 IsUnit a ** have : IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a)) := by exists u\u207b\u00b9 * v; rw [\u2190 hv, Units.val_mul] ** \u03b1 : Type u M\u271d : Type u_1 N : Type u_2 M : Type u_3 inst\u271d : Monoid M u : M\u02e3 a : M x\u271d : IsUnit (\u2191u * a) v : M\u02e3 hv : \u2191v = \u2191u * a this : IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a)) \u22a2 IsUnit a ** rwa [\u2190 mul_assoc, Units.inv_mul, one_mul] at this ** \u03b1 : Type u M\u271d : Type u_1 N : Type u_2 M : Type u_3 inst\u271d : Monoid M u : M\u02e3 a : M x\u271d : IsUnit (\u2191u * a) v : M\u02e3 hv : \u2191v = \u2191u * a \u22a2 IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a)) ** exists u\u207b\u00b9 * v ** \u03b1 : Type u M\u271d : Type u_1 N : Type u_2 M : Type u_3 inst\u271d : Monoid M u : M\u02e3 a : M x\u271d : IsUnit (\u2191u * a) v : M\u02e3 hv : \u2191v = \u2191u * a \u22a2 \u2191(u\u207b\u00b9 * v) = \u2191u\u207b\u00b9 * (\u2191u * a) ** rw [\u2190 hv, Units.val_mul] ** Qed", + "informal": "" + }, + { + "formal": "Set.Ioc_union_Ioc_union_Ioc_cycle ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LinearOrder \u03b2 f : \u03b1 \u2192 \u03b2 a a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1 \u22a2 Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc (min a (min b c)) (max a (max b c)) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Commute.mul_zpow ** \u03b1 : Type u_1 M : Type u N : Type v G : Type w H : Type x A : Type y B : Type z R : Type u\u2081 S : Type u\u2082 inst\u271d : DivisionMonoid \u03b1 a b : \u03b1 h : Commute a b n : \u2115 \u22a2 (a * b) ^ \u2191n = a ^ \u2191n * b ^ \u2191n ** simp [zpow_ofNat, h.mul_pow n] ** \u03b1 : Type u_1 M : Type u N : Type v G : Type w H : Type x A : Type y B : Type z R : Type u\u2081 S : Type u\u2082 inst\u271d : DivisionMonoid \u03b1 a b : \u03b1 h : Commute a b n : \u2115 \u22a2 (a * b) ^ Int.negSucc n = a ^ Int.negSucc n * b ^ Int.negSucc n ** simp [h.mul_pow, (h.pow_pow _ _).eq, mul_inv_rev] ** Qed", + "informal": "" + }, + { + "formal": "Lagrange.nodal_insert_eq_nodal ** R : Type u_1 inst\u271d\u00b9 : CommRing R \u03b9 : Type u_2 s : Finset \u03b9 v : \u03b9 \u2192 R inst\u271d : DecidableEq \u03b9 i : \u03b9 hi : \u00aci \u2208 s \u22a2 nodal (insert i s) v = (X - \u2191C (v i)) * nodal s v ** simp_rw [nodal, prod_insert hi] ** Qed", + "informal": "" + }, + { + "formal": "Complex.betaIntegral_convergent ** u v : \u2102 hu : 0 < u.re hv : 0 < v.re \u22a2 IntervalIntegrable (fun x => \u2191x ^ (u - 1) * (1 - \u2191x) ^ (v - 1)) volume 0 1 ** refine' (betaIntegral_convergent_left hu v).trans _ ** u v : \u2102 hu : 0 < u.re hv : 0 < v.re \u22a2 IntervalIntegrable (fun x => \u2191x ^ (u - 1) * (1 - \u2191x) ^ (v - 1)) volume (1 / 2) 1 ** rw [IntervalIntegrable.iff_comp_neg] ** u v : \u2102 hu : 0 < u.re hv : 0 < v.re \u22a2 IntervalIntegrable (fun x => \u2191(-x) ^ (u - 1) * (1 - \u2191(-x)) ^ (v - 1)) volume (-(1 / 2)) (-1) ** convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1 ** case h.e'_3 u v : \u2102 hu : 0 < u.re hv : 0 < v.re \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) ** ext1 x ** case h.e'_3.h u v : \u2102 hu : 0 < u.re hv : 0 < v.re x : \u211d \u22a2 \u2191(-x) ^ (u - 1) * (1 - \u2191(-x)) ^ (v - 1) = \u2191(x + 1) ^ (v - 1) * (1 - \u2191(x + 1)) ^ (u - 1) ** conv_lhs => rw [mul_comm] ** case h.e'_3.h.e_a.e_a u v : \u2102 hu : 0 < u.re hv : 0 < v.re x : \u211d \u22a2 \u2191(-x) = 1 - \u2191(x + 1) ** push_cast ** case h.e'_3.h.e_a.e_a u v : \u2102 hu : 0 < u.re hv : 0 < v.re x : \u211d \u22a2 -\u2191x = 1 - (\u2191x + 1) ** ring ** case h.e'_5 u v : \u2102 hu : 0 < u.re hv : 0 < v.re \u22a2 -(1 / 2) = 1 / 2 - 1 ** norm_num ** case h.e'_6 u v : \u2102 hu : 0 < u.re hv : 0 < v.re \u22a2 -1 = 0 - 1 ** norm_num ** Qed", + "informal": "" + }, + { + "formal": "Matrix.represents_iff' ** \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A : Matrix \u03b9 \u03b9 R f : Module.End R M \u22a2 Represents b A f \u2194 \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j) ** constructor ** case mp \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A : Matrix \u03b9 \u03b9 R f : Module.End R M \u22a2 Represents b A f \u2192 \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j) ** intro h i ** case mp \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A : Matrix \u03b9 \u03b9 R f : Module.End R M h : Represents b A f i : \u03b9 \u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i) ** have := LinearMap.congr_fun h (Pi.single i 1) ** case mp \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A : Matrix \u03b9 \u03b9 R f : Module.End R M h : Represents b A f i : \u03b9 this : \u2191(\u2191(PiToModule.fromMatrix R b) A) (Pi.single i 1) = \u2191(\u2191(PiToModule.fromEnd R b) f) (Pi.single i 1) \u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i) ** rwa [PiToModule.fromEnd_apply_single_one, PiToModule.fromMatrix_apply_single_one] at this ** case mpr \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A : Matrix \u03b9 \u03b9 R f : Module.End R M \u22a2 (\u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)) \u2192 Represents b A f ** intro h ** case mpr \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A : Matrix \u03b9 \u03b9 R f : Module.End R M h : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j) \u22a2 Represents b A f ** refine LinearMap.pi_ext' (fun i => LinearMap.ext_ring ?_) ** case mpr \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A : Matrix \u03b9 \u03b9 R f : Module.End R M h : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j) i : \u03b9 \u22a2 \u2191(LinearMap.comp (\u2191(PiToModule.fromMatrix R b) A) (LinearMap.single i)) 1 = \u2191(LinearMap.comp (\u2191(PiToModule.fromEnd R b) f) (LinearMap.single i)) 1 ** simp_rw [LinearMap.comp_apply, LinearMap.coe_single, PiToModule.fromEnd_apply_single_one,\n PiToModule.fromMatrix_apply_single_one] ** case mpr \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A : Matrix \u03b9 \u03b9 R f : Module.End R M h : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j) i : \u03b9 \u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i) ** apply h ** Qed", + "informal": "" + }, + { + "formal": "Finset.univ_map_equiv_to_embedding ** \u03b1\u271d : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : Fintype \u03b1\u271d s t : Finset \u03b1\u271d \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u03b2 e : \u03b1 \u2243 \u03b2 b : \u03b2 \u22a2 \u2191(Equiv.toEmbedding e) (\u2191e.symm b) = b ** simp ** Qed", + "informal": "" + }, + { + "formal": "Nat.noZeroSMulDivisors ** \u03b1 : Type u_1 R : Type u_2 k : Type u_3 S : Type u_4 M : Type u_5 M\u2082 : Type u_6 M\u2083 : Type u_7 \u03b9 : Type u_8 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroSMulDivisors R M inst\u271d : CharZero R \u22a2 \u2200 {c : \u2115} {x : M}, c \u2022 x = 0 \u2192 c = 0 \u2228 x = 0 ** intro c x ** \u03b1 : Type u_1 R : Type u_2 k : Type u_3 S : Type u_4 M : Type u_5 M\u2082 : Type u_6 M\u2083 : Type u_7 \u03b9 : Type u_8 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroSMulDivisors R M inst\u271d : CharZero R c : \u2115 x : M \u22a2 c \u2022 x = 0 \u2192 c = 0 \u2228 x = 0 ** rw [nsmul_eq_smul_cast R, smul_eq_zero] ** \u03b1 : Type u_1 R : Type u_2 k : Type u_3 S : Type u_4 M : Type u_5 M\u2082 : Type u_6 M\u2083 : Type u_7 \u03b9 : Type u_8 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroSMulDivisors R M inst\u271d : CharZero R c : \u2115 x : M \u22a2 \u2191c = 0 \u2228 x = 0 \u2192 c = 0 \u2228 x = 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Finite.one_lt_card_iff_nontrivial ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Finite \u03b1 \u22a2 1 < Nat.card \u03b1 \u2194 Nontrivial \u03b1 ** haveI := Fintype.ofFinite \u03b1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Finite \u03b1 this : Fintype \u03b1 \u22a2 1 < Nat.card \u03b1 \u2194 Nontrivial \u03b1 ** simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial] ** Qed", + "informal": "" + }, + { + "formal": "LowerSet.symm_map ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03ba : \u03b9 \u2192 Sort u_5 inst\u271d\u00b2 : Preorder \u03b1 inst\u271d\u00b9 : Preorder \u03b2 inst\u271d : Preorder \u03b3 f\u271d : \u03b1 \u2243o \u03b2 s\u271d t : LowerSet \u03b1 a : \u03b1 b : \u03b2 f : \u03b1 \u2243o \u03b2 s : LowerSet \u03b2 \u22a2 \u2191(\u2191(OrderIso.symm (map f)) s) = \u2191(\u2191(map (OrderIso.symm f)) s) ** convert Set.preimage_equiv_eq_image_symm s f.toEquiv ** Qed", + "informal": "" + }, + { + "formal": "Int.bit_negSucc ** b : Bool n : \u2115 \u22a2 bit b -[n+1] = -[Nat.bit (!b) n+1] ** rw [bit_val, Nat.bit_val] ** b : Bool n : \u2115 \u22a2 (2 * -[n+1] + bif b then 1 else 0) = -[2 * n + bif !b then 1 else 0+1] ** cases b <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "Cubic.of_d_eq_zero ** R : Type u_1 S : Type u_2 F : Type u_3 K : Type u_4 P Q : Cubic R a b c d a' b' c' d' : R inst\u271d : Semiring R ha : P.a = 0 hb : P.b = 0 hc : P.c = 0 hd : P.d = 0 \u22a2 toPoly P = 0 ** rw [of_c_eq_zero ha hb hc, hd, C_0] ** Qed", + "informal": "" + }, + { + "formal": "PrimeMultiset.coePNat_prime ** v : PrimeMultiset p : \u2115+ h : p \u2208 toPNatMultiset v \u22a2 PNat.Prime p ** rcases Multiset.mem_map.mp h with \u27e8\u27e8_, hp'\u27e9, \u27e8_, h_eq\u27e9\u27e9 ** case intro.mk.intro v : PrimeMultiset p : \u2115+ h : p \u2208 toPNatMultiset v val\u271d : \u2115 hp' : Nat.Prime val\u271d left\u271d : { val := val\u271d, property := hp' } \u2208 v h_eq : Coe.coe { val := val\u271d, property := hp' } = p \u22a2 PNat.Prime p ** exact h_eq \u25b8 hp' ** Qed", + "informal": "" + }, + { + "formal": "StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 P (g \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) \u2194 P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x) ** rw [\u2190 hG.right_invariance (compatible_of_mem_maximalAtlas he he')] ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e'))) (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) (\u2191(LocalHomeomorph.symm e \u226b\u2095 e') (\u2191e x)) \u2194 P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x) H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 \u2191e x \u2208 (LocalHomeomorph.symm e \u226b\u2095 e').toLocalEquiv.source ** swap ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e'))) (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) (\u2191(LocalHomeomorph.symm e \u226b\u2095 e') (\u2191e x)) \u2194 P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x) ** simp_rw [LocalHomeomorph.trans_apply, e.left_inv xe] ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e'))) (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) (\u2191e' x) \u2194 P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x) ** rw [hG.congr_iff] ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 \u2191e x \u2208 (LocalHomeomorph.symm e \u226b\u2095 e').toLocalEquiv.source ** simp only [xe, xe', mfld_simps] ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \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 P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x) ** refine' hG.congr_set _ ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) =\u1da0[\ud835\udcdd (\u2191e' x)] \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s ** refine' (eventually_of_mem _ fun y (hy : y \u2208 e'.symm \u207b\u00b9' e.source) \u21a6 _).set_eq ** case refine'_2 H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source y : H hy : y \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' e.source \u22a2 y \u2208 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2194 y \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s ** simp_rw [mem_preimage, LocalHomeomorph.coe_trans_symm, LocalHomeomorph.symm_symm,\n Function.comp_apply, e.left_inv hy] ** case refine'_1 H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' e.source \u2208 \ud835\udcdd (\u2191e' x) ** refine' (e'.symm.continuousAt <| e'.mapsTo xe').preimage_mem_nhds (e.open_source.mem_nhds _) ** case refine'_1 H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 \u2191(LocalHomeomorph.symm e') (\u2191e' x) \u2208 e.source ** simp_rw [e'.left_inv xe', xe] ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source \u22a2 (g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) =\u1da0[\ud835\udcdd (\u2191e' x)] g \u2218 \u2191(LocalHomeomorph.symm e') ** refine' ((e'.eventually_nhds' _ xe').mpr <| e.eventually_left_inverse xe).mono fun y hy \u21a6 _ ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source y : H hy : \u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y)) = \u2191(LocalHomeomorph.symm e') y \u22a2 ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e'))) y = (g \u2218 \u2191(LocalHomeomorph.symm e')) y ** simp only [mfld_simps] ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' e e' : LocalHomeomorph M H f f' : LocalHomeomorph M' H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop g\u271d g' : M \u2192 M' s t : Set M x : M Q : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop hG : LocalInvariantProp G G' P g : M \u2192 H' he : e \u2208 maximalAtlas M G xe : x \u2208 e.source he' : e' \u2208 maximalAtlas M G xe' : x \u2208 e'.source y : H hy : \u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y)) = \u2191(LocalHomeomorph.symm e') y \u22a2 g (\u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y))) = g (\u2191(LocalHomeomorph.symm e') y) ** rw [hy] ** Qed", + "informal": "" + }, + { + "formal": "Set.pow_mem_pow ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : Monoid \u03b1 s t : Set \u03b1 a : \u03b1 m n : \u2115 ha : a \u2208 s \u22a2 a ^ 0 \u2208 s ^ 0 ** rw [pow_zero] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : Monoid \u03b1 s t : Set \u03b1 a : \u03b1 m n : \u2115 ha : a \u2208 s \u22a2 1 \u2208 s ^ 0 ** exact one_mem_one ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : Monoid \u03b1 s t : Set \u03b1 a : \u03b1 m n\u271d : \u2115 ha : a \u2208 s n : \u2115 \u22a2 a ^ (n + 1) \u2208 s ^ (n + 1) ** rw [pow_succ] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : Monoid \u03b1 s t : Set \u03b1 a : \u03b1 m n\u271d : \u2115 ha : a \u2208 s n : \u2115 \u22a2 a * a ^ n \u2208 s ^ (n + 1) ** exact mul_mem_mul ha (pow_mem_pow ha _) ** Qed", + "informal": "" + }, + { + "formal": "Filter.exists_subset_subsingleton_mem_of_forall_separating ** \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s hs : s \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l \u22a2 \u2203 t, t \u2286 s \u2227 Set.Subsingleton t \u2227 t \u2208 l ** rcases h.1 with \u27e8S, hSc, hSp, hS\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s hs : s \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 \u22a2 \u2203 t, t \u2286 s \u2227 Set.Subsingleton t \u2227 t \u2208 l ** refine \u27e8s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 (U \u2208 S) (_ : U\u1d9c \u2208 l), U\u1d9c, ?_, ?_, ?_\u27e9 ** case intro.intro.intro.refine_1 \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s hs : s \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 \u22a2 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c \u2286 s ** exact fun _ h \u21a6 h.1.1 ** case intro.intro.intro.refine_2 \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s hs : s \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 \u22a2 Set.Subsingleton (s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c) ** intro x hx y hy ** case intro.intro.intro.refine_2 \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s hs : s \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 x : \u03b1 hx : x \u2208 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c y : \u03b1 hy : y \u2208 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c \u22a2 x = y ** simp only [mem_sInter, mem_inter_iff, mem_iInter, mem_compl_iff] at hx hy ** case intro.intro.intro.refine_2 \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s hs : s \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 x y : \u03b1 hx : (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 hy : (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 \u22a2 x = y ** refine hS x hx.1.1 y hy.1.1 (fun s hsS \u21a6 ?_) ** case intro.intro.intro.refine_2 \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s\u271d : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s\u271d hs : s\u271d \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 x y : \u03b1 hx : (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 hy : (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 s : Set \u03b1 hsS : s \u2208 S \u22a2 x \u2208 s \u2194 y \u2208 s ** 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] ** case intro.intro.intro.refine_2.inl \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s\u271d : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s\u271d hs : s\u271d \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 x y : \u03b1 hx : (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 hy : (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 s : Set \u03b1 hsS : s \u2208 S hsl : s \u2208 l \u22a2 x \u2208 s \u2194 y \u2208 s ** simp only [hx.1.2 s \u27e8hsS, hsl\u27e9, hy.1.2 s \u27e8hsS, hsl\u27e9] ** case intro.intro.intro.refine_2.inr \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s\u271d : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s\u271d hs : s\u271d \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 x y : \u03b1 hx : (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 hy : (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 s : Set \u03b1 hsS : s \u2208 S hsl : s\u1d9c \u2208 l \u22a2 x \u2208 s \u2194 y \u2208 s ** simp only [hx.2 s hsS hsl, hy.2 s hsS hsl] ** case intro.intro.intro.refine_3 \u03b1 : Type u_1 \u03b2 : Sort ?u.5035 l : Filter \u03b1 inst\u271d : CountableInterFilter l f g : \u03b1 \u2192 \u03b2 p : Set \u03b1 \u2192 Prop s : Set \u03b1 h : HasCountableSeparatingOn \u03b1 p s hs : s \u2208 l hl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l S : Set (Set \u03b1) hSc : Set.Countable S hSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s hS : \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 \u22a2 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c \u2208 l ** 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) ** Qed", + "informal": "" + }, + { + "formal": "normSq_eq_of_mem_circle ** z : { x // x \u2208 circle } \u22a2 \u2191normSq \u2191z = 1 ** simp [normSq_eq_abs] ** Qed", + "informal": "" + }, + { + "formal": "ZNum.dvd_to_int ** \u03b1 : Type u_1 m n : ZNum x\u271d : \u2191m \u2223 \u2191n k : \u2124 e : \u2191n = \u2191m * k \u22a2 n = m * \u2191k ** rw [\u2190 of_to_int n, e] ** \u03b1 : Type u_1 m n : ZNum x\u271d : \u2191m \u2223 \u2191n k : \u2124 e : \u2191n = \u2191m * k \u22a2 \u2191(\u2191m * k) = m * \u2191k ** simp ** \u03b1 : Type u_1 m n : ZNum x\u271d : m \u2223 n k : ZNum e : n = m * k \u22a2 \u2191n = \u2191m * \u2191k ** simp [e] ** Qed", + "informal": "" + }, + { + "formal": "List.get?_set_eq_of_lt ** \u03b1 : Type u_1 a : \u03b1 n : Nat l : List \u03b1 h : n < length l \u22a2 get? (set l n a) n = some a ** rw [get?_set_eq, get?_eq_get h] ** \u03b1 : Type u_1 a : \u03b1 n : Nat l : List \u03b1 h : n < length l \u22a2 (fun x => a) <$> some (get l { val := n, isLt := h }) = some a ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Egorov.measure_inter_notConvergentSeq_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) n : \u2115 \u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 j, notConvergentSeq f g n j) = 0 ** simp_rw [Metric.tendsto_atTop, ae_iff] at hfg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 n : \u2115 hfg : \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 \u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 j, notConvergentSeq f g n j) = 0 ** rw [\u2190 nonpos_iff_eq_zero, \u2190 hfg] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 n : \u2115 hfg : \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 \u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 j, notConvergentSeq f g n j) \u2264 \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)} ** refine' measure_mono fun x => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 n : \u2115 hfg : \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 x : \u03b1 \u22a2 x \u2208 s \u2229 \u22c2 j, notConvergentSeq f g n j \u2192 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)} ** simp only [Set.mem_inter_iff, Set.mem_iInter, ge_iff_le, mem_notConvergentSeq_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 n : \u2115 hfg : \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 x : \u03b1 \u22a2 (x \u2208 s \u2227 \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)) \u2192 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)} ** push_neg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 n : \u2115 hfg : \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 x : \u03b1 \u22a2 (x \u2208 s \u2227 \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)) \u2192 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)} ** rintro \u27e8hmem, hx\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 n : \u2115 hfg : \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 x : \u03b1 hmem : x \u2208 s hx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x) \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)} ** refine' \u27e8hmem, 1 / (n + 1 : \u211d), Nat.one_div_pos_of_nat, fun N => _\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 n : \u2115 hfg : \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 x : \u03b1 hmem : x \u2208 s hx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x) N : \u03b9 \u22a2 \u2203 n_1, N \u2264 n_1 \u2227 1 / (\u2191n + 1) \u2264 dist (f n_1 x) (g x) ** obtain \u27e8n, hn\u2081, hn\u2082\u27e9 := hx N ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d\u00b9 : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b9 inst\u271d : Nonempty \u03b9 n\u271d : \u2115 hfg : \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 x : \u03b1 hmem : x \u2208 s hx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n\u271d + 1) < dist (f k x) (g x) N n : \u03b9 hn\u2081 : N \u2264 n hn\u2082 : 1 / (\u2191n\u271d + 1) < dist (f n x) (g x) \u22a2 \u2203 n, N \u2264 n \u2227 1 / (\u2191n\u271d + 1) \u2264 dist (f n x) (g x) ** exact \u27e8n, hn\u2081, hn\u2082.le\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "IsPrimitiveRoot.minpoly_dvd_x_pow_sub_one ** n : \u2115 K : Type u_1 inst\u271d\u00b2 : CommRing K \u03bc : K h : IsPrimitiveRoot \u03bc n inst\u271d\u00b9 : IsDomain K inst\u271d : CharZero K \u22a2 minpoly \u2124 \u03bc \u2223 X ^ n - 1 ** rcases n.eq_zero_or_pos with (rfl | h0) ** case inr n : \u2115 K : Type u_1 inst\u271d\u00b2 : CommRing K \u03bc : K h : IsPrimitiveRoot \u03bc n inst\u271d\u00b9 : IsDomain K inst\u271d : CharZero K h0 : n > 0 \u22a2 minpoly \u2124 \u03bc \u2223 X ^ n - 1 ** apply minpoly.isIntegrallyClosed_dvd (isIntegral h h0) ** case inr n : \u2115 K : Type u_1 inst\u271d\u00b2 : CommRing K \u03bc : K h : IsPrimitiveRoot \u03bc n inst\u271d\u00b9 : IsDomain K inst\u271d : CharZero K h0 : n > 0 \u22a2 \u2191(Polynomial.aeval \u03bc) (X ^ n - 1) = 0 ** 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] ** case inl K : Type u_1 inst\u271d\u00b2 : CommRing K \u03bc : K inst\u271d\u00b9 : IsDomain K inst\u271d : CharZero K h : IsPrimitiveRoot \u03bc 0 \u22a2 minpoly \u2124 \u03bc \u2223 X ^ 0 - 1 ** simp ** Qed", + "informal": "" + }, + { + "formal": "WittVector.coeff_add_of_disjoint ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 \u22a2 coeff (x + y) n = coeff x n + coeff y n ** let P : \u2115 \u2192 Prop := fun n => y.coeff n = 0 ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 \u22a2 coeff (x + y) n = coeff x n + coeff y n ** haveI : DecidablePred P := Classical.decPred P ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P \u22a2 coeff (x + y) n = coeff x n + coeff y n ** set z := mk p fun n => if P n then x.coeff n else y.coeff n ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n \u22a2 select P z = x ** ext1 n ** case h p : \u2115 hp : Fact (Nat.Prime p) n\u271d : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n n : \u2115 \u22a2 coeff (select P z) n = coeff x n ** rw [select, coeff_mk, coeff_mk] ** case h p : \u2115 hp : Fact (Nat.Prime p) n\u271d : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n n : \u2115 \u22a2 (if P n then if P n then coeff x n else coeff y n else 0) = coeff x n ** split_ifs with hn ** case pos p : \u2115 hp : Fact (Nat.Prime p) n\u271d : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n n : \u2115 hn : P n \u22a2 coeff x n = coeff x n ** rfl ** case neg p : \u2115 hp : Fact (Nat.Prime p) n\u271d : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n n : \u2115 hn : \u00acP n \u22a2 0 = coeff x n ** rw [(h n).resolve_right hn] ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x \u22a2 select (fun i => \u00acP i) z = y ** ext1 n ** case h p : \u2115 hp : Fact (Nat.Prime p) n\u271d : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x n : \u2115 \u22a2 coeff (select (fun i => \u00acP i) z) n = coeff y n ** rw [select, coeff_mk, coeff_mk] ** case h p : \u2115 hp : Fact (Nat.Prime p) n\u271d : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x n : \u2115 \u22a2 (if \u00acP n then if P n then coeff x n else coeff y n else 0) = coeff y n ** split_ifs with hn ** case pos p : \u2115 hp : Fact (Nat.Prime p) n\u271d : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x n : \u2115 hn : P n \u22a2 0 = coeff y n ** exact hn.symm ** case neg p : \u2115 hp : Fact (Nat.Prime p) n\u271d : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x n : \u2115 hn : \u00acP n \u22a2 coeff y n = coeff y n ** rfl ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x hy : select (fun i => \u00acP i) z = y \u22a2 coeff (x + y) n = coeff z n ** rw [\u2190 hx, \u2190 hy, select_add_select_not P z] ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x hy : select (fun i => \u00acP i) z = y \u22a2 coeff z n = coeff x n + coeff y n ** simp only [mk._eq_1] ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x hy : select (fun i => \u00acP i) z = y \u22a2 (if coeff y n = 0 then coeff x n else coeff y n) = coeff x n + coeff y n ** split_ifs with y0 ** case pos p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x hy : select (fun i => \u00acP i) z = y y0 : coeff y n = 0 \u22a2 coeff x n = coeff x n + coeff y n ** rw [y0, add_zero] ** case neg p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R P\u271d : \u2115 \u2192 Prop x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0 P : \u2115 \u2192 Prop := fun n => coeff y n = 0 this : DecidablePred P z : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n hx : select P z = x hy : select (fun i => \u00acP i) z = y y0 : \u00accoeff y n = 0 \u22a2 coeff y n = coeff x n + coeff y n ** rw [h n |>.resolve_right y0, zero_add] ** Qed", + "informal": "" + }, + { + "formal": "Int.even_or_odd' ** m n\u271d n : \u2124 \u22a2 \u2203 k, n = 2 * k \u2228 n = 2 * k + 1 ** simpa only [two_mul, exists_or, Odd, Even] using even_or_odd n ** Qed", + "informal": "" + }, + { + "formal": "Finset.subset_insert_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t\u271d u v : Finset \u03b1 a\u271d b a : \u03b1 s t : Finset \u03b1 \u22a2 s \u2286 insert a t \u2194 erase s a \u2286 t ** simp only [subset_iff, or_iff_not_imp_left, mem_erase, mem_insert, and_imp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t\u271d u v : Finset \u03b1 a\u271d b a : \u03b1 s t : Finset \u03b1 \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 ** exact forall_congr' fun x => forall_swap ** Qed", + "informal": "" + }, + { + "formal": "PMF.toOuterMeasure_caratheodory ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 \u22a2 OuterMeasure.caratheodory (toOuterMeasure p) = \u22a4 ** refine' eq_top_iff.2 <| le_trans (le_sInf fun x hx => _) (le_sum_caratheodory _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 x : MeasurableSpace \u03b1 hx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i) \u22a2 \u22a4 \u2264 x ** have \u27e8y, hy\u27e9 := hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 x : MeasurableSpace \u03b1 hx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i) y : \u03b1 hy : (fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)) y = x \u22a2 \u22a4 \u2264 x ** exact\n ((le_of_eq (dirac_caratheodory y).symm).trans (le_smul_caratheodory _ _)).trans (le_of_eq hy) ** Qed", + "informal": "" + }, + { + "formal": "FreeMonoid.hom_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 M : Type u_4 inst\u271d\u00b9 : Monoid M N : Type u_5 inst\u271d : Monoid N f g : FreeMonoid \u03b1 \u2192* M h : \u2200 (x : \u03b1), \u2191f (of x) = \u2191g (of x) l : FreeMonoid \u03b1 x : \u03b1 xs : FreeMonoid \u03b1 hxs : \u2191f xs = \u2191g xs \u22a2 \u2191f (of x * xs) = \u2191g (of x * xs) ** simp only [h, hxs, MonoidHom.map_mul] ** Qed", + "informal": "" + }, + { + "formal": "Turing.PartrecToTM2.unrev_ok ** q : \u039b' s : Option \u0393' S : K' \u2192 List \u0393' \u22a2 rev \u2260 main ** decide ** Qed", + "informal": "" + }, + { + "formal": "Finset.Equiv.prod_comp_finset ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : CommMonoid \u03b2 \u03b9' : Type u_2 inst\u271d : DecidableEq \u03b9 e : \u03b9 \u2243 \u03b9' f : \u03b9' \u2192 \u03b2 s' : Finset \u03b9' s : Finset \u03b9 h : s = image (\u2191e.symm) s' \u22a2 \u220f i' in s', f i' = \u220f i in s, f (\u2191e i) ** rw [h] ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : CommMonoid \u03b2 \u03b9' : Type u_2 inst\u271d : DecidableEq \u03b9 e : \u03b9 \u2243 \u03b9' f : \u03b9' \u2192 \u03b2 s' : Finset \u03b9' s : Finset \u03b9 h : s = image (\u2191e.symm) s' \u22a2 \u220f i' in s', f i' = \u220f i in image (\u2191e.symm) s', f (\u2191e i) ** 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 ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a\u271d : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : CommMonoid \u03b2 \u03b9' : Type u_2 inst\u271d : DecidableEq \u03b9 e : \u03b9 \u2243 \u03b9' f : \u03b9' \u2192 \u03b2 s' : Finset \u03b9' s : Finset \u03b9 h : s = image (\u2191e.symm) s' a : \u03b9 ha : a \u2208 image (\u2191e.symm) s' \u22a2 (fun i _hi => \u2191e i) a ha \u2208 s' ** rcases Finset.mem_image.mp ha with \u27e8i', hi', rfl\u27e9 ** case intro.intro \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : CommMonoid \u03b2 \u03b9' : Type u_2 inst\u271d : DecidableEq \u03b9 e : \u03b9 \u2243 \u03b9' f : \u03b9' \u2192 \u03b2 s' : Finset \u03b9' s : Finset \u03b9 h : s = image (\u2191e.symm) s' i' : \u03b9' hi' : i' \u2208 s' ha : \u2191e.symm i' \u2208 image (\u2191e.symm) s' \u22a2 (fun i _hi => \u2191e i) (\u2191e.symm i') ha \u2208 s' ** dsimp only ** case intro.intro \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : CommMonoid \u03b2 \u03b9' : Type u_2 inst\u271d : DecidableEq \u03b9 e : \u03b9 \u2243 \u03b9' f : \u03b9' \u2192 \u03b2 s' : Finset \u03b9' s : Finset \u03b9 h : s = image (\u2191e.symm) s' i' : \u03b9' hi' : i' \u2208 s' ha : \u2191e.symm i' \u2208 image (\u2191e.symm) s' \u22a2 \u2191e (\u2191e.symm i') \u2208 s' ** rwa [e.apply_symm_apply] ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a\u271d : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : CommMonoid \u03b2 \u03b9' : Type u_2 inst\u271d : DecidableEq \u03b9 e : \u03b9 \u2243 \u03b9' f : \u03b9' \u2192 \u03b2 s' : Finset \u03b9' s : Finset \u03b9 h : s = image (\u2191e.symm) s' a : \u03b9' _ha : a \u2208 s' \u22a2 f a = f (\u2191e ((fun i' _hi' => \u2191e.symm i') a _ha)) ** simp_rw [e.apply_symm_apply] ** Qed", + "informal": "" + }, + { + "formal": "NormedRing.inverse_add_nth_order ** R : Type u_1 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R\u02e3 n : \u2115 hzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0) \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) ** filter_upwards [inverse_add x, hzero.eventually (inverse_one_sub_nth_order n)] with t ht ht' ** case h R : Type u_1 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R\u02e3 n : \u2115 hzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0) t : R ht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9 ht' : 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) \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) ** rw [neg_mul, sub_neg_eq_add] at ht' ** case h R : Type u_1 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R\u02e3 n : \u2115 hzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0) t : R ht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9 ht' : 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) \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) ** conv_lhs => rw [ht, ht', add_mul, \u2190 neg_mul, mul_assoc] ** case h R : Type u_1 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R\u02e3 n : \u2115 hzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0) t : R ht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9 ht' : 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) \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) = (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (\u2191x + t) ** rw [ht] ** Qed", + "informal": "" + }, + { + "formal": "Nat.mul_div_le_mul_div_assoc ** m n k a b c : \u2115 hc0 : c = 0 \u22a2 a * (b / c) \u2264 a * b / c ** simp [hc0] ** m n k a b c : \u2115 hc0 : \u00acc = 0 \u22a2 a * (b / c) * c \u2264 a * b ** rw [mul_assoc] ** m n k a b c : \u2115 hc0 : \u00acc = 0 \u22a2 a * (b / c * c) \u2264 a * b ** exact Nat.mul_le_mul_left _ (Nat.div_mul_le_self _ _) ** Qed", + "informal": "" + }, + { + "formal": "isAdjointPair_toBilin ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 inst\u271d\u00b9 : Fintype n b : Basis n R\u2083 M\u2083 J J\u2083 A A' : Matrix n n R\u2083 inst\u271d : DecidableEq n \u22a2 BilinForm.IsAdjointPair (\u2191(toBilin b) J) (\u2191(toBilin b) J\u2083) (\u2191(toLin b b) A) (\u2191(toLin b b) A') \u2194 IsAdjointPair J J\u2083 A A' ** rw [BilinForm.isAdjointPair_iff_compLeft_eq_compRight] ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 inst\u271d\u00b9 : Fintype n b : Basis n R\u2083 M\u2083 J J\u2083 A A' : Matrix n n R\u2083 inst\u271d : DecidableEq n h : \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B' \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 IsAdjointPair J J\u2083 A A' ** rw [h, BilinForm.toMatrix_compLeft, BilinForm.toMatrix_compRight, LinearMap.toMatrix_toLin,\n LinearMap.toMatrix_toLin, BilinForm.toMatrix_toBilin, BilinForm.toMatrix_toBilin] ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 inst\u271d\u00b9 : Fintype n b : Basis n R\u2083 M\u2083 J J\u2083 A A' : Matrix n n R\u2083 inst\u271d : DecidableEq n h : \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B' \u22a2 A\u1d40 * J\u2083 = J * A' \u2194 IsAdjointPair J J\u2083 A A' ** rfl ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 inst\u271d\u00b9 : Fintype n b : Basis n R\u2083 M\u2083 J J\u2083 A A' : Matrix n n R\u2083 inst\u271d : DecidableEq n \u22a2 \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B' ** intro B B' ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B\u271d : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 inst\u271d\u00b9 : Fintype n b : Basis n R\u2083 M\u2083 J J\u2083 A A' : Matrix n n R\u2083 inst\u271d : DecidableEq n B B' : BilinForm R\u2083 M\u2083 \u22a2 B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B' ** constructor <;> intro h ** case mp R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B\u271d : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 inst\u271d\u00b9 : Fintype n b : Basis n R\u2083 M\u2083 J J\u2083 A A' : Matrix n n R\u2083 inst\u271d : DecidableEq n B B' : BilinForm R\u2083 M\u2083 h : B = B' \u22a2 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B' ** rw [h] ** case mpr R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B\u271d : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 inst\u271d\u00b9 : Fintype n b : Basis n R\u2083 M\u2083 J J\u2083 A A' : Matrix n n R\u2083 inst\u271d : DecidableEq n B B' : BilinForm R\u2083 M\u2083 h : \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B' \u22a2 B = B' ** exact (BilinForm.toMatrix b).injective h ** Qed", + "informal": "" + }, + { + "formal": "Rat.mul_add ** a b c : \u211a \u22a2 a * (b + c) = a * b + a * c ** rw [Rat.mul_comm, Rat.add_mul, Rat.mul_comm, Rat.mul_comm c a] ** Qed", + "informal": "" + }, + { + "formal": "SubfieldClass.rat_smul_mem ** K : Type u L : Type v M : Type w inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field M S : Type u_1 inst\u271d : SetLike S K h : SubfieldClass S K s : S a : \u211a x : { x // x \u2208 s } \u22a2 a \u2022 \u2191x \u2208 s ** simpa only [Rat.smul_def] using mul_mem (coe_rat_mem s a) x.prop ** Qed", + "informal": "" + }, + { + "formal": "Basis.equivFun_symm_apply ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2075 : Semiring R inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : AddCommMonoid M' inst\u271d\u00b9 : Module R M' inst\u271d : Fintype \u03b9 b : Basis \u03b9 R M x : \u03b9 \u2192 R \u22a2 \u2191(LinearEquiv.symm (equivFun b)) x = \u2211 i : \u03b9, x i \u2022 \u2191b i ** simp [Basis.equivFun, Finsupp.total_apply, Finsupp.sum_fintype, Finsupp.equivFunOnFinite] ** Qed", + "informal": "" + }, + { + "formal": "pow_three ** \u03b1 : Type u_1 M : Type u N : Type v G : Type w H : Type x A : Type y B : Type z R : Type u\u2081 S : Type u\u2082 inst\u271d\u00b9 : Monoid M inst\u271d : AddMonoid A a : M \u22a2 a ^ 3 = a * (a * a) ** rw [pow_succ, pow_two] ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.ker_map ** R : Type u S : Type u_1 \u03c3 : Type v M : Type w inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M f : R \u2192+* S \u22a2 RingHom.ker (map f) = Ideal.map C (RingHom.ker f) ** ext ** case h R : Type u S : Type u_1 \u03c3 : Type v M : Type w inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M f : R \u2192+* S x\u271d : MvPolynomial \u03c3 R \u22a2 x\u271d \u2208 RingHom.ker (map f) \u2194 x\u271d \u2208 Ideal.map C (RingHom.ker f) ** rw [MvPolynomial.mem_map_C_iff, RingHom.mem_ker, MvPolynomial.ext_iff] ** case h R : Type u S : Type u_1 \u03c3 : Type v M : Type w inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M f : R \u2192+* S x\u271d : MvPolynomial \u03c3 R \u22a2 (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) x\u271d) = coeff m 0) \u2194 \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m x\u271d \u2208 RingHom.ker f ** simp_rw [coeff_map, coeff_zero, RingHom.mem_ker] ** Qed", + "informal": "" + }, + { + "formal": "Ordnode.Valid'.dual ** \u03b1 : Type u_1 inst\u271d : Preorder \u03b1 l : Ordnode \u03b1 x : \u03b1 r : Ordnode \u03b1 o\u2081 : WithBot \u03b1 o\u2082 : WithTop \u03b1 ol : Bounded l o\u2081 \u2191x Or : Bounded r (\u2191x) o\u2082 sl : Sized l sr : Sized r b : BalancedSz (size l) (size r) bl : Balanced l br : Balanced r ol' : Bounded (Ordnode.dual l) (\u2191x) o\u2081 sl' : Sized (Ordnode.dual l) bl' : Balanced (Ordnode.dual l) or' : Bounded (Ordnode.dual r) o\u2082 \u2191x sr' : Sized (Ordnode.dual r) br' : Balanced (Ordnode.dual r) \u22a2 size l + size r + 1 = size (Ordnode.dual r) + size (Ordnode.dual l) + 1 ** simp [size_dual, add_comm] ** \u03b1 : Type u_1 inst\u271d : Preorder \u03b1 l : Ordnode \u03b1 x : \u03b1 r : Ordnode \u03b1 o\u2081 : WithBot \u03b1 o\u2082 : WithTop \u03b1 ol : Bounded l o\u2081 \u2191x Or : Bounded r (\u2191x) o\u2082 sl : Sized l sr : Sized r b : BalancedSz (size l) (size r) bl : Balanced l br : Balanced r ol' : Bounded (Ordnode.dual l) (\u2191x) o\u2081 sl' : Sized (Ordnode.dual l) bl' : Balanced (Ordnode.dual l) or' : Bounded (Ordnode.dual r) o\u2082 \u2191x sr' : Sized (Ordnode.dual r) br' : Balanced (Ordnode.dual r) \u22a2 BalancedSz (size (Ordnode.dual r)) (size (Ordnode.dual l)) ** rw [size_dual, size_dual] ** \u03b1 : Type u_1 inst\u271d : Preorder \u03b1 l : Ordnode \u03b1 x : \u03b1 r : Ordnode \u03b1 o\u2081 : WithBot \u03b1 o\u2082 : WithTop \u03b1 ol : Bounded l o\u2081 \u2191x Or : Bounded r (\u2191x) o\u2082 sl : Sized l sr : Sized r b : BalancedSz (size l) (size r) bl : Balanced l br : Balanced r ol' : Bounded (Ordnode.dual l) (\u2191x) o\u2081 sl' : Sized (Ordnode.dual l) bl' : Balanced (Ordnode.dual l) or' : Bounded (Ordnode.dual r) o\u2082 \u2191x sr' : Sized (Ordnode.dual r) br' : Balanced (Ordnode.dual r) \u22a2 BalancedSz (size r) (size l) ** exact b.symm ** Qed", + "informal": "" + }, + { + "formal": "Set.PartiallyWellOrderedOn.image_of_monotone_on ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 r : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop f : \u03b1 \u2192 \u03b2 s t : Set \u03b1 a : \u03b1 hs : PartiallyWellOrderedOn s r hf : \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) \u22a2 PartiallyWellOrderedOn (f '' s) r' ** intro g' hg' ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 r : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop f : \u03b1 \u2192 \u03b2 s t : Set \u03b1 a : \u03b1 hs : PartiallyWellOrderedOn s r hf : \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) g' : \u2115 \u2192 \u03b2 hg' : \u2200 (n : \u2115), g' n \u2208 f '' s \u22a2 \u2203 m n, m < n \u2227 r' (g' m) (g' n) ** choose g hgs heq using hg' ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 r : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop f : \u03b1 \u2192 \u03b2 s t : Set \u03b1 a : \u03b1 hs : PartiallyWellOrderedOn s r hf : \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) g' : \u2115 \u2192 \u03b2 g : \u2115 \u2192 \u03b1 hgs : \u2200 (n : \u2115), g n \u2208 s heq : \u2200 (n : \u2115), f (g n) = g' n \u22a2 \u2203 m n, m < n \u2227 r' (g' m) (g' n) ** obtain rfl : f \u2218 g = g' ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 r : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop f : \u03b1 \u2192 \u03b2 s t : Set \u03b1 a : \u03b1 hs : PartiallyWellOrderedOn s r hf : \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) g' : \u2115 \u2192 \u03b2 g : \u2115 \u2192 \u03b1 hgs : \u2200 (n : \u2115), g n \u2208 s heq : \u2200 (n : \u2115), f (g n) = g' n \u22a2 f \u2218 g = g' \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 r : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop f : \u03b1 \u2192 \u03b2 s t : Set \u03b1 a : \u03b1 hs : PartiallyWellOrderedOn s r hf : \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) g : \u2115 \u2192 \u03b1 hgs : \u2200 (n : \u2115), g n \u2208 s heq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n \u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n) ** exact funext heq ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 r : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop f : \u03b1 \u2192 \u03b2 s t : Set \u03b1 a : \u03b1 hs : PartiallyWellOrderedOn s r hf : \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) g : \u2115 \u2192 \u03b1 hgs : \u2200 (n : \u2115), g n \u2208 s heq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n \u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n) ** obtain \u27e8m, n, hlt, hmn\u27e9 := hs g hgs ** case intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 r : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop f : \u03b1 \u2192 \u03b2 s t : Set \u03b1 a : \u03b1 hs : PartiallyWellOrderedOn s r hf : \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) g : \u2115 \u2192 \u03b1 hgs : \u2200 (n : \u2115), g n \u2208 s heq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n m n : \u2115 hlt : m < n hmn : r (g m) (g n) \u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n) ** exact \u27e8m, n, hlt, hf _ (hgs m) _ (hgs n) hmn\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.single_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : Zero M a a' : \u03b1 b : M \u22a2 single a b = 0 \u2194 b = 0 ** simp [FunLike.ext_iff, single_eq_set_indicator] ** Qed", + "informal": "" + }, + { + "formal": "Submonoid.LocalizationMap.lift_eq ** M : Type u_1 inst\u271d\u00b2 : CommMonoid M S : Submonoid M N : Type u_2 inst\u271d\u00b9 : CommMonoid N P : Type u_3 inst\u271d : CommMonoid P f : LocalizationMap S N g : M \u2192* P hg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y) x : M \u22a2 \u2191(lift f hg) (\u2191(toMap f) x) = \u2191g x ** rw [lift_spec, \u2190 g.map_mul] ** M : Type u_1 inst\u271d\u00b2 : CommMonoid M S : Submonoid M N : Type u_2 inst\u271d\u00b9 : CommMonoid N P : Type u_3 inst\u271d : CommMonoid P f : LocalizationMap S N g : M \u2192* P hg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y) x : M \u22a2 \u2191g (sec f (\u2191(toMap f) x)).1 = \u2191g (\u2191(sec f (\u2191(toMap f) x)).2 * x) ** exact f.eq_of_eq hg (by rw [sec_spec', f.toMap.map_mul]) ** M : Type u_1 inst\u271d\u00b2 : CommMonoid M S : Submonoid M N : Type u_2 inst\u271d\u00b9 : CommMonoid N P : Type u_3 inst\u271d : CommMonoid P f : LocalizationMap S N g : M \u2192* P hg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y) x : M \u22a2 \u2191(toMap f) (sec f (\u2191(toMap f) x)).1 = \u2191(toMap f) (\u2191(sec f (\u2191(toMap f) x)).2 * x) ** rw [sec_spec', f.toMap.map_mul] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.AEEqFun.inf_le_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 inst\u271d\u00b2 : TopologicalSpace \u03b4 inst\u271d\u00b9 : SemilatticeInf \u03b2 inst\u271d : ContinuousInf \u03b2 f g : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 f \u2293 g \u2264 g ** rw [\u2190 coeFn_le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 inst\u271d\u00b2 : TopologicalSpace \u03b4 inst\u271d\u00b9 : SemilatticeInf \u03b2 inst\u271d : ContinuousInf \u03b2 f g : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 \u2191(f \u2293 g) \u2264\u1d50[\u03bc] \u2191g ** filter_upwards [coeFn_inf f g] with _ ha ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 inst\u271d\u00b2 : TopologicalSpace \u03b4 inst\u271d\u00b9 : SemilatticeInf \u03b2 inst\u271d : ContinuousInf \u03b2 f g : \u03b1 \u2192\u2098[\u03bc] \u03b2 a\u271d : \u03b1 ha : \u2191(f \u2293 g) a\u271d = \u2191f a\u271d \u2293 \u2191g a\u271d \u22a2 \u2191(f \u2293 g) a\u271d \u2264 \u2191g a\u271d ** rw [ha] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 inst\u271d\u00b2 : TopologicalSpace \u03b4 inst\u271d\u00b9 : SemilatticeInf \u03b2 inst\u271d : ContinuousInf \u03b2 f g : \u03b1 \u2192\u2098[\u03bc] \u03b2 a\u271d : \u03b1 ha : \u2191(f \u2293 g) a\u271d = \u2191f a\u271d \u2293 \u2191g a\u271d \u22a2 \u2191f a\u271d \u2293 \u2191g a\u271d \u2264 \u2191g a\u271d ** exact inf_le_right ** Qed", + "informal": "" + }, + { + "formal": "sdiff_triangle ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : GeneralizedCoheytingAlgebra \u03b1 a\u271d b\u271d c\u271d d a b c : \u03b1 \u22a2 a \\ c \u2264 a \\ b \u2294 b \\ c ** rw [sdiff_le_iff, sup_left_comm, \u2190 sdiff_le_iff] ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : GeneralizedCoheytingAlgebra \u03b1 a\u271d b\u271d c\u271d d a b c : \u03b1 \u22a2 a \\ (a \\ b) \u2264 c \u2294 b \\ c ** exact sdiff_sdiff_le.trans le_sup_sdiff ** Qed", + "informal": "" + }, + { + "formal": "Monotone.map_iSup_of_continuousAt ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 : CompleteLinearOrder \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : OrderTopology \u03b1 inst\u271d\u00b3 : CompleteLinearOrder \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : Nonempty \u03b3 \u03b9 : Sort u_1 f : \u03b1 \u2192 \u03b2 g : \u03b9 \u2192 \u03b1 Cf : ContinuousAt f (iSup g) Mf : Monotone f fbot : f \u22a5 = \u22a5 \u22a2 f (\u2a06 i, g i) = \u2a06 i, f (g i) ** rw [iSup, Mf.map_sSup_of_continuousAt Cf fbot, \u2190 range_comp, iSup] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 : CompleteLinearOrder \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : OrderTopology \u03b1 inst\u271d\u00b3 : CompleteLinearOrder \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : Nonempty \u03b3 \u03b9 : Sort u_1 f : \u03b1 \u2192 \u03b2 g : \u03b9 \u2192 \u03b1 Cf : ContinuousAt f (iSup g) Mf : Monotone f fbot : f \u22a5 = \u22a5 \u22a2 sSup (range (f \u2218 g)) = sSup (range fun i => f (g i)) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "isComplete_iff_ultrafilter ** \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 s : Set \u03b1 \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 ** refine' \u27e8fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => _\u27e9 ** \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 s : Set \u03b1 H : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x l : Filter \u03b1 hl : Cauchy l hls : l \u2264 \ud835\udcdf s \u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l ** haveI := hl.1 ** \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 s : Set \u03b1 H : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x l : Filter \u03b1 hl : Cauchy l hls : l \u2264 \ud835\udcdf s this : NeBot l \u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l ** rcases H (Ultrafilter.of l) hl.ultrafilter_of ((Ultrafilter.of_le l).trans hls) with \u27e8x, hxs, hxl\u27e9 ** case intro.intro \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 s : Set \u03b1 H : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x l : Filter \u03b1 hl : Cauchy l hls : l \u2264 \ud835\udcdf s this : NeBot l x : \u03b1 hxs : x \u2208 s hxl : \u2191(Ultrafilter.of l) \u2264 \ud835\udcdd x \u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l ** exact \u27e8x, hxs, (ClusterPt.of_le_nhds hxl).mono (Ultrafilter.of_le l)\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "eq_div_iff_mul_eq'' ** \u03b1 : Type u_1 \u03b2 : Type u_2 G : Type u_3 inst\u271d : CommGroup G a b c d : G \u22a2 a = b / c \u2194 c * a = b ** rw [eq_div_iff_mul_eq', mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "HasStrictDerivAt.const_smul ** \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F E : Type w inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c R : Type u_1 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F inst\u271d : ContinuousConstSMul R F c : R hf : HasStrictDerivAt f f' x \u22a2 HasStrictDerivAt (fun y => c \u2022 f y) (c \u2022 f') x ** simpa using (hf.rst.imnst_smul c).hasStrictDerivAt ** Qed", + "informal": "" + }, + { + "formal": "Filter.compl_mem_coprod\u1d62 ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 f f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i) s\u271d : (i : \u03b9) \u2192 Set (\u03b1 i) s : Set ((i : \u03b9) \u2192 \u03b1 i) \u22a2 s\u1d9c \u2208 Filter.coprod\u1d62 f \u2194 \u2200 (i : \u03b9), (eval i '' s)\u1d9c \u2208 f i ** simp only [Filter.coprod\u1d62, mem_iSup, compl_mem_comap] ** Qed", + "informal": "" + }, + { + "formal": "Nat.Primrec.add ** p : \u2115 \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 = unpaired (fun x x_1 => x + x_1) p ** simp ** p : \u2115 \u22a2 Nat.rec (unpair p).1 (fun y IH => succ IH) (unpair p).2 = (unpair p).1 + (unpair p).2 ** induction p.unpair.2 <;> simp [*, add_succ] ** Qed", + "informal": "" + }, + { + "formal": "ContDiffBump.one_lt_rOut_div_rIn ** E : Type u_1 X : Type u_2 c : E f : ContDiffBump c \u22a2 1 < f.rOut / f.rIn ** rw [one_lt_div f.rIn_pos] ** E : Type u_1 X : Type u_2 c : E f : ContDiffBump c \u22a2 f.rIn < f.rOut ** exact f.rIn_lt_rOut ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.exists_frequently_lt_of_liminf_ne_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e \u03b9 : Type u_4 l : Filter \u03b9 x : \u03b9 \u2192 \u211d hx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4 \u22a2 \u2203 R, \u2203\u1da0 (n : \u03b9) in l, x n < R ** by_contra h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e \u03b9 : Type u_4 l : Filter \u03b9 x : \u03b9 \u2192 \u211d hx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4 h : \u00ac\u2203 R, \u2203\u1da0 (n : \u03b9) in l, x n < R \u22a2 False ** simp_rw [not_exists, not_frequently, not_lt] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e \u03b9 : Type u_4 l : Filter \u03b9 x : \u03b9 \u2192 \u211d hx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4 h : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2 \u22a2 False ** refine hx (ENNReal.eq_top_of_forall_nnreal_le fun r => le_limsInf_of_le (by isBoundedDefault) ?_) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r\u271d p q : \u211d\u22650 x\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e \u03b9 : Type u_4 l : Filter \u03b9 x : \u03b9 \u2192 \u211d hx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4 h : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2 r : \u211d\u22650 \u22a2 \u2200\u1da0 (n : \u211d\u22650\u221e) in map (fun n => \u2191(\u2191Real.nnabs (x n))) l, \u2191r \u2264 n ** simp only [eventually_map, ENNReal.coe_le_coe] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r\u271d p q : \u211d\u22650 x\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e \u03b9 : Type u_4 l : Filter \u03b9 x : \u03b9 \u2192 \u211d hx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4 h : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2 r : \u211d\u22650 \u22a2 \u2200\u1da0 (a : \u03b9) in l, r \u2264 \u2191Real.nnabs (x a) ** filter_upwards [h r] with i hi using hi.trans (le_abs_self (x i)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r\u271d p q : \u211d\u22650 x\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e \u03b9 : Type u_4 l : Filter \u03b9 x : \u03b9 \u2192 \u211d hx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4 h : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2 r : \u211d\u22650 \u22a2 IsCobounded (fun x x_1 => x \u2265 x_1) (map (fun n => \u2191(\u2191Real.nnabs (x n))) l) ** isBoundedDefault ** Qed", + "informal": "" + }, + { + "formal": "isUnit_of_mul_isUnit_left ** \u03b1 : Type u M : Type u_1 N : Type u_2 inst\u271d : CommMonoid M x y : M hu : IsUnit (x * y) z : M hz : x * y * z = 1 \u22a2 x * (y * z) = 1 ** rwa [\u2190 mul_assoc] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.CoverDense.ext ** C : Type u_1 inst\u271d\u00b3 : Category.{u_5, u_1} C D : Type u_2 inst\u271d\u00b2 : Category.{u_6, u_2} D E : Type u_3 inst\u271d\u00b9 : Category.{?u.5707, u_3} E J : GrothendieckTopology C K : GrothendieckTopology D L : GrothendieckTopology E A : Type u_4 inst\u271d : Category.{?u.5759, u_4} A G : C \u2964 D H\u271d H : CoverDense K G \u2131 : SheafOfTypes K X : D s t : \u2131.val.obj (op X) h : \u2200 \u2983Y : C\u2984 (f : G.obj Y \u27f6 X), \u2131.val.map f.op s = \u2131.val.map f.op t \u22a2 s = t ** apply (\u2131.cond (Sieve.coverByImage G X) (H.is_cover X)).isSeparatedFor.ext ** C : Type u_1 inst\u271d\u00b3 : Category.{u_5, u_1} C D : Type u_2 inst\u271d\u00b2 : Category.{u_6, u_2} D E : Type u_3 inst\u271d\u00b9 : Category.{?u.5707, u_3} E J : GrothendieckTopology C K : GrothendieckTopology D L : GrothendieckTopology E A : Type u_4 inst\u271d : Category.{?u.5759, u_4} A G : C \u2964 D H\u271d H : CoverDense K G \u2131 : SheafOfTypes K X : D s t : \u2131.val.obj (op X) h : \u2200 \u2983Y : C\u2984 (f : G.obj Y \u27f6 X), \u2131.val.map f.op s = \u2131.val.map f.op t \u22a2 \u2200 \u2983Y : D\u2984 \u2983f : Y \u27f6 X\u2984, (Sieve.coverByImage G X).arrows f \u2192 \u2131.val.map f.op s = \u2131.val.map f.op t ** rintro Y _ \u27e8Z, f\u2081, f\u2082, \u27e8rfl\u27e9\u27e9 ** case intro.mk.refl C : Type u_1 inst\u271d\u00b3 : Category.{u_5, u_1} C D : Type u_2 inst\u271d\u00b2 : Category.{u_6, u_2} D E : Type u_3 inst\u271d\u00b9 : Category.{?u.5707, u_3} E J : GrothendieckTopology C K : GrothendieckTopology D L : GrothendieckTopology E A : Type u_4 inst\u271d : Category.{?u.5759, u_4} A G : C \u2964 D H\u271d H : CoverDense K G \u2131 : SheafOfTypes K X : D s t : \u2131.val.obj (op X) h : \u2200 \u2983Y : C\u2984 (f : G.obj Y \u27f6 X), \u2131.val.map f.op s = \u2131.val.map f.op t Y : D Z : C f\u2081 : Y \u27f6 G.obj Z f\u2082 : G.obj Z \u27f6 X \u22a2 \u2131.val.map (f\u2081 \u226b f\u2082).op s = \u2131.val.map (f\u2081 \u226b f\u2082).op t ** simp [h f\u2082] ** Qed", + "informal": "" + }, + { + "formal": "closedBall_div_singleton ** E : Type u_1 inst\u271d : SeminormedCommGroup E \u03b5 \u03b4 : \u211d s t : Set E x y : E \u22a2 closedBall x \u03b4 / {y} = closedBall (x / y) \u03b4 ** simp [div_eq_mul_inv] ** Qed", + "informal": "" + }, + { + "formal": "invOf_div ** \u03b1 : Type u inst\u271d\u00b3 : GroupWithZero \u03b1 a b : \u03b1 inst\u271d\u00b2 : Invertible a inst\u271d\u00b9 : Invertible b inst\u271d : Invertible (a / b) \u22a2 a / b * (b / a) = 1 ** simp [\u2190 mul_div_assoc] ** Qed", + "informal": "" + }, + { + "formal": "Nat.prod_factors_toFinset_of_squarefree ** n : \u2115 hn : Squarefree n \u22a2 \u220f p in List.toFinset (factors n), p = n ** rw [List.prod_toFinset _ hn.nodup_factors, List.map_id'', Nat.prod_factors hn.ne_zero] ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_map ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 s : Finset \u03b1 e : \u03b1 \u21aa \u03b3 f : \u03b3 \u2192 \u03b2 \u22a2 \u220f x in map e s, f x = \u220f x in s, f (\u2191e x) ** rw [Finset.prod, Finset.map_val, Multiset.map_map] ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 s : Finset \u03b1 e : \u03b1 \u21aa \u03b3 f : \u03b3 \u2192 \u03b2 \u22a2 Multiset.prod (Multiset.map ((fun x => f x) \u2218 \u2191e) s.val) = \u220f x in s, f (\u2191e x) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "tendsto_ceil_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : FloorRing \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : OrderClosedTopology \u03b1 n : \u2124 \u22a2 pure (IntCast.intCast (n + 1)) \u2264 \ud835\udcdd[Ici (\u2191n + 1)] (\u2191n + 1) ** rw [\u2190 @cast_one \u03b1, \u2190 cast_add] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : FloorRing \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : OrderClosedTopology \u03b1 n : \u2124 \u22a2 pure (IntCast.intCast (n + 1)) \u2264 \ud835\udcdd[Ici \u2191(n + 1)] \u2191(n + 1) ** exact pure_le_nhdsWithin le_rfl ** Qed", + "informal": "" + }, + { + "formal": "segment_eq_image_lineMap ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 inst\u271d\u2075 : OrderedRing \ud835\udd5c inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : AddCommGroup G inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y : E \u22a2 [x-[\ud835\udd5c]y] = \u2191(AffineMap.lineMap x y) '' Icc 0 1 ** convert segment_eq_image \ud835\udd5c x y using 2 ** case h.e'_3.h \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 \u03b9 : Type u_5 \u03c0 : \u03b9 \u2192 Type u_6 inst\u271d\u2075 : OrderedRing \ud835\udd5c inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : AddCommGroup G inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y : E a\u271d\u00b9 : \ud835\udd5c a\u271d : a\u271d\u00b9 \u2208 Icc 0 1 \u22a2 \u2191(AffineMap.lineMap x y) a\u271d\u00b9 = (1 - a\u271d\u00b9) \u2022 x + a\u271d\u00b9 \u2022 y ** exact AffineMap.lineMap_apply_module _ _ _ ** Qed", + "informal": "" + }, + { + "formal": "Set.ncard_add_ncard_compl ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d hsc : autoParam (Set.Finite s\u1d9c) _auto\u271d \u22a2 ncard s + ncard s\u1d9c = Nat.card \u03b1 ** rw [\u2190 ncard_univ, \u2190 ncard_union_eq (@disjoint_compl_right _ _ s) hs hsc, union_compl_self] ** Qed", + "informal": "" + }, + { + "formal": "Std.AssocList.find?_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : BEq \u03b1 a : \u03b1 l : AssocList \u03b1 \u03b2 \u22a2 find? a l = Option.map (fun x => x.snd) (List.find? (fun x => x.fst == a) (toList l)) ** simp [find?_eq_findEntry?] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Subsingleton.stronglyMeasurable' ** \u03b1\u271d : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 \u03b1 : Type u_5 \u03b2 : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : Subsingleton \u03b1 f : \u03b1 \u2192 \u03b2 x y : \u03b1 \u22a2 f x = f y ** rw [Subsingleton.elim x y] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.PresheafedSpace.ext ** C : Type u_1 inst\u271d : Category.{?u.77087, u_1} C X Y : PresheafedSpace C \u03b1 \u03b2 : X \u27f6 Y w : \u03b1.base = \u03b2.base \u22a2 (Opens.map \u03b1.base).op = (Opens.map \u03b2.base).op ** rw [w] ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_le_one' ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d\u00b9 : CommMonoid M inst\u271d : OrderedCommMonoid N f g : \u03b9 \u2192 N s t : Finset \u03b9 h : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 1 \u22a2 \u220f i in s, 1 = 1 ** rw [prod_const_one] ** Qed", + "informal": "" + }, + { + "formal": "Filter.frequently_iff ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f : Filter \u03b1 P : \u03b1 \u2192 Prop \u22a2 (\u2203\u1da0 (x : \u03b1) in f, P x) \u2194 \u2200 {U : Set \u03b1}, U \u2208 f \u2192 \u2203 x, x \u2208 U \u2227 P x ** simp only [frequently_iff_forall_eventually_exists_and, exists_prop, @and_comm (P _)] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f : Filter \u03b1 P : \u03b1 \u2192 Prop \u22a2 (\u2200 {q : \u03b1 \u2192 Prop}, (\u2200\u1da0 (x : \u03b1) in f, q x) \u2192 \u2203 x, q x \u2227 P x) \u2194 \u2200 {U : Set \u03b1}, U \u2208 f \u2192 \u2203 x, x \u2208 U \u2227 P x ** rfl ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top' ** C : Type u inst\u271d\u00b2 : Category.{v, u} C J : GrothendieckTopology C A : Type u' inst\u271d\u00b9 : Category.{v', u'} A inst\u271d : ConcreteCategory A F G : C\u1d52\u1d56 \u2964 Type w f : F \u27f6 G \u22a2 IsLocallySurjective J f \u2194 Subpresheaf.sheafify J (imagePresheaf f) = \u22a4 ** 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] ** C : Type u inst\u271d\u00b2 : Category.{v, u} C J : GrothendieckTopology C A : Type u' inst\u271d\u00b9 : Category.{v', u'} A inst\u271d : ConcreteCategory A F G : C\u1d52\u1d56 \u2964 Type w f : F \u27f6 G \u22a2 IsLocallySurjective J f \u2194 \u2200 (a : C\u1d52\u1d56) (x : G.obj a), x \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f)) a ** exact \u27e8fun H U => H (unop U), fun H U => H (op U)\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "strictConvexOn_pow ** n : \u2115 hn : 2 \u2264 n \u22a2 StrictConvexOn \u211d (Ici 0) fun x => x ^ n ** apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _) ** n : \u2115 hn : 2 \u2264 n \u22a2 StrictMonoOn (deriv fun x => x ^ n) (interior (Ici 0)) ** rw [deriv_pow', interior_Ici] ** n : \u2115 hn : 2 \u2264 n \u22a2 StrictMonoOn (fun x => \u2191n * x ^ (n - 1)) (Ioi 0) ** 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) ** Qed", + "informal": "" + }, + { + "formal": "AlternatingMap.curryLeft_compLinearMap ** R : Type u_1 inst\u271d\u00b9\u2079 : Semiring R M : Type u_2 inst\u271d\u00b9\u2078 : AddCommMonoid M inst\u271d\u00b9\u2077 : Module R M N : Type u_3 inst\u271d\u00b9\u2076 : AddCommMonoid N inst\u271d\u00b9\u2075 : Module R N P : Type u_4 inst\u271d\u00b9\u2074 : AddCommMonoid P inst\u271d\u00b9\u00b3 : Module R P M' : Type u_5 inst\u271d\u00b9\u00b2 : AddCommGroup M' inst\u271d\u00b9\u00b9 : Module R M' N' : Type u_6 inst\u271d\u00b9\u2070 : AddCommGroup N' inst\u271d\u2079 : Module R N' \u03b9 : Type u_7 \u03b9' : Type u_8 \u03b9'' : Type u_9 R' : Type u_10 M'' : Type u_11 M\u2082'' : Type u_12 N'' : Type u_13 N\u2082'' : Type u_14 inst\u271d\u2078 : CommSemiring R' inst\u271d\u2077 : AddCommMonoid M'' inst\u271d\u2076 : AddCommMonoid M\u2082'' inst\u271d\u2075 : AddCommMonoid N'' inst\u271d\u2074 : AddCommMonoid N\u2082'' inst\u271d\u00b3 : Module R' M'' inst\u271d\u00b2 : Module R' M\u2082'' inst\u271d\u00b9 : Module R' N'' inst\u271d : Module R' N\u2082'' n : \u2115 g : M\u2082'' \u2192\u2097[R'] M'' f : AlternatingMap R' M'' N'' (Fin (Nat.succ n)) m : M\u2082'' v : Fin n \u2192 M\u2082'' \u22a2 \u2200 (x : Fin (Nat.succ 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 ** refine' Fin.cases _ _ ** case refine'_1 R : Type u_1 inst\u271d\u00b9\u2079 : Semiring R M : Type u_2 inst\u271d\u00b9\u2078 : AddCommMonoid M inst\u271d\u00b9\u2077 : Module R M N : Type u_3 inst\u271d\u00b9\u2076 : AddCommMonoid N inst\u271d\u00b9\u2075 : Module R N P : Type u_4 inst\u271d\u00b9\u2074 : AddCommMonoid P inst\u271d\u00b9\u00b3 : Module R P M' : Type u_5 inst\u271d\u00b9\u00b2 : AddCommGroup M' inst\u271d\u00b9\u00b9 : Module R M' N' : Type u_6 inst\u271d\u00b9\u2070 : AddCommGroup N' inst\u271d\u2079 : Module R N' \u03b9 : Type u_7 \u03b9' : Type u_8 \u03b9'' : Type u_9 R' : Type u_10 M'' : Type u_11 M\u2082'' : Type u_12 N'' : Type u_13 N\u2082'' : Type u_14 inst\u271d\u2078 : CommSemiring R' inst\u271d\u2077 : AddCommMonoid M'' inst\u271d\u2076 : AddCommMonoid M\u2082'' inst\u271d\u2075 : AddCommMonoid N'' inst\u271d\u2074 : AddCommMonoid N\u2082'' inst\u271d\u00b3 : Module R' M'' inst\u271d\u00b2 : Module R' M\u2082'' inst\u271d\u00b9 : Module R' N'' inst\u271d : Module R' N\u2082'' n : \u2115 g : M\u2082'' \u2192\u2097[R'] M'' f : AlternatingMap R' M'' N'' (Fin (Nat.succ n)) m : M\u2082'' v : Fin n \u2192 M\u2082'' \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 ** rfl ** case refine'_2 R : Type u_1 inst\u271d\u00b9\u2079 : Semiring R M : Type u_2 inst\u271d\u00b9\u2078 : AddCommMonoid M inst\u271d\u00b9\u2077 : Module R M N : Type u_3 inst\u271d\u00b9\u2076 : AddCommMonoid N inst\u271d\u00b9\u2075 : Module R N P : Type u_4 inst\u271d\u00b9\u2074 : AddCommMonoid P inst\u271d\u00b9\u00b3 : Module R P M' : Type u_5 inst\u271d\u00b9\u00b2 : AddCommGroup M' inst\u271d\u00b9\u00b9 : Module R M' N' : Type u_6 inst\u271d\u00b9\u2070 : AddCommGroup N' inst\u271d\u2079 : Module R N' \u03b9 : Type u_7 \u03b9' : Type u_8 \u03b9'' : Type u_9 R' : Type u_10 M'' : Type u_11 M\u2082'' : Type u_12 N'' : Type u_13 N\u2082'' : Type u_14 inst\u271d\u2078 : CommSemiring R' inst\u271d\u2077 : AddCommMonoid M'' inst\u271d\u2076 : AddCommMonoid M\u2082'' inst\u271d\u2075 : AddCommMonoid N'' inst\u271d\u2074 : AddCommMonoid N\u2082'' inst\u271d\u00b3 : Module R' M'' inst\u271d\u00b2 : Module R' M\u2082'' inst\u271d\u00b9 : Module R' N'' inst\u271d : Module R' N\u2082'' n : \u2115 g : M\u2082'' \u2192\u2097[R'] M'' f : AlternatingMap R' M'' N'' (Fin (Nat.succ n)) m : M\u2082'' v : Fin n \u2192 M\u2082'' \u22a2 \u2200 (i : Fin n), \u2191((fun x => g) (Fin.succ i)) (Matrix.vecCons m v (Fin.succ i)) = Matrix.vecCons (\u2191g m) (fun i => \u2191((fun x => g) i) (v i)) (Fin.succ i) ** simp ** Qed", + "informal": "" + }, + { + "formal": "smul_lt_smul_of_neg ** k : Type u_1 M : Type u_2 N : Type u_3 inst\u271d\u00b3 : OrderedRing k inst\u271d\u00b2 : OrderedAddCommGroup M inst\u271d\u00b9 : Module k M inst\u271d : OrderedSMul k M a b : M c : k h : a < b hc : c < 0 \u22a2 c \u2022 b < c \u2022 a ** rw [\u2190 neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff] ** k : Type u_1 M : Type u_2 N : Type u_3 inst\u271d\u00b3 : OrderedRing k inst\u271d\u00b2 : OrderedAddCommGroup M inst\u271d\u00b9 : Module k M inst\u271d : OrderedSMul k M a b : M c : k h : a < b hc : c < 0 \u22a2 -c \u2022 a < -c \u2022 b ** exact smul_lt_smul_of_pos h (neg_pos_of_neg hc) ** Qed", + "informal": "" + }, + { + "formal": "convexOn_of_deriv2_nonneg ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F D : Set \u211d hD : Convex \u211d D f : \u211d \u2192 \u211d hf : ContinuousOn f D hf' : DifferentiableOn \u211d f (interior D) hf'' : DifferentiableOn \u211d (deriv f) (interior D) hf''_nonneg : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 \u2264 deriv^[2] f x \u22a2 DifferentiableOn \u211d (deriv f) (interior (interior D)) ** rwa [interior_interior] ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F D : Set \u211d hD : Convex \u211d D f : \u211d \u2192 \u211d hf : ContinuousOn f D hf' : DifferentiableOn \u211d f (interior D) hf'' : DifferentiableOn \u211d (deriv f) (interior D) hf''_nonneg : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 \u2264 deriv^[2] f x \u22a2 \u2200 (x : \u211d), x \u2208 interior (interior D) \u2192 0 \u2264 deriv (deriv f) x ** rwa [interior_interior] ** Qed", + "informal": "" + }, + { + "formal": "nhds_induced ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 f\u271d : \u03b1 \u2192 \u03b2 \u03b9 : Sort u_2 T : TopologicalSpace \u03b1 f : \u03b2 \u2192 \u03b1 a : \u03b2 \u22a2 \ud835\udcdd a = comap f (\ud835\udcdd (f a)) ** ext s ** case a \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 f\u271d : \u03b1 \u2192 \u03b2 \u03b9 : Sort u_2 T : TopologicalSpace \u03b1 f : \u03b2 \u2192 \u03b1 a : \u03b2 s : Set \u03b2 \u22a2 s \u2208 \ud835\udcdd a \u2194 s \u2208 comap f (\ud835\udcdd (f a)) ** rw [mem_nhds_induced, mem_comap] ** Qed", + "informal": "" + }, + { + "formal": "LinearLocallyFiniteOrder.le_succFn ** \u03b9 : Type u_1 inst\u271d : LinearOrder \u03b9 i : \u03b9 \u22a2 i \u2264 succFn i ** rw [le_isGLB_iff (succFn_spec i), mem_lowerBounds] ** \u03b9 : Type u_1 inst\u271d : LinearOrder \u03b9 i : \u03b9 \u22a2 \u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 i \u2264 x ** exact fun x hx \u21a6 le_of_lt hx ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 s : Set \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_2 m' : MeasurableSpace \u03b2 \u03bc' : Measure \u03b2 s' : Set \u03b2 g : \u03b1 \u2192 \u03b2 hg : MeasurePreserving g hf : PreErgodic f f' : \u03b2 \u2192 \u03b2 h_comm : g \u2218 f = f' \u2218 g \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 ** intro s hs\u2080 hs\u2081 ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_2 m' : MeasurableSpace \u03b2 \u03bc' : Measure \u03b2 s' : Set \u03b2 g : \u03b1 \u2192 \u03b2 hg : MeasurePreserving g hf : PreErgodic f f' : \u03b2 \u2192 \u03b2 h_comm : g \u2218 f = f' \u2218 g s : Set \u03b2 hs\u2080 : MeasurableSet s hs\u2081 : f' \u207b\u00b9' s = s \u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ ** replace hs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_2 m' : MeasurableSpace \u03b2 \u03bc' : Measure \u03b2 s' : Set \u03b2 g : \u03b1 \u2192 \u03b2 hg : MeasurePreserving g hf : PreErgodic f f' : \u03b2 \u2192 \u03b2 h_comm : g \u2218 f = f' \u2218 g s : Set \u03b2 hs\u2080 : MeasurableSet s hs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s \u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ ** cases' hf.ae_empty_or_univ (hg.measurable hs\u2080) hs\u2081 with hs\u2082 hs\u2082 <;> [left; right] ** case hs\u2081 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_2 m' : MeasurableSpace \u03b2 \u03bc' : Measure \u03b2 s' : Set \u03b2 g : \u03b1 \u2192 \u03b2 hg : MeasurePreserving g hf : PreErgodic f f' : \u03b2 \u2192 \u03b2 h_comm : g \u2218 f = f' \u2218 g s : Set \u03b2 hs\u2080 : MeasurableSet s hs\u2081 : f' \u207b\u00b9' s = s \u22a2 f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s ** rw [\u2190 preimage_comp, h_comm, preimage_comp, hs\u2081] ** case inl.h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_2 m' : MeasurableSpace \u03b2 \u03bc' : Measure \u03b2 s' : Set \u03b2 g : \u03b1 \u2192 \u03b2 hg : MeasurePreserving g hf : PreErgodic f f' : \u03b2 \u2192 \u03b2 h_comm : g \u2218 f = f' \u2218 g s : Set \u03b2 hs\u2080 : MeasurableSet s hs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s hs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] \u2205 \u22a2 s =\u1da0[ae \u03bc'] \u2205 ** simpa only [ae_eq_empty, hg.measure_preimage hs\u2080] using hs\u2082 ** case inr.h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_2 m' : MeasurableSpace \u03b2 \u03bc' : Measure \u03b2 s' : Set \u03b2 g : \u03b1 \u2192 \u03b2 hg : MeasurePreserving g hf : PreErgodic f f' : \u03b2 \u2192 \u03b2 h_comm : g \u2218 f = f' \u2218 g s : Set \u03b2 hs\u2080 : MeasurableSet s hs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s hs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] univ \u22a2 s =\u1da0[ae \u03bc'] univ ** simpa only [ae_eq_univ, \u2190 preimage_compl, hg.measure_preimage hs\u2080.compl] using hs\u2082 ** Qed", + "informal": "" + }, + { + "formal": "UniqueFactorizationMonoid.normalizedFactors_mul ** \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 \u22a2 normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y ** have h : (normalize : \u03b1 \u2192 \u03b1) = Associates.out \u2218 Associates.mk := by\n ext\n rw [Function.comp_apply, Associates.out_mk] ** \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk \u22a2 normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y ** 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] ** \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk \u22a2 Multiset.map Associates.out (Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y)) = Multiset.map Associates.out (Multiset.map Associates.mk (normalizedFactors (x * y))) ** refine' congr rfl _ ** \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk \u22a2 Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y) = Multiset.map Associates.mk (normalizedFactors (x * y)) ** apply Multiset.map_mk_eq_map_mk_of_rel ** case hst \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk \u22a2 Multiset.Rel Setoid.r (normalizedFactors x + normalizedFactors y) (normalizedFactors (x * y)) ** apply factors_unique ** \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 \u22a2 \u2191normalize = Associates.out \u2218 Associates.mk ** ext ** case h \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 x\u271d : \u03b1 \u22a2 \u2191normalize x\u271d = (Associates.out \u2218 Associates.mk) x\u271d ** rw [Function.comp_apply, Associates.out_mk] ** case hst.hf \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk \u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 normalizedFactors x + normalizedFactors y \u2192 Irreducible x_1 ** intro x hx ** case hst.hf \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x\u271d y : \u03b1 hx\u271d : x\u271d \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk x : \u03b1 hx : x \u2208 normalizedFactors x\u271d + normalizedFactors y \u22a2 Irreducible x ** rcases Multiset.mem_add.1 hx with (hx | hx) <;> exact irreducible_of_normalized_factor x hx ** case hst.hg \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk \u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 normalizedFactors (x * y) \u2192 Irreducible x_1 ** exact irreducible_of_normalized_factor ** case hst.h \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk \u22a2 Multiset.prod (normalizedFactors x + normalizedFactors y) ~\u1d64 Multiset.prod (normalizedFactors (x * y)) ** rw [Multiset.prod_add] ** case hst.h \u03b1 : Type u_1 inst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : NormalizationMonoid \u03b1 inst\u271d : UniqueFactorizationMonoid \u03b1 x y : \u03b1 hx : x \u2260 0 hy : y \u2260 0 h : \u2191normalize = Associates.out \u2218 Associates.mk \u22a2 Multiset.prod (normalizedFactors x) * Multiset.prod (normalizedFactors y) ~\u1d64 Multiset.prod (normalizedFactors (x * y)) ** exact\n ((normalizedFactors_prod hx).mul_mul (normalizedFactors_prod hy)).trans\n (normalizedFactors_prod (mul_ne_zero hx hy)).symm ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.aemeasurable_withDensity_ennreal_iff ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 AEMeasurable g \u2194 AEMeasurable fun x => \u2191(f x) * g x ** constructor ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 AEMeasurable g \u2192 AEMeasurable fun x => \u2191(f x) * g x ** rintro \u27e8g', g'meas, hg'\u27e9 ** case mp.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g' \u22a2 AEMeasurable fun x => \u2191(f x) * g x ** have A : MeasurableSet { x : \u03b1 | f x \u2260 0 } := (hf (measurableSet_singleton 0)).compl ** case mp.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g' A : MeasurableSet {x | f x \u2260 0} \u22a2 AEMeasurable fun x => \u2191(f x) * g x ** refine' \u27e8fun x => f x * g' x, hf.rst.ime_nnreal_ennreal.smul g'meas, _\u27e9 ** case mp.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g' A : MeasurableSet {x | f x \u2260 0} \u22a2 (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] fun x => \u2191(f x) * g' x ** apply ae_of_ae_restrict_of_ae_restrict_compl { x | f x \u2260 0 } ** case mp.intro.intro.ht \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g' A : MeasurableSet {x | f x \u2260 0} \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 ** rw [EventuallyEq, ae_withDensity_iff hf.rst.ime_nnreal_ennreal] at hg' ** case mp.intro.intro.ht \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x A : MeasurableSet {x | f x \u2260 0} \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 ** rw [ae_restrict_iff' A] ** case mp.intro.intro.ht \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x A : MeasurableSet {x | f x \u2260 0} \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 ** filter_upwards [hg'] ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x A : MeasurableSet {x | f x \u2260 0} \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 ** intro a ha h'a ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x A : MeasurableSet {x | f x \u2260 0} a : \u03b1 ha : \u2191(f a) \u2260 0 \u2192 g a = g' a h'a : f a \u2260 0 \u22a2 \u2191(f a) * g a = \u2191(f a) * g' a ** have : (f a : \u211d\u22650\u221e) \u2260 0 := by simpa only [Ne.def, coe_eq_zero] using h'a ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x A : MeasurableSet {x | f x \u2260 0} a : \u03b1 ha : \u2191(f a) \u2260 0 \u2192 g a = g' a h'a : f a \u2260 0 this : \u2191(f a) \u2260 0 \u22a2 \u2191(f a) * g a = \u2191(f a) * g' a ** rw [ha this] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x A : MeasurableSet {x | f x \u2260 0} a : \u03b1 ha : \u2191(f a) \u2260 0 \u2192 g a = g' a h'a : f a \u2260 0 \u22a2 \u2191(f a) \u2260 0 ** simpa only [Ne.def, coe_eq_zero] using h'a ** case mp.intro.intro.htc \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g' A : MeasurableSet {x | f x \u2260 0} \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 ** filter_upwards [ae_restrict_mem A.compl] ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g' A : MeasurableSet {x | f x \u2260 0} \u22a2 \u2200 (a : \u03b1), a \u2208 {x | f x \u2260 0}\u1d9c \u2192 \u2191(f a) * g a = \u2191(f a) * g' a ** intro x hx ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g' A : MeasurableSet {x | f x \u2260 0} x : \u03b1 hx : x \u2208 {x | f x \u2260 0}\u1d9c \u22a2 \u2191(f x) * g x = \u2191(f x) * g' x ** simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g' A : MeasurableSet {x | f x \u2260 0} x : \u03b1 hx : f x = 0 \u22a2 \u2191(f x) * g x = \u2191(f x) * g' x ** simp [hx] ** case mpr \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 (AEMeasurable fun x => \u2191(f x) * g x) \u2192 AEMeasurable g ** rintro \u27e8g', g'meas, hg'\u27e9 ** case mpr.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g' \u22a2 AEMeasurable g ** refine' \u27e8fun x => ((f x)\u207b\u00b9 : \u211d\u22650\u221e) * g' x, hf.rst.ime_nnreal_ennreal.inv.smul g'meas, _\u27e9 ** case mpr.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g' \u22a2 g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] fun x => (\u2191(f x))\u207b\u00b9 * g' x ** rw [EventuallyEq, ae_withDensity_iff hf.rst.ime_nnreal_ennreal] ** case mpr.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g' \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = (\u2191(f x))\u207b\u00b9 * g' x ** filter_upwards [hg'] ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g' \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 ** intro x hx h'x ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650 hf : Measurable f g g' : \u03b1 \u2192 \u211d\u22650\u221e g'meas : Measurable g' hg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g' x : \u03b1 hx : \u2191(f x) * g x = g' x h'x : \u2191(f x) \u2260 0 \u22a2 g x = (\u2191(f x))\u207b\u00b9 * g' x ** rw [\u2190 hx, \u2190 mul_assoc, ENNReal.inv_mul_cancel h'x ENNReal.coe_ne_top, one_mul] ** Qed", + "informal": "" + }, + { + "formal": "strictConcaveOn_cos_Icc ** \u22a2 StrictConcaveOn \u211d (Icc (-(\u03c0 / 2)) (\u03c0 / 2)) cos ** apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_cos fun x hx => ?_ ** x : \u211d hx : x \u2208 interior (Icc (-(\u03c0 / 2)) (\u03c0 / 2)) \u22a2 deriv^[2] cos x < 0 ** rw [interior_Icc] at hx ** x : \u211d hx : x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2) \u22a2 deriv^[2] cos x < 0 ** simp [cos_pos_of_mem_Ioo hx] ** Qed", + "informal": "" + }, + { + "formal": "KaehlerDifferential.derivationQuotKerTotal_lift_comp_total ** R : Type u S : Type v inst\u271d\u2076 : CommRing R inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S M : Type u_1 inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower R S M \u22a2 LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S)) (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S)) = Submodule.mkQ (kerTotal R S) ** apply Finsupp.lhom_ext ** case h R : Type u S : Type v inst\u271d\u2076 : CommRing R inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S M : Type u_1 inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower R S M \u22a2 \u2200 (a b : S), (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S)) (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S))) fun\u2080 | a => b) = b\ud835\udda3a ** intro a b ** case h R : Type u S : Type v inst\u271d\u2076 : CommRing R inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S M : Type u_1 inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower R S M a b : S \u22a2 (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S)) (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S))) fun\u2080 | a => b) = b\ud835\udda3a ** conv_rhs => rw [\u2190 Finsupp.smul_single_one a b, LinearMap.map_smul] ** case h R : Type u S : Type v inst\u271d\u2076 : CommRing R inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S M : Type u_1 inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower R S M a b : S \u22a2 (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S)) (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S))) fun\u2080 | a => b) = b \u2022 1\ud835\udda3a ** simp [KaehlerDifferential.derivationQuotKerTotal_apply] ** Qed", + "informal": "" + }, + { + "formal": "Cubic.natDegree_of_c_eq_zero ** R : Type u_1 S : Type u_2 F : Type u_3 K : Type u_4 P Q : Cubic R a b c d a' b' c' d' : R inst\u271d : Semiring R ha : P.a = 0 hb : P.b = 0 hc : P.c = 0 \u22a2 natDegree (toPoly P) = 0 ** rw [of_c_eq_zero ha hb hc, natDegree_C] ** Qed", + "informal": "" + }, + { + "formal": "WittVector.mem_ker_truncate ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R x : \ud835\udd4e R \u22a2 x \u2208 RingHom.ker (truncate n) \u2194 \u2200 (i : \u2115), i < n \u2192 coeff x i = 0 ** simp only [RingHom.mem_ker, truncate, truncateFun, RingHom.coe_mk, TruncatedWittVector.ext_iff,\n TruncatedWittVector.coeff_mk, coeff_zero] ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R x : \ud835\udd4e R \u22a2 (\u2200 (i : Fin n), TruncatedWittVector.coeff i (\u2191{ toOneHom := { toFun := truncateFun n, map_one' := (_ : truncateFun n 1 = 1) }, map_mul' := (_ : \u2200 (x y : \ud835\udd4e R), truncateFun n (x * y) = truncateFun n x * truncateFun n y) } x) = 0) \u2194 \u2200 (i : \u2115), i < n \u2192 coeff x i = 0 ** exact Fin.forall_iff ** Qed", + "informal": "" + }, + { + "formal": "Multiset.countP_filter ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 p : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p q : \u03b1 \u2192 Prop inst\u271d : DecidablePred q s : Multiset \u03b1 \u22a2 countP p (filter q s) = countP (fun a => p a \u2227 q a) s ** simp [countP_eq_card_filter] ** Qed", + "informal": "" + }, + { + "formal": "Int.clog_natCast ** R : Type u_1 inst\u271d\u00b9 : LinearOrderedSemifield R inst\u271d : FloorSemiring R b n : \u2115 \u22a2 clog b \u2191n = \u2191(Nat.clog b n) ** cases' n with n ** case zero R : Type u_1 inst\u271d\u00b9 : LinearOrderedSemifield R inst\u271d : FloorSemiring R b : \u2115 \u22a2 clog b \u2191Nat.zero = \u2191(Nat.clog b Nat.zero) ** simp [clog_of_right_le_one] ** case succ R : Type u_1 inst\u271d\u00b9 : LinearOrderedSemifield R inst\u271d : FloorSemiring R b n : \u2115 \u22a2 clog b \u2191(Nat.succ n) = \u2191(Nat.clog b (Nat.succ n)) ** rw [clog_of_one_le_right, (Nat.ceil_eq_iff (Nat.succ_ne_zero n)).mpr] <;> simp ** Qed", + "informal": "" + }, + { + "formal": "IsPGroup.to_sup_of_normal_right ** p : \u2115 G : Type u_1 inst\u271d\u00b9 : Group G H K : Subgroup G hH : IsPGroup p { x // x \u2208 H } hK : IsPGroup p { x // x \u2208 K } inst\u271d : Subgroup.Normal K \u22a2 IsPGroup p { x // x \u2208 H \u2294 K } ** rw [\u2190 QuotientGroup.ker_mk' K, \u2190 Subgroup.comap_map_eq] ** p : \u2115 G : Type u_1 inst\u271d\u00b9 : Group G H K : Subgroup G hH : IsPGroup p { x // x \u2208 H } hK : IsPGroup p { x // x \u2208 K } inst\u271d : Subgroup.Normal K \u22a2 IsPGroup p { x // x \u2208 Subgroup.comap (QuotientGroup.mk' K) (Subgroup.map (QuotientGroup.mk' K) H) } ** apply (hH.map (QuotientGroup.mk' K)).comap_of_ker_isPGroup ** case h\u03d5 p : \u2115 G : Type u_1 inst\u271d\u00b9 : Group G H K : Subgroup G hH : IsPGroup p { x // x \u2208 H } hK : IsPGroup p { x // x \u2208 K } inst\u271d : Subgroup.Normal K \u22a2 IsPGroup p { x // x \u2208 MonoidHom.ker (QuotientGroup.mk' K) } ** rwa [QuotientGroup.ker_mk'] ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.degreeOf_X ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d\u00b9 : DecidableEq \u03c3 i j : \u03c3 inst\u271d : Nontrivial R \u22a2 degreeOf i (X j) = if i = j then 1 else 0 ** by_cases c : i = j ** case neg R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d\u00b9 : DecidableEq \u03c3 i j : \u03c3 inst\u271d : Nontrivial R c : \u00aci = j \u22a2 degreeOf i (X j) = if i = j then 1 else 0 ** simp [c, if_false, degreeOf_def, degrees_X] ** case pos R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d\u00b9 : DecidableEq \u03c3 i j : \u03c3 inst\u271d : Nontrivial R c : i = j \u22a2 degreeOf i (X j) = if i = j then 1 else 0 ** simp only [c, if_true, eq_self_iff_true, degreeOf_def, degrees_X, Multiset.count_singleton] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_sub_eq_neg_right_of_lt ** R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Ring R p q : R[X] df : natDegree p < n \u22a2 coeff (p - q) n = -coeff q n ** rwa [sub_eq_add_neg, coeff_add_eq_right_of_lt, coeff_neg] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.ennrealToMeasure ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2075 : AddCommMonoid L inst\u271d\u2074 : TopologicalSpace L inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e \u22a2 (\u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0) \u2194 v \u226a\u1d65 \u03bc ** constructor <;> intro h ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2075 : AddCommMonoid L inst\u271d\u2074 : TopologicalSpace L inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0 \u22a2 v \u226a\u1d65 \u03bc ** refine' mk fun s hmeas hs => h _ ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2075 : AddCommMonoid L inst\u271d\u2074 : TopologicalSpace L inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0 s : Set \u03b1 hmeas : MeasurableSet s hs : \u2191\u03bc s = 0 \u22a2 \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 ** rw [\u2190 hs, ennrealToMeasure_apply hmeas] ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2075 : AddCommMonoid L inst\u271d\u2074 : TopologicalSpace L inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : v \u226a\u1d65 \u03bc \u22a2 \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0 ** intro s hs ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2075 : AddCommMonoid L inst\u271d\u2074 : TopologicalSpace L inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : v \u226a\u1d65 \u03bc s : Set \u03b1 hs : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u22a2 \u2191v s = 0 ** by_cases hmeas : MeasurableSet s ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2075 : AddCommMonoid L inst\u271d\u2074 : TopologicalSpace L inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : v \u226a\u1d65 \u03bc s : Set \u03b1 hs : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 hmeas : MeasurableSet s \u22a2 \u2191v s = 0 ** rw [ennrealToMeasure_apply hmeas] at hs ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2075 : AddCommMonoid L inst\u271d\u2074 : TopologicalSpace L inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : v \u226a\u1d65 \u03bc s : Set \u03b1 hs : \u2191\u03bc s = 0 hmeas : MeasurableSet s \u22a2 \u2191v s = 0 ** exact h hs ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2075 : AddCommMonoid L inst\u271d\u2074 : TopologicalSpace L inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid N inst\u271d : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : v \u226a\u1d65 \u03bc s : Set \u03b1 hs : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 hmeas : \u00acMeasurableSet s \u22a2 \u2191v s = 0 ** exact not_measurable v hmeas ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.pullback_diagonal_map_snd_fst_fst ** C : Type u_1 inst\u271d\u00b9 : Category.{?u.9398, u_1} C X Y Z : C inst\u271d : HasPullbacks C U V\u2081 V\u2082 : C f : X \u27f6 Y i : U \u27f6 Y i\u2081 : V\u2081 \u27f6 pullback f i i\u2082 : V\u2082 \u27f6 pullback f i \u22a2 (i\u2081 \u226b snd) \u226b i = (i\u2081 \u226b fst) \u226b f ** simp [condition] ** C : Type u_1 inst\u271d\u00b9 : Category.{?u.9398, u_1} C X Y Z : C inst\u271d : HasPullbacks C U V\u2081 V\u2082 : C f : X \u27f6 Y i : U \u27f6 Y i\u2081 : V\u2081 \u27f6 pullback f i i\u2082 : V\u2082 \u27f6 pullback f i \u22a2 (i\u2082 \u226b snd) \u226b i = (i\u2082 \u226b fst) \u226b f ** simp [condition] ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C X Y Z : C inst\u271d : HasPullbacks C U V\u2081 V\u2082 : C f : X \u27f6 Y i : U \u27f6 Y i\u2081 : V\u2081 \u27f6 pullback f i i\u2082 : V\u2082 \u27f6 pullback f i \u22a2 snd \u226b fst \u226b i\u2081 \u226b fst = fst ** conv_rhs => rw [\u2190 Category.comp_id pullback.fst] ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C X Y Z : C inst\u271d : HasPullbacks C U V\u2081 V\u2082 : C f : X \u27f6 Y i : U \u27f6 Y i\u2081 : V\u2081 \u27f6 pullback f i i\u2082 : V\u2082 \u27f6 pullback f i \u22a2 snd \u226b fst \u226b i\u2081 \u226b fst = fst \u226b \ud835\udfd9 X ** rw [\u2190 diagonal_fst f, pullback.condition_assoc, pullback.lift_fst] ** Qed", + "informal": "" + }, + { + "formal": "Complex.sinh_two_mul ** x y : \u2102 \u22a2 sinh (2 * x) = 2 * sinh x * cosh x ** rw [two_mul, sinh_add] ** x y : \u2102 \u22a2 sinh x * cosh x + cosh x * sinh x = 2 * sinh x * cosh x ** ring ** Qed", + "informal": "" + }, + { + "formal": "IsLocallyConstant.range_finite ** X : Type u_1 Y : Type u_2 Z : Type u_3 \u03b1 : Type u_4 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : CompactSpace X f : X \u2192 Y hf : IsLocallyConstant f \u22a2 Set.Finite (range f) ** letI : TopologicalSpace Y := \u22a5 ** X : Type u_1 Y : Type u_2 Z : Type u_3 \u03b1 : Type u_4 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : CompactSpace X f : X \u2192 Y hf : IsLocallyConstant f this : TopologicalSpace Y := \u22a5 \u22a2 Set.Finite (range f) ** haveI := discreteTopology_bot Y ** X : Type u_1 Y : Type u_2 Z : Type u_3 \u03b1 : Type u_4 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : CompactSpace X f : X \u2192 Y hf : IsLocallyConstant f this\u271d : TopologicalSpace Y := \u22a5 this : DiscreteTopology Y \u22a2 Set.Finite (range f) ** exact (isCompact_range hf.rst.imntinuous).finite_of_discrete ** Qed", + "informal": "" + }, + { + "formal": "Stream'.mem_cycle ** \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w a : \u03b1 l : List \u03b1 h : l \u2260 [] ainl : a \u2208 l \u22a2 a \u2208 cycle l h ** rw [cycle_eq] ** \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w a : \u03b1 l : List \u03b1 h : l \u2260 [] ainl : a \u2208 l \u22a2 a \u2208 l ++\u209b cycle l h ** exact mem_append_stream_left _ ainl ** Qed", + "informal": "" + }, + { + "formal": "Set.Intersecting.disjoint_map_compl ** \u03b1 : Type u_1 inst\u271d : BooleanAlgebra \u03b1 s : Finset \u03b1 hs : Intersecting \u2191s \u22a2 Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s) ** rw [Finset.disjoint_left] ** \u03b1 : Type u_1 inst\u271d : BooleanAlgebra \u03b1 s : Finset \u03b1 hs : Intersecting \u2191s \u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 \u00aca \u2208 map { toFun := compl, inj' := (_ : Function.Injective compl) } s ** rintro x hx hxc ** \u03b1 : Type u_1 inst\u271d : BooleanAlgebra \u03b1 s : Finset \u03b1 hs : Intersecting \u2191s x : \u03b1 hx : x \u2208 s hxc : x \u2208 map { toFun := compl, inj' := (_ : Function.Injective compl) } s \u22a2 False ** obtain \u27e8x, hx', rfl\u27e9 := mem_map.mp hxc ** case intro.intro \u03b1 : Type u_1 inst\u271d : BooleanAlgebra \u03b1 s : Finset \u03b1 hs : Intersecting \u2191s x : \u03b1 hx' : x \u2208 s hx : \u2191{ toFun := compl, inj' := (_ : Function.Injective compl) } x \u2208 s hxc : \u2191{ toFun := compl, inj' := (_ : Function.Injective compl) } x \u2208 map { toFun := compl, inj' := (_ : Function.Injective compl) } s \u22a2 False ** exact hs.not_compl_mem hx' hx ** Qed", + "informal": "" + }, + { + "formal": "SlimCheck.InjectiveFunction.applyId_injective ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys \u22a2 Injective (applyId (List.zip xs ys)) ** intro x y h ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y \u22a2 x = y ** by_cases hx : x \u2208 xs <;> by_cases hy : y \u2208 xs ** case pos \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : x \u2208 xs hy : y \u2208 xs \u22a2 x = y ** rw [List.mem_iff_get?] at hx hy ** case pos \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : \u2203 n, List.get? xs n = some x hy : \u2203 n, List.get? xs n = some y \u22a2 x = y ** cases' hx with i hx ** case pos.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hy : \u2203 n, List.get? xs n = some y i : \u2115 hx : List.get? xs i = some x \u22a2 x = y ** cases' hy with j hy ** case pos.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y i : \u2115 hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y \u22a2 x = y ** suffices some x = some y by injection this ** case pos.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y i : \u2115 hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y \u22a2 some x = some y ** have h\u2082 := h\u2081.length_eq ** case pos.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y i : \u2115 hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 some x = some y ** rw [List.applyId_zip_eq h\u2080 h\u2082 _ _ _ hx] at h ** case pos.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 some x = some y ** rw [\u2190 hx, \u2190 hy] ** case pos.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 List.get? xs i = List.get? xs j ** congr ** case pos.intro.intro.e_a \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 i = j ** apply List.get?_injective _ (h\u2081.nodup_iff.1 h\u2080) ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y i : \u2115 hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y this : some x = some y \u22a2 x = y ** injection this ** case pos.intro.intro.e_a \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 List.get? ys i = List.get? ys j ** symm ** case pos.intro.intro.e_a \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 List.get? ys j = List.get? ys i ** rw [h] ** case pos.intro.intro.e_a \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 List.get? ys j = some (applyId (List.zip xs ys) y) ** rw [\u2190 List.applyId_zip_eq] <;> assumption ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 i < List.length ys ** rw [\u2190 h\u2081.length_eq] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : List.get? xs i = some x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 i < List.length xs ** rw [List.get?_eq_some] at hx ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) hx : \u2203 h, List.get xs { val := i, isLt := h } = x j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys \u22a2 i < List.length xs ** cases' hx with hx hx' ** case intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 i : \u2115 h : List.get? ys i = some (applyId (List.zip xs ys) y) j : \u2115 hy : List.get? xs j = some y h\u2082 : List.length xs = List.length ys hx : i < List.length xs hx' : List.get xs { val := i, isLt := hx } = x \u22a2 i < List.length xs ** exact hx ** case neg \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : x \u2208 xs hy : \u00acy \u2208 xs \u22a2 x = y ** rw [\u2190 applyId_mem_iff h\u2080 h\u2081] at hx hy ** case neg \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : applyId (List.zip xs ys) x \u2208 ys hy : \u00acapplyId (List.zip xs ys) y \u2208 ys \u22a2 x = y ** rw [h] at hx ** case neg \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : applyId (List.zip xs ys) y \u2208 ys hy : \u00acapplyId (List.zip xs ys) y \u2208 ys \u22a2 x = y ** contradiction ** case pos \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : \u00acx \u2208 xs hy : y \u2208 xs \u22a2 x = y ** rw [\u2190 applyId_mem_iff h\u2080 h\u2081] at hx hy ** case pos \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : \u00acapplyId (List.zip xs ys) x \u2208 ys hy : applyId (List.zip xs ys) y \u2208 ys \u22a2 x = y ** rw [h] at hx ** case pos \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : \u00acapplyId (List.zip xs ys) y \u2208 ys hy : applyId (List.zip xs ys) y \u2208 ys \u22a2 x = y ** contradiction ** case neg \u03b1 : Type u \u03b2 : Type v \u03b3 : Sort w inst\u271d : DecidableEq \u03b1 xs ys : List \u03b1 h\u2080 : List.Nodup xs h\u2081 : xs ~ ys x y : \u03b1 h : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y hx : \u00acx \u2208 xs hy : \u00acy \u2208 xs \u22a2 x = y ** rwa [List.applyId_eq_self, List.applyId_eq_self] at h <;> assumption ** Qed", + "informal": "" + }, + { + "formal": "RingQuot.neg_quot ** R\u271d : Type uR inst\u271d\u2074 : Semiring R\u271d S : Type uS inst\u271d\u00b3 : CommSemiring S T : Type uT A : Type uA inst\u271d\u00b2 : Semiring A inst\u271d\u00b9 : Algebra S A r\u271d : R\u271d \u2192 R\u271d \u2192 Prop R : Type uR inst\u271d : Ring R r : R \u2192 R \u2192 Prop a : R \u22a2 -{ toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) } ** show neg r _ = _ ** R\u271d : Type uR inst\u271d\u2074 : Semiring R\u271d S : Type uS inst\u271d\u00b3 : CommSemiring S T : Type uT A : Type uA inst\u271d\u00b2 : Semiring A inst\u271d\u00b9 : Algebra S A r\u271d : R\u271d \u2192 R\u271d \u2192 Prop R : Type uR inst\u271d : Ring R r : R \u2192 R \u2192 Prop a : R \u22a2 RingQuot.neg r { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) } ** rw [neg_def] ** R\u271d : Type uR inst\u271d\u2074 : Semiring R\u271d S : Type uS inst\u271d\u00b3 : CommSemiring S T : Type uT A : Type uA inst\u271d\u00b2 : Semiring A inst\u271d\u00b9 : Algebra S A r\u271d : R\u271d \u2192 R\u271d \u2192 Prop R : Type uR inst\u271d : Ring R r : R \u2192 R \u2192 Prop a : R \u22a2 (match { toQuot := Quot.mk (Rel r) a } with | { toQuot := a } => { toQuot := Quot.map (fun a => -a) (_ : \u2200 \u2983a b : R\u2984, Rel r a b \u2192 Rel r (-a) (-b)) a }) = { toQuot := Quot.mk (Rel r) (-a) } ** rfl ** Qed", + "informal": "" + }, + { + "formal": "intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hab : a \u2264 b hcont : ContinuousOn f (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x f'int : IntervalIntegrable f' volume a b \u22a2 \u222b (y : \u211d) in a..b, f' y = f b - f a ** refine' (NormedSpace.eq_iff_forall_dual_eq \u211d).2 fun g => _ ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hab : a \u2264 b hcont : ContinuousOn f (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x f'int : IntervalIntegrable f' volume a b g : NormedSpace.Dual \u211d E \u22a2 \u2191g (\u222b (y : \u211d) in a..b, f' y) = \u2191g (f b - f a) ** rw [\u2190 g.intervalIntegral_comp_comm f'int, g.map_sub] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hab : a \u2264 b hcont : ContinuousOn f (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x f'int : IntervalIntegrable f' volume a b g : NormedSpace.Dual \u211d E \u22a2 \u222b (x : \u211d) in a..b, \u2191g (f' x) = \u2191g (f b) - \u2191g (f a) ** 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)) ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.univ_le_of_forall_fin_meas_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : Set \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C h_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) \u22a2 f univ \u2264 C ** let S := @spanningSets _ m (\u03bc.trim hm) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : Set \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C h_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) S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) \u22a2 f univ \u2264 C ** have hS_mono : Monotone S := @monotone_spanningSets _ m (\u03bc.trim hm) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : Set \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C h_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) S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_mono : Monotone S \u22a2 f univ \u2264 C ** have hS_meas : \u2200 n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (\u03bc.trim hm) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : Set \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C h_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) S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_mono : Monotone S hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) \u22a2 f univ \u2264 C ** rw [\u2190 @iUnion_spanningSets _ m (\u03bc.trim hm)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : Set \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C h_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) S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_mono : Monotone S hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) \u22a2 f (\u22c3 i, spanningSets (Measure.trim \u03bc hm) i) \u2264 C ** refine' (h_F_lim S hS_meas hS_mono).trans _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : Set \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C h_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) S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_mono : Monotone S hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) \u22a2 \u2a06 n, f (S n) \u2264 C ** refine' iSup_le fun n => hf (S n) (hS_meas n) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : Set \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C h_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) S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_mono : Monotone S hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) n : \u2115 \u22a2 \u2191\u2191\u03bc (S n) \u2260 \u22a4 ** exact ((le_trim hm).trans_lt (@measure_spanningSets_lt_top _ m (\u03bc.trim hm) _ n)).ne ** Qed", + "informal": "" + }, + { + "formal": "sub_pow_char_pow_of_commute ** R : Type u_1 inst\u271d\u00b2 : Ring R p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP R p n : \u2115 x y : R h : Commute x y \u22a2 (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n ** 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 ** case zero R : Type u_1 inst\u271d\u00b2 : Ring R p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP R p x y : R h : Commute x y \u22a2 (x - y) ^ p ^ Nat.zero = x ^ p ^ Nat.zero - y ^ p ^ Nat.zero ** simp ** case succ R : Type u_1 inst\u271d\u00b2 : Ring R p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP R p x y : R h : Commute x y n : \u2115 n_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n \u22a2 (x - y) ^ p ^ Nat.succ n = x ^ p ^ Nat.succ n - y ^ p ^ Nat.succ n ** rw [pow_succ', pow_mul, pow_mul, pow_mul, n_ih] ** case succ R : Type u_1 inst\u271d\u00b2 : Ring R p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP R p x y : R h : Commute x y n : \u2115 n_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n \u22a2 (x ^ p ^ n - y ^ p ^ n) ^ p = (x ^ p ^ n) ^ p - (y ^ p ^ n) ^ p ** apply sub_pow_char_of_commute ** case succ.h R : Type u_1 inst\u271d\u00b2 : Ring R p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP R p x y : R h : Commute x y n : \u2115 n_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n \u22a2 Commute (x ^ p ^ n) (y ^ p ^ n) ** apply Commute.pow_pow h ** Qed", + "informal": "" + }, + { + "formal": "interior_Ioc ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : DenselyOrdered \u03b1 a\u271d b\u271d : \u03b1 s : Set \u03b1 inst\u271d : NoMaxOrder \u03b1 a b : \u03b1 \u22a2 interior (Ioc a b) = Ioo a b ** rw [\u2190 Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio] ** Qed", + "informal": "" + }, + { + "formal": "NonUnitalStarAlgHom.fst_prod ** R : Type u_1 A : Type u_2 B : Type u_3 C : Type u_4 inst\u271d\u2079 : Monoid R inst\u271d\u2078 : NonUnitalNonAssocSemiring A inst\u271d\u2077 : DistribMulAction R A inst\u271d\u2076 : Star A inst\u271d\u2075 : NonUnitalNonAssocSemiring B inst\u271d\u2074 : DistribMulAction R B inst\u271d\u00b3 : Star B inst\u271d\u00b2 : NonUnitalNonAssocSemiring C inst\u271d\u00b9 : DistribMulAction R C inst\u271d : Star C f : A \u2192\u22c6\u2099\u2090[R] B g : A \u2192\u22c6\u2099\u2090[R] C \u22a2 comp (fst R B C) (prod f g) = f ** ext ** case h R : Type u_1 A : Type u_2 B : Type u_3 C : Type u_4 inst\u271d\u2079 : Monoid R inst\u271d\u2078 : NonUnitalNonAssocSemiring A inst\u271d\u2077 : DistribMulAction R A inst\u271d\u2076 : Star A inst\u271d\u2075 : NonUnitalNonAssocSemiring B inst\u271d\u2074 : DistribMulAction R B inst\u271d\u00b3 : Star B inst\u271d\u00b2 : NonUnitalNonAssocSemiring C inst\u271d\u00b9 : DistribMulAction R C inst\u271d : Star C f : A \u2192\u22c6\u2099\u2090[R] B g : A \u2192\u22c6\u2099\u2090[R] C x\u271d : A \u22a2 \u2191(comp (fst R B C) (prod f g)) x\u271d = \u2191f x\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "IsCompact.elim_nhds_subcover ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s t\u271d : Set \u03b1 hs : IsCompact s U : \u03b1 \u2192 Set \u03b1 hU : \u2200 (x : \u03b1), x \u2208 s \u2192 U x \u2208 \ud835\udcdd x t : Finset \u2191s ht : s \u2286 \u22c3 x \u2208 t, U \u2191x \u22a2 s \u2286 \u22c3 x \u2208 Finset.image Subtype.val t, U x ** rwa [Finset.set_biUnion_finset_image] ** Qed", + "informal": "" + }, + { + "formal": "Substring.Valid.next_stop ** x\u271d : Substring h\u271d : Valid x\u271d l m r : List Char h : ValidFor l m r x\u271d \u22a2 Substring.next x\u271d { byteIdx := Substring.bsize x\u271d } = { byteIdx := Substring.bsize x\u271d } ** simp [h.bsize, h.next_stop] ** Qed", + "informal": "" + }, + { + "formal": "TopCat.Presheaf.covering_presieve_eq_self ** X : TopCat Y : Opens \u2191X R : Presieve Y \u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R ** funext Z ** case h X : TopCat Y : Opens \u2191X R : Presieve Y Z : Opens \u2191X \u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R ** ext f ** case h.h X : TopCat Y : Opens \u2191X R : Presieve Y Z : Opens \u2191X f : Z \u27f6 Y \u22a2 f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y \u2194 f \u2208 R ** 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 ** X : TopCat Y : Opens \u2191X R : Presieve Y Z : Opens \u2191X f : Z \u27f6 Y x\u271d : f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y f' : Z \u27f6 Y h : R f' \u22a2 f \u2208 R ** rwa [Subsingleton.elim f f'] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.degree_sum_eq_of_disjoint ** R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s : Finset S h : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u22a2 degree (Finset.sum s f) = sup s fun i => degree (f i) ** induction' s using Finset.induction_on with x s hx IH ** case empty R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s : Finset S h\u271d : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) h : Set.Pairwise {i | i \u2208 \u2205 \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u22a2 degree (Finset.sum \u2205 f) = sup \u2205 fun i => degree (f i) ** simp ** case insert R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s IH : 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) h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u22a2 degree (Finset.sum (insert x s) f) = sup (insert x s) fun i => degree (f i) ** simp only [hx, Finset.sum_insert, not_false_iff, Finset.sup_insert] ** case insert R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s IH : 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) h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** specialize IH (h.mono fun _ => by simp (config := { contextual := true })) ** case insert R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** rcases lt_trichotomy (degree (f x)) (degree (s.sum f)) with (H | H | H) ** R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s IH : 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) h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) x\u271d : S \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} ** simp (config := { contextual := true }) ** case insert.inl R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) H : degree (f x) < degree (Finset.sum s f) \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** rw [\u2190 IH, sup_eq_right.mpr H.le, degree_add_eq_right_of_degree_lt H] ** case insert.inr.inl R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) H : degree (f x) = degree (Finset.sum s f) \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** rcases s.eq_empty_or_nonempty with (rfl | hs) ** case insert.inr.inl.inr R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) H : degree (f x) = degree (Finset.sum s f) hs : Finset.Nonempty s \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** obtain \u27e8y, hy, hy'\u27e9 := Finset.exists_mem_eq_sup s hs fun i => degree (f i) ** case insert.inr.inl.inr.intro.intro R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) H : degree (f x) = degree (Finset.sum s f) hs : Finset.Nonempty s y : S hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** rw [IH, hy'] at H ** case insert.inr.inl.inr.intro.intro R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S H : degree (f x) = degree (f y) hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** by_cases hx0 : f x = 0 ** case neg R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S H : degree (f x) = degree (f y) hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) hx0 : \u00acf x = 0 \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** have hy0 : f y \u2260 0 := by\n contrapose! H\n simpa [H, degree_eq_bot] using hx0 ** case neg R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S H : degree (f x) = degree (f y) hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) hx0 : \u00acf x = 0 hy0 : f y \u2260 0 \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** refine' absurd H (h _ _ fun H => hx _) ** case insert.inr.inl.inl R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s : Finset S h\u271d : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S hx : \u00acx \u2208 \u2205 h : Set.Pairwise {i | i \u2208 insert x \u2205 \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum \u2205 f) = sup \u2205 fun i => degree (f i) H : degree (f x) = degree (Finset.sum \u2205 f) \u22a2 degree (f x + Finset.sum \u2205 fun x => f x) = degree (f x) \u2294 sup \u2205 fun i => degree (f i) ** simp ** case pos R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S H : degree (f x) = degree (f y) hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) hx0 : f x = 0 \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** simp [hx0, IH] ** R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S H : degree (f x) = degree (f y) hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) hx0 : \u00acf x = 0 \u22a2 f y \u2260 0 ** contrapose! H ** R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) hx0 : \u00acf x = 0 H : f y = 0 \u22a2 degree (f x) \u2260 degree (f y) ** simpa [H, degree_eq_bot] using hx0 ** case neg.refine'_1 R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S H : degree (f x) = degree (f y) hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) hx0 : \u00acf x = 0 hy0 : f y \u2260 0 \u22a2 x \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0} ** simp [hx0] ** case neg.refine'_2 R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S H : degree (f x) = degree (f y) hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) hx0 : \u00acf x = 0 hy0 : f y \u2260 0 \u22a2 y \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0} ** simp [hy, hy0] ** case neg.refine'_3 R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) hs : Finset.Nonempty s y : S H\u271d : degree (f x) = degree (f y) hy : y \u2208 s hy' : (sup s fun i => degree (f i)) = degree (f y) hx0 : \u00acf x = 0 hy0 : f y \u2260 0 H : x = y \u22a2 x \u2208 s ** exact H.symm \u25b8 hy ** case insert.inr.inr R : Type u S : Type v \u03b9 : Type w a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] f : S \u2192 R[X] s\u271d : Finset S h\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f) x : S s : Finset S hx : \u00acx \u2208 s h : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f) IH : degree (Finset.sum s f) = sup s fun i => degree (f i) H : degree (Finset.sum s f) < degree (f x) \u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i) ** rw [\u2190 IH, sup_eq_left.mpr H.le, degree_add_eq_left_of_degree_lt H] ** Qed", + "informal": "" + }, + { + "formal": "Metric.closure_subset_cthickening ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d : PseudoEMetricSpace \u03b1 \u03b4\u271d \u03b5 : \u211d s t : Set \u03b1 x : \u03b1 \u03b4 : \u211d E : Set \u03b1 \u22a2 closure E \u2286 cthickening \u03b4 E ** rw [\u2190 cthickening_of_nonpos (min_le_right \u03b4 0)] ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d : PseudoEMetricSpace \u03b1 \u03b4\u271d \u03b5 : \u211d s t : Set \u03b1 x : \u03b1 \u03b4 : \u211d E : Set \u03b1 \u22a2 cthickening (min \u03b4 0) E \u2286 cthickening \u03b4 E ** exact cthickening_mono (min_le_left \u03b4 0) E ** Qed", + "informal": "" + }, + { + "formal": "AddMonoidAlgebra.mul_of'_modOf ** k : Type u_1 G : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : AddCancelCommMonoid G x : k[G] g : G \u22a2 x * of' k G g %\u1d52\u1da0 g = 0 ** refine Finsupp.ext fun g' => ?_ ** k : Type u_1 G : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : AddCancelCommMonoid G x : k[G] g g' : G \u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) g' = \u21910 g' ** rw [Finsupp.zero_apply] ** k : Type u_1 G : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : AddCancelCommMonoid G x : k[G] g g' : G \u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) g' = 0 ** obtain \u27e8d, rfl\u27e9 | h := em (\u2203 d, g' = g + d) ** case inl.intro k : Type u_1 G : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : AddCancelCommMonoid G x : k[G] g d : G \u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) (g + d) = 0 ** rw [modOf_apply_self_add] ** case inr k : Type u_1 G : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : AddCancelCommMonoid G x : k[G] g g' : G h : \u00ac\u2203 d, g' = g + d \u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) g' = 0 ** rw [modOf_apply_of_not_exists_add _ _ _ h, of'_apply, mul_single_apply_of_not_exists_add] ** case inr.h k : Type u_1 G : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : AddCancelCommMonoid G x : k[G] g g' : G h : \u00ac\u2203 d, g' = g + d \u22a2 \u00ac\u2203 d, g' = d + g ** simpa only [add_comm] using h ** Qed", + "informal": "" + }, + { + "formal": "finsuppLequivDFinsupp_apply_apply ** \u03b9 : Type u_1 R : Type u_2 M : Type u_3 inst\u271d\u2074 : DecidableEq \u03b9 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : (m : M) \u2192 Decidable (m \u2260 0) inst\u271d : Module R M \u22a2 \u2191(finsuppLequivDFinsupp R) = Finsupp.toDFinsupp ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Set.preimage_sInter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 f : \u03b1 \u2192 \u03b2 s : Set (Set \u03b2) \u22a2 f \u207b\u00b9' \u22c2\u2080 s = \u22c2 t \u2208 s, f \u207b\u00b9' t ** rw [sInter_eq_biInter, preimage_iInter\u2082] ** Qed", + "informal": "" + }, + { + "formal": "circleMap_not_mem_ball ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R \u03b8 : \u211d \u22a2 \u00accircleMap c R \u03b8 \u2208 ball c R ** simp [dist_eq, le_abs_self] ** Qed", + "informal": "" + }, + { + "formal": "Ideal.basisSpanSingleton_apply ** \u03b9 : Type u_1 R : Type u_2 S : Type u_3 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : IsDomain S inst\u271d : Algebra R S b : Basis \u03b9 R S x : S hx : x \u2260 0 i : \u03b9 \u22a2 \u2191(\u2191(basisSpanSingleton b hx) i) = x * \u2191b i ** 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'] ** \u03b9 : Type u_1 R : Type u_2 S : Type u_3 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : IsDomain S inst\u271d : Algebra R S b : Basis \u03b9 R S x : S hx : x \u2260 0 i : \u03b9 \u22a2 \u2191(\u2191(LinearEquiv.ofEq (LinearMap.range (\u2191(Algebra.lmul R S) x)) (Submodule.restrictScalars R (span {x})) (_ : LinearMap.range (\u2191(Algebra.lmul R S) x) = Submodule.restrictScalars R (span {x}))) (\u2191(LinearEquiv.ofInjective (\u2191(Algebra.lmul R S) x) (_ : Function.Injective \u2191(\u2191(LinearMap.mul R S) x))) (\u2191b i))) = x * \u2191b i ** erw [LinearEquiv.coe_ofEq_apply, LinearEquiv.ofInjective_apply, Algebra.coe_lmul_eq_mul,\n LinearMap.mul_apply'] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousMultilinearMap.norm_mkPiAlgebraFin ** \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedSpace \ud835\udd5c G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u00b2 : NormedRing A inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A inst\u271d : NormOneClass A \u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c n A\u2016 = 1 ** cases n ** case zero \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedSpace \ud835\udd5c G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u2074 : NormedRing A inst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A inst\u271d\u00b2 : NormOneClass A Ei : Fin (Nat.succ Nat.zero) \u2192 Type wEi inst\u271d\u00b9 : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedAddCommGroup (Ei i) inst\u271d : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedSpace \ud835\udd5c (Ei i) \u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c Nat.zero A\u2016 = 1 ** rw [norm_mkPiAlgebraFin_zero] ** case zero \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedSpace \ud835\udd5c G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u2074 : NormedRing A inst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A inst\u271d\u00b2 : NormOneClass A Ei : Fin (Nat.succ Nat.zero) \u2192 Type wEi inst\u271d\u00b9 : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedAddCommGroup (Ei i) inst\u271d : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedSpace \ud835\udd5c (Ei i) \u22a2 \u20161\u2016 = 1 ** simp ** case succ \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedSpace \ud835\udd5c G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u2074 : NormedRing A inst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A inst\u271d\u00b2 : NormOneClass A n\u271d : \u2115 Ei : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi inst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i) inst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i) \u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A\u2016 = 1 ** refine' le_antisymm norm_mkPiAlgebraFin_succ_le _ ** case succ \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedSpace \ud835\udd5c G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u2074 : NormedRing A inst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A inst\u271d\u00b2 : NormOneClass A n\u271d : \u2115 Ei : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi inst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i) inst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i) \u22a2 1 \u2264 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A\u2016 ** convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ _) A)\n fun _ => 1 ** case h.e'_3 \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedSpace \ud835\udd5c G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u2074 : NormedRing A inst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A inst\u271d\u00b2 : NormOneClass A n\u271d : \u2115 Ei : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi inst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i) inst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i) \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 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Filter.le_cofinite_iff_compl_singleton_mem ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 l : Filter \u03b1 \u22a2 l \u2264 cofinite \u2194 \u2200 (x : \u03b1), {x}\u1d9c \u2208 l ** refine' \u27e8fun h x => h (finite_singleton x).compl_mem_cofinite, fun h s (hs : s\u1d9c.Finite) => _\u27e9 ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 l : Filter \u03b1 h : \u2200 (x : \u03b1), {x}\u1d9c \u2208 l s : Set \u03b1 hs : Set.Finite s\u1d9c \u22a2 s \u2208 l ** rw [\u2190 compl_compl s, \u2190 biUnion_of_singleton s\u1d9c, compl_iUnion\u2082, Filter.biInter_mem hs] ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 l : Filter \u03b1 h : \u2200 (x : \u03b1), {x}\u1d9c \u2208 l s : Set \u03b1 hs : Set.Finite s\u1d9c \u22a2 \u2200 (i : \u03b1), i \u2208 s\u1d9c \u2192 {i}\u1d9c \u2208 l ** exact fun x _ => h x ** Qed", + "informal": "" + }, + { + "formal": "vsub_right_cancel ** G : Type u_1 P : Type u_2 inst\u271d : AddGroup G T : AddTorsor G P p1 p2 p : P h : p -\u1d65 p1 = p -\u1d65 p2 \u22a2 p1 = p2 ** refine' vadd_left_cancel (p -\u1d65 p2) _ ** G : Type u_1 P : Type u_2 inst\u271d : AddGroup G T : AddTorsor G P p1 p2 p : P h : p -\u1d65 p1 = p -\u1d65 p2 \u22a2 p -\u1d65 p2 +\u1d65 p1 = p -\u1d65 p2 +\u1d65 p2 ** rw [vsub_vadd, \u2190 h, vsub_vadd] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.neg_mul ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 m' : o \u2192 Type u_5 n' : o \u2192 Type u_6 R : Type u_7 S : Type u_8 \u03b1 : Type v \u03b2 : Type w \u03b3 : Type u_9 inst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1 inst\u271d : Fintype n M : Matrix m n \u03b1 N : Matrix n o \u03b1 \u22a2 -M * N = -(M * N) ** ext ** case a.h l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 m' : o \u2192 Type u_5 n' : o \u2192 Type u_6 R : Type u_7 S : Type u_8 \u03b1 : Type v \u03b2 : Type w \u03b3 : Type u_9 inst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1 inst\u271d : Fintype n M : Matrix m n \u03b1 N : Matrix n o \u03b1 i\u271d : m x\u271d : o \u22a2 (-M * N) i\u271d x\u271d = (-(M * N)) i\u271d x\u271d ** apply neg_dotProduct ** Qed", + "informal": "" + }, + { + "formal": "List.rotate'_nil ** \u03b1 : Type u n : \u2115 \u22a2 rotate' [] n = [] ** cases n <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "neg_div ** \u03b1 : Type u_1 \u03b2 : Type u_2 K : Type u_3 inst\u271d\u00b9 : DivisionMonoid K inst\u271d : HasDistribNeg K a\u271d b\u271d a b : K \u22a2 -b / a = -(b / a) ** rw [neg_eq_neg_one_mul, mul_div_assoc, \u2190 neg_eq_neg_one_mul] ** Qed", + "informal": "" + }, + { + "formal": "LipschitzWith.const_min ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d : PseudoEMetricSpace \u03b1 f g : \u03b1 \u2192 \u211d Kf Kg : \u211d\u22650 hf : LipschitzWith Kf f a : \u211d \u22a2 LipschitzWith Kf fun x => min a (f x) ** simpa only [min_comm] using hf.min_const a ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.mem_rightTransversals_iff_existsUnique_quotient_mk''_eq ** G : Type u_1 inst\u271d : Group G H K : Subgroup G S T : Set G \u22a2 S \u2208 rightTransversals \u2191H \u2194 \u2200 (q : Quotient (QuotientGroup.rightRel H)), \u2203! s, Quotient.mk'' \u2191s = q ** simp_rw [mem_rightTransversals_iff_existsUnique_mul_inv_mem, SetLike.mem_coe, \u2190\n QuotientGroup.rightRel_apply, \u2190 Quotient.eq''] ** G : Type u_1 inst\u271d : Group G H K : Subgroup G S T : Set G \u22a2 (\u2200 (g : G), \u2203! s, Quotient.mk'' \u2191s = Quotient.mk'' g) \u2194 \u2200 (q : Quotient (QuotientGroup.rightRel H)), \u2203! s, Quotient.mk'' \u2191s = q ** exact \u27e8fun h q => Quotient.inductionOn' q h, fun h g => h (Quotient.mk'' g)\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Sum.denselyOrdered_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 x\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2) a b : \u03b1 h : a < b \u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b ** obtain \u27e8c | c, ha, hb\u27e9 := @exists_between (Sum \u03b1 \u03b2) _ _ _ _ (inl_lt_inl_iff.2 h) ** case intro.inl.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 x\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2) a b : \u03b1 h : a < b c : \u03b1 ha : inl a < inl c hb : inl c < inl b \u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b ** exact \u27e8c, inl_lt_inl_iff.1 ha, inl_lt_inl_iff.1 hb\u27e9 ** case intro.inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 x\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2) a b : \u03b1 h : a < b c : \u03b2 ha : inl a < inr c hb : inr c < inl b \u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b ** exact (not_inl_lt_inr ha).elim ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 x\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2) a b : \u03b2 h : a < b \u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b ** obtain \u27e8c | c, ha, hb\u27e9 := @exists_between (Sum \u03b1 \u03b2) _ _ _ _ (inr_lt_inr_iff.2 h) ** case intro.inl.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 x\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2) a b : \u03b2 h : a < b c : \u03b1 ha : inr a < inl c hb : inl c < inr b \u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b ** exact (not_inl_lt_inr hb).elim ** case intro.inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 x\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2) a b : \u03b2 h : a < b c : \u03b2 ha : inr a < inr c hb : inr c < inr b \u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b ** exact \u27e8c, inr_lt_inr_iff.1 ha, inr_lt_inr_iff.1 hb\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "map_uniformity_set_coe ** \u03b1 : Type ua \u03b2 : Type ub \u03b3 : Type uc \u03b4 : Type ud \u03b9 : Sort u_1 s : Set \u03b1 inst\u271d : UniformSpace \u03b1 \u22a2 map (Prod.map Subtype.val Subtype.val) (\ud835\udce4 \u2191s) = \ud835\udce4 \u03b1 \u2293 \ud835\udcdf (s \u00d7\u02e2 s) ** rw [uniformity_setCoe, map_comap, range_prod_map, Subtype.range_val] ** Qed", + "informal": "" + }, + { + "formal": "Homeomorph.isPreconnected_image ** X : Type u_1 Y : Type u_2 Z : Type u_3 inst\u271d\u2074 : TopologicalSpace X inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : TopologicalSpace Z X' : Type u_4 Y' : Type u_5 inst\u271d\u00b9 : TopologicalSpace X' inst\u271d : TopologicalSpace Y' s : Set X h : X \u2243\u209c Y hs : IsPreconnected (\u2191h '' s) \u22a2 IsPreconnected s ** simpa only [image_symm, preimage_image]\nusing hs.image _ h.symm.continuous.continuousOn ** Qed", + "informal": "" + }, + { + "formal": "TrivSqZeroExt.nhds_def ** \u03b1 : Type u_1 S : Type u_2 R : Type u_3 M : Type u_4 inst\u271d\u00b9 : TopologicalSpace R inst\u271d : TopologicalSpace M x : tsze R M \u22a2 nhds x = Filter.prod (nhds (fst x)) (nhds (snd x)) ** cases x ** case mk \u03b1 : Type u_1 S : Type u_2 R : Type u_3 M : Type u_4 inst\u271d\u00b9 : TopologicalSpace R inst\u271d : TopologicalSpace M fst\u271d : R snd\u271d : M \u22a2 nhds (fst\u271d, snd\u271d) = Filter.prod (nhds (fst (fst\u271d, snd\u271d))) (nhds (snd (fst\u271d, snd\u271d))) ** exact nhds_prod_eq ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.degree_sub_le ** R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Ring R p\u271d q\u271d p q : R[X] \u22a2 degree (p - q) \u2264 max (degree p) (degree q) ** simpa only [degree_neg q] using degree_add_le p (-q) ** Qed", + "informal": "" + }, + { + "formal": "inv_mul_lt_iff' ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedSemifield \u03b1 a b c d e : \u03b1 m n : \u2124 h : 0 < b \u22a2 b\u207b\u00b9 * a < c \u2194 a < c * b ** rw [inv_mul_lt_iff h, mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "LinearPMap.inverse_range ** R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G f : E \u2192\u2097.[R] F hf : LinearMap.ker f.toFun = \u22a5 \u22a2 LinearMap.range (inverse f).toFun = f.domain ** 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] ** R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G f : E \u2192\u2097.[R] F hf : LinearMap.ker f.toFun = \u22a5 \u22a2 Submodule.map (LinearMap.snd R F E) (Submodule.map (LinearEquiv.prodComm R E F) (graph f)) = Submodule.map (LinearMap.snd R F E) (Submodule.map (\u2191(LinearEquiv.prodComm R E F)) (graph f)) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Zsqrtd.norm_int_cast ** d n : \u2124 \u22a2 norm \u2191n = n * n ** simp [norm] ** Qed", + "informal": "" + }, + { + "formal": "lTensorHomEquivHomLTensor_apply ** \u03b9 : Type w R : Type u M : Type v\u2081 N : Type v\u2082 P : Type v\u2083 Q : Type v\u2084 inst\u271d\u00b9\u00b3 : CommRing R inst\u271d\u00b9\u00b2 : AddCommGroup M inst\u271d\u00b9\u00b9 : AddCommGroup N inst\u271d\u00b9\u2070 : AddCommGroup P inst\u271d\u2079 : AddCommGroup Q inst\u271d\u2078 : Module R M inst\u271d\u2077 : Module R N inst\u271d\u2076 : Module R P inst\u271d\u2075 : Module R Q inst\u271d\u2074 : Free R M inst\u271d\u00b3 : Module.Finite R M inst\u271d\u00b2 : Free R N inst\u271d\u00b9 : Module.Finite R N inst\u271d : Nontrivial R x : P \u2297[R] (M \u2192\u2097[R] Q) \u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) x = \u2191(lTensorHomToHomLTensor R M P Q) x ** rw [\u2190 LinearEquiv.coe_toLinearMap, lTensorHomEquivHomLTensor_toLinearMap] ** Qed", + "informal": "" + }, + { + "formal": "Set.disjoint_toFinset ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s t : Set \u03b1 inst\u271d\u00b9 : Fintype \u2191s inst\u271d : Fintype \u2191t \u22a2 Disjoint (toFinset s) (toFinset t) \u2194 Disjoint s t ** simp only [\u2190 disjoint_coe, coe_toFinset] ** Qed", + "informal": "" + }, + { + "formal": "continuousOn_Ioc_extendFrom_Ioo ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : LinearOrder \u03b1 inst\u271d\u00b3 : DenselyOrdered \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : RegularSpace \u03b2 f : \u03b1 \u2192 \u03b2 a b : \u03b1 lb : \u03b2 hab : a < b hf : ContinuousOn f (Ioo a b) hb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lb) \u22a2 ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b) ** have := @continuousOn_Ico_extendFrom_Ioo \u03b1\u1d52\u1d48 _ _ _ _ _ _ _ f _ _ lb hab ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : LinearOrder \u03b1 inst\u271d\u00b3 : DenselyOrdered \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : RegularSpace \u03b2 f : \u03b1 \u2192 \u03b2 a b : \u03b1 lb : \u03b2 hab : a < b hf : ContinuousOn f (Ioo a b) hb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lb) this : 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) \u22a2 ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b) ** erw [dual_Ico, dual_Ioi, dual_Ioo] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : LinearOrder \u03b1 inst\u271d\u00b3 : DenselyOrdered \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : RegularSpace \u03b2 f : \u03b1 \u2192 \u03b2 a b : \u03b1 lb : \u03b2 hab : a < b hf : ContinuousOn f (Ioo a b) hb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lb) this : ContinuousOn f (\u2191OrderDual.ofDual \u207b\u00b9' Ioo a b) \u2192 Tendsto f (\ud835\udcdd[\u2191OrderDual.ofDual \u207b\u00b9' Iio b] b) (\ud835\udcdd lb) \u2192 ContinuousOn (extendFrom (\u2191OrderDual.ofDual \u207b\u00b9' Ioo a b) f) (\u2191OrderDual.ofDual \u207b\u00b9' Ioc a b) \u22a2 ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b) ** exact this hf hb ** Qed", + "informal": "" + }, + { + "formal": "sq_lt_sq ** \u03b2 : Type u_1 A : Type u_2 G : Type u_3 M : Type u_4 R : Type u_5 inst\u271d : LinearOrderedRing R x y : R \u22a2 x ^ 2 < y ^ 2 \u2194 |x| < |y| ** simpa only [sq_abs] using\n (@strictMonoOn_pow R _ _ two_pos).lt_iff_lt (abs_nonneg x) (abs_nonneg y) ** Qed", + "informal": "" + }, + { + "formal": "disjoint_assoc ** \u03b1 : Type u_1 inst\u271d\u00b9 : SemilatticeInf \u03b1 inst\u271d : OrderBot \u03b1 a b c d : \u03b1 \u22a2 Disjoint (a \u2293 b) c \u2194 Disjoint a (b \u2293 c) ** rw [disjoint_iff_inf_le, disjoint_iff_inf_le, inf_assoc] ** Qed", + "informal": "" + }, + { + "formal": "Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E R : \u211d c w : \u2102 f : \u2102 \u2192 E s : Set \u2102 hs : Set.Countable s hw : w \u2208 ball c R hc : ContinuousOn f (closedBall c R) hd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w ** 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] ** case hc E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E R : \u211d c w : \u2102 f : \u2102 \u2192 E s : Set \u2102 hs : Set.Countable s hw : w \u2208 ball c R hc : ContinuousOn f (closedBall c R) hd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x \u22a2 2 * \u2191\u03c0 * I \u2260 0 ** simp [Real.pi_ne_zero, I_ne_zero] ** Qed", + "informal": "" + }, + { + "formal": "LatticeOrderedGroup.sup_div_inf_eq_abs_div ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \u22a2 (a \u2294 b) / (a \u2293 b) = (a \u2294 b) * (a\u207b\u00b9 \u2294 b\u207b\u00b9) ** rw [div_eq_mul_inv, \u2190 inv_inf_eq_sup_inv] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \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) ** rw [mul_sup, sup_mul, sup_mul] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \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) ** rw [mul_right_inv, mul_right_inv, \u2190div_eq_mul_inv, \u2190div_eq_mul_inv] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \u22a2 1 \u2294 b / a \u2294 (a / b \u2294 1) = 1 \u2294 b / a \u2294 (1 / (b / a) \u2294 1) ** rw [one_div_div] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \u22a2 1 \u2294 b / a \u2294 (1 / (b / a) \u2294 1) = 1 \u2294 b / a \u2294 ((b / a)\u207b\u00b9 \u2294 1) ** rw [inv_eq_one_div] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \u22a2 1 \u2294 b / a \u2294 ((b / a)\u207b\u00b9 \u2294 1) = 1 \u2294 (b / a \u2294 (b / a)\u207b\u00b9 \u2294 1) ** rw [sup_assoc, sup_assoc] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \u22a2 1 \u2294 (b / a \u2294 (b / a)\u207b\u00b9 \u2294 1) = 1 \u2294 (|b / a| \u2294 1) ** rw [abs_eq_sup_inv] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \u22a2 1 \u2294 (|b / a| \u2294 1) = 1 \u2294 |b / a| ** rw [\u2190 m_pos_part_def, m_pos_abs] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \u22a2 1 \u2294 |b / a| = |b / a| \u2294 1 ** rw [sup_comm] ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b3 : Lattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 \u22a2 |b / a| \u2294 1 = |b / a| ** rw [\u2190 m_pos_part_def, m_pos_abs] ** Qed", + "informal": "" + }, + { + "formal": "Nat.dist_eq_max_sub_min ** i j : \u2115 \u22a2 i < j \u2192 dist i j = max i j - min i j ** intro h ** i j : \u2115 h : i < j \u22a2 dist i j = max i j - min i j ** rw [max_eq_right_of_lt h, min_eq_left_of_lt h, dist_eq_sub_of_le (Nat.le_of_lt h)] ** i j : \u2115 \u22a2 i \u2265 j \u2192 dist i j = max i j - min i j ** intro h ** i j : \u2115 h : i \u2265 j \u22a2 dist i j = max i j - min i j ** rw [max_eq_left h, min_eq_right h, dist_eq_sub_of_le_right h] ** Qed", + "informal": "" + }, + { + "formal": "Equiv.symm_apply_eq ** \u03b1\u271d : Sort u \u03b2\u271d : Sort v \u03b3 : Sort w \u03b1 : Sort u_1 \u03b2 : Sort u_2 e : \u03b1 \u2243 \u03b2 x : \u03b2 y : (fun x => \u03b1) x H : \u2191e.symm x = y \u22a2 x = \u2191e y ** simp [H.symm] ** \u03b1\u271d : Sort u \u03b2\u271d : Sort v \u03b3 : Sort w \u03b1 : Sort u_1 \u03b2 : Sort u_2 e : \u03b1 \u2243 \u03b2 x : \u03b2 y : (fun x => \u03b1) x H : x = \u2191e y \u22a2 \u2191e.symm x = y ** simp [H] ** Qed", + "informal": "" + }, + { + "formal": "QuadraticForm.polar_sub_right ** S : Type u_1 T : Type u_2 R : Type u_3 M : Type u_4 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M Q : QuadraticForm R M x y y' : M \u22a2 polar (\u2191Q) x (y - y') = polar (\u2191Q) x y - polar (\u2191Q) x y' ** rw [sub_eq_add_neg, sub_eq_add_neg, polar_add_right, polar_neg_right] ** Qed", + "informal": "" + }, + { + "formal": "Rel.dom_inv ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : Rel \u03b1 \u03b2 \u22a2 dom (inv r) = codom r ** ext x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : Rel \u03b1 \u03b2 x : \u03b2 \u22a2 x \u2208 dom (inv r) \u2194 x \u2208 codom r ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Real.Wallis.W_pos ** k : \u2115 \u22a2 0 < W k ** induction' k with k hk ** case zero \u22a2 0 < W Nat.zero ** unfold W ** case zero \u22a2 0 < \u220f i in range Nat.zero, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) ** simp ** case succ k : \u2115 hk : 0 < W k \u22a2 0 < W (Nat.succ k) ** rw [W_succ] ** case succ k : \u2115 hk : 0 < W k \u22a2 0 < W k * ((2 * \u2191k + 2) / (2 * \u2191k + 1) * ((2 * \u2191k + 2) / (2 * \u2191k + 3))) ** refine' mul_pos hk (mul_pos (div_pos _ _) (div_pos _ _)) <;> positivity ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.multiplicity_finite_of_degree_pos_of_monic ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r \u22a2 False ** have hp0 : p \u2260 0 := fun hp0 => by simp [hp0] at hp ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 \u22a2 False ** have hr0 : r \u2260 0 := fun hr0 => by subst hr0; simp [hq] at hr ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 \u22a2 False ** have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp] ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 \u22a2 False ** have hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 := hpn1.symm \u25b8 zn0.symm ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 \u22a2 False ** 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] ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 \u22a2 False ** have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by\n rw [\u2190 degree_eq_natDegree hp0]; exact hp ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 hnp : 0 < natDegree p \u22a2 False ** have := congr_arg natDegree hr ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 hnp : 0 < natDegree p this : natDegree q = natDegree (p ^ (natDegree q + 1) * r) \u22a2 False ** rw [natDegree_mul' hpnr0, natDegree_pow' hpn0', add_mul, add_assoc] at this ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 hnp : 0 < natDegree p this : natDegree q = natDegree q * natDegree p + (1 * natDegree p + natDegree r) \u22a2 False ** 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 ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p = 0 \u22a2 False ** simp [hp0] at hp ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r = 0 \u22a2 False ** subst hr0 ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q hp0 : p \u2260 0 hr : q = p ^ (natDegree q + 1) * 0 \u22a2 False ** simp [hq] at hr ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 \u22a2 leadingCoeff p ^ (natDegree q + 1) = 1 ** simp [show _ = _ from hmp] ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 \u22a2 leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 ** simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,\n hr0] ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 \u22a2 \u21910 < \u2191(natDegree p) ** rw [\u2190 degree_eq_natDegree hp0] ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 \u22a2 \u21910 < degree p R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 \u22a2 PartialOrder (WithBot \u2115) ** exact hp ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommSemiring R p q : R[X] hp : 0 < degree p hmp : Monic p hq : q \u2260 0 zn0 : 0 \u2260 1 x\u271d : p ^ (natDegree q + 1) \u2223 q r : R[X] hr : q = p ^ (natDegree q + 1) * r hp0 : p \u2260 0 hr0 : r \u2260 0 hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 hnp : 0 < natDegree p this : natDegree q = natDegree q * natDegree p + (1 * natDegree p + natDegree r) \u22a2 0 < 1 * natDegree p ** rwa [one_mul] ** Qed", + "informal": "" + }, + { + "formal": "contMDiffOn_iff_source_of_mem_maximalAtlas ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s s\u2081 t : Set M x : M m n : \u2115\u221e he : e \u2208 maximalAtlas I M hs : s \u2286 e.source \u22a2 ContMDiffOn I I' n f s \u2194 ContMDiffOn \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I))) (\u2191(LocalHomeomorph.extend e I) '' s) ** simp_rw [ContMDiffOn, Set.ball_image_iff] ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s s\u2081 t : Set M x : M m n : \u2115\u221e he : e \u2208 maximalAtlas I M hs : s \u2286 e.source \u22a2 (\u2200 (x : M), x \u2208 s \u2192 ContMDiffWithinAt I I' n f s x) \u2194 \u2200 (x : M), x \u2208 s \u2192 ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I))) (\u2191(LocalHomeomorph.extend e I) '' s) (\u2191(LocalHomeomorph.extend e I) x) ** refine' forall\u2082_congr fun x hx => _ ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s s\u2081 t : Set M x\u271d : M m n : \u2115\u221e he : e \u2208 maximalAtlas I M hs : s \u2286 e.source x : M hx : x \u2208 s \u22a2 ContMDiffWithinAt I I' n f s x \u2194 ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I))) (\u2191(LocalHomeomorph.extend e I) '' s) (\u2191(LocalHomeomorph.extend e I) x) ** rw [contMDiffWithinAt_iff_source_of_mem_maximalAtlas he (hs hx)] ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s s\u2081 t : Set M x\u271d : M m n : \u2115\u221e he : e \u2208 maximalAtlas I M hs : s \u2286 e.source x : M hx : x \u2208 s \u22a2 ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I))) (\u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(LocalHomeomorph.extend e I) x) \u2194 ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I))) (\u2191(LocalHomeomorph.extend e I) '' s) (\u2191(LocalHomeomorph.extend e I) x) ** apply contMDiffWithinAt_congr_nhds ** case hst \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s s\u2081 t : Set M x\u271d : M m n : \u2115\u221e he : e \u2208 maximalAtlas I M hs : s \u2286 e.source x : M hx : x \u2208 s \u22a2 \ud835\udcdd[\u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(LocalHomeomorph.extend e I) x = \ud835\udcdd[\u2191(LocalHomeomorph.extend e I) '' s] \u2191(LocalHomeomorph.extend e I) x ** simp_rw [nhdsWithin_eq_iff_eventuallyEq,\n e.extend_symm_preimage_inter_range_eventuallyEq I hs (hs hx)] ** Qed", + "informal": "" + }, + { + "formal": "Set.toFinset_congr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s\u271d t\u271d s t : Set \u03b1 inst\u271d\u00b9 : Fintype \u2191s inst\u271d : Fintype \u2191t h : s = t \u22a2 toFinset s = toFinset t ** subst h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s\u271d t s : Set \u03b1 inst\u271d\u00b9 inst\u271d : Fintype \u2191s \u22a2 toFinset s = toFinset s ** congr ** case h.e_3.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s\u271d t s : Set \u03b1 inst\u271d\u00b9 inst\u271d : Fintype \u2191s \u22a2 inst\u271d\u00b9 = inst\u271d ** exact Subsingleton.elim _ _ ** Qed", + "informal": "" + }, + { + "formal": "Finset.card_sigmaLift ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 \u03b3 : \u03b9 \u2192 Type u_4 inst\u271d : DecidableEq \u03b9 f g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i) a : (i : \u03b9) \u00d7 \u03b1 i b : (i : \u03b9) \u00d7 \u03b2 i \u22a2 card (sigmaLift f a b) = if h : a.fst = b.fst then card (f (h \u25b8 a.snd) b.snd) else 0 ** simp_rw [sigmaLift] ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 \u03b3 : \u03b9 \u2192 Type u_4 inst\u271d : DecidableEq \u03b9 f g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i) a : (i : \u03b9) \u00d7 \u03b1 i b : (i : \u03b9) \u00d7 \u03b2 i \u22a2 card (if h : a.fst = b.fst then map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) else \u2205) = if h : a.fst = b.fst then card (f (h \u25b8 a.snd) b.snd) else 0 ** split_ifs with h <;> simp [h] ** Qed", + "informal": "" + }, + { + "formal": "Set.pi_update_of_not_mem ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 inst\u271d : DecidableEq \u03b9 hi : \u00aci \u2208 s f : (j : \u03b9) \u2192 \u03b1 j a : \u03b1 i t : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j) j : \u03b9 hj : j \u2208 s \u22a2 t j (update f i a j) = t j (f j) ** rw [update_noteq] ** case h \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 inst\u271d : DecidableEq \u03b9 hi : \u00aci \u2208 s f : (j : \u03b9) \u2192 \u03b1 j a : \u03b1 i t : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j) j : \u03b9 hj : j \u2208 s \u22a2 j \u2260 i ** exact fun h => hi (h \u25b8 hj) ** Qed", + "informal": "" + }, + { + "formal": "hasSum_single ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : AddCommMonoid \u03b1 inst\u271d : TopologicalSpace \u03b1 f\u271d g : \u03b2 \u2192 \u03b1 a b\u271d : \u03b1 s : Finset \u03b2 f : \u03b2 \u2192 \u03b1 b : \u03b2 hf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0 this : HasSum f (\u2211 b' in {b}, f b') \u22a2 HasSum f (f b) ** simpa using this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : AddCommMonoid \u03b1 inst\u271d : TopologicalSpace \u03b1 f\u271d g : \u03b2 \u2192 \u03b1 a b\u271d : \u03b1 s : Finset \u03b2 f : \u03b2 \u2192 \u03b1 b : \u03b2 hf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0 \u22a2 \u2200 (b_1 : \u03b2), \u00acb_1 \u2208 {b} \u2192 f b_1 = 0 ** simpa [hf] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SimpleFunc.integrable_approxOn_range ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : BorelSpace E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 fmeas : Measurable f inst\u271d : SeparableSpace \u2191(Set.range f \u222a {0}) hf : Integrable f n : \u2115 \u22a2 0 \u2208 Set.range f \u222a {0} ** simp ** Qed", + "informal": "" + }, + { + "formal": "Cycle.Chain.eq_nil_of_irrefl ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : Cycle \u03b1 inst\u271d\u00b9 : IsTrans \u03b1 r inst\u271d : IsIrrefl \u03b1 r h : Chain r s \u22a2 s = Cycle.nil ** induction' s using Cycle.induction_on with a l _ h ** case H0 \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : Cycle \u03b1 inst\u271d\u00b9 : IsTrans \u03b1 r inst\u271d : IsIrrefl \u03b1 r h\u271d : Chain r s h : Chain r Cycle.nil \u22a2 Cycle.nil = Cycle.nil ** rfl ** case HI \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : Cycle \u03b1 inst\u271d\u00b9 : IsTrans \u03b1 r inst\u271d : IsIrrefl \u03b1 r h\u271d : Chain r s a : \u03b1 l : List \u03b1 a\u271d : Chain r \u2191l \u2192 \u2191l = Cycle.nil h : Chain r \u2191(a :: l) \u22a2 \u2191(a :: l) = Cycle.nil ** have ha := mem_cons_self a l ** case HI \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : Cycle \u03b1 inst\u271d\u00b9 : IsTrans \u03b1 r inst\u271d : IsIrrefl \u03b1 r h\u271d : Chain r s a : \u03b1 l : List \u03b1 a\u271d : Chain r \u2191l \u2192 \u2191l = Cycle.nil h : Chain r \u2191(a :: l) ha : a \u2208 a :: l \u22a2 \u2191(a :: l) = Cycle.nil ** exact (irrefl_of r a <| chain_iff_pairwise.1 h a ha a ha).elim ** Qed", + "informal": "" + }, + { + "formal": "intervalIntegral.integral_eq_zero_iff_of_nonneg_ae ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f g : \u211d \u2192 \u211d a b : \u211d \u03bc : Measure \u211d hf : 0 \u2264\u1d50[Measure.restrict \u03bc (Ioc a b \u222a Ioc b a)] f hfi : IntervalIntegrable f \u03bc a b \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 ** cases' le_total a b with hab hab <;>\n simp only [Ioc_eq_empty hab.not_lt, empty_union, union_empty] at hf \u22a2 ** case inl \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f g : \u211d \u2192 \u211d a b : \u211d \u03bc : Measure \u211d hfi : IntervalIntegrable f \u03bc a b hab : a \u2264 b hf : 0 \u2264\u1d50[Measure.restrict \u03bc (Ioc a b)] f \u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = 0 \u2194 f =\u1d50[Measure.restrict \u03bc (Ioc a b)] 0 ** exact integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f g : \u211d \u2192 \u211d a b : \u211d \u03bc : Measure \u211d hfi : IntervalIntegrable f \u03bc a b hab : b \u2264 a hf : 0 \u2264\u1d50[Measure.restrict \u03bc (Ioc b a)] f \u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = 0 \u2194 f =\u1d50[Measure.restrict \u03bc (Ioc b a)] 0 ** rw [integral_symm, neg_eq_zero, integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi.symm] ** Qed", + "informal": "" + }, + { + "formal": "Cycle.length_nontrivial ** \u03b1 : Type u_1 s : Cycle \u03b1 h : Nontrivial s \u22a2 2 \u2264 length s ** obtain \u27e8x, y, hxy, hx, hy\u27e9 := h ** case intro.intro.intro.intro \u03b1 : Type u_1 s : Cycle \u03b1 x y : \u03b1 hxy : x \u2260 y hx : x \u2208 s hy : y \u2208 s \u22a2 2 \u2264 length s ** induction' s using Quot.inductionOn with l ** case intro.intro.intro.intro.h \u03b1 : Type u_1 s : Cycle \u03b1 x y : \u03b1 hxy : x \u2260 y hx\u271d : x \u2208 s hy\u271d : y \u2208 s l : List \u03b1 hx : x \u2208 Quot.mk Setoid.r l hy : y \u2208 Quot.mk Setoid.r l \u22a2 2 \u2264 length (Quot.mk Setoid.r l) ** rcases l with (_ | \u27e8hd, _ | \u27e8hd', tl\u27e9\u27e9) ** case intro.intro.intro.intro.h.nil \u03b1 : Type u_1 s : Cycle \u03b1 x y : \u03b1 hxy : x \u2260 y hx\u271d : x \u2208 s hy\u271d : y \u2208 s hx : x \u2208 Quot.mk Setoid.r [] hy : y \u2208 Quot.mk Setoid.r [] \u22a2 2 \u2264 length (Quot.mk Setoid.r []) ** simp at hx ** case intro.intro.intro.intro.h.cons.nil \u03b1 : Type u_1 s : Cycle \u03b1 x y : \u03b1 hxy : x \u2260 y hx\u271d : x \u2208 s hy\u271d : y \u2208 s hd : \u03b1 hx : x \u2208 Quot.mk Setoid.r [hd] hy : y \u2208 Quot.mk Setoid.r [hd] \u22a2 2 \u2264 length (Quot.mk Setoid.r [hd]) ** simp only [mem_coe_iff, mk_eq_coe, mem_singleton] at hx hy ** case intro.intro.intro.intro.h.cons.nil \u03b1 : Type u_1 s : Cycle \u03b1 x y : \u03b1 hxy : x \u2260 y hx\u271d : x \u2208 s hy\u271d : y \u2208 s hd : \u03b1 hx : x = hd hy : y = hd \u22a2 2 \u2264 length (Quot.mk Setoid.r [hd]) ** simp [hx, hy] at hxy ** case intro.intro.intro.intro.h.cons.cons \u03b1 : Type u_1 s : Cycle \u03b1 x y : \u03b1 hxy : x \u2260 y hx\u271d : x \u2208 s hy\u271d : y \u2208 s hd hd' : \u03b1 tl : List \u03b1 hx : x \u2208 Quot.mk Setoid.r (hd :: hd' :: tl) hy : y \u2208 Quot.mk Setoid.r (hd :: hd' :: tl) \u22a2 2 \u2264 length (Quot.mk Setoid.r (hd :: hd' :: tl)) ** simp [Nat.succ_le_succ_iff] ** Qed", + "informal": "" + }, + { + "formal": "AddSubgroup.exists_isLeast_pos ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) \u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b ** 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 ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) \u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b ** classical rcases Nat.findX this with \u27e8n, \u27e8x, hxH, hnx, hxn\u27e9, hmin\u27e9 ** case mk.intro.intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) n : \u2115 hmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a)) x : G hxH : x \u2208 \u2191H hnx : n \u2022 a < x hxn : x \u2264 (n + 1) \u2022 a \u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b ** by_contra hxmin ** case mk.intro.intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) n : \u2115 hmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a)) x : G hxH : x \u2208 \u2191H hnx : n \u2022 a < x hxn : x \u2264 (n + 1) \u2022 a hxmin : \u00ac\u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b \u22a2 False ** simp only [IsLeast, not_and, mem_setOf_eq, mem_lowerBounds, not_exists, not_forall,\n not_le] at hxmin ** case mk.intro.intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) n : \u2115 hmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a)) x : G hxH : x \u2208 \u2191H hnx : n \u2022 a < x hxn : x \u2264 (n + 1) \u2022 a hxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x \u22a2 False ** rcases hxmin x \u27e8hxH, (nsmul_nonneg h\u2080.le _).trans_lt hnx\u27e9 with \u27e8y, \u27e8hyH, hy\u2080\u27e9, hxy\u27e9 ** case mk.intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) n : \u2115 hmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a)) x : G hxH : x \u2208 \u2191H hnx : n \u2022 a < x hxn : x \u2264 (n + 1) \u2022 a hxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x y : G hxy : y < x hyH : y \u2208 H hy\u2080 : 0 < y \u22a2 False ** rcases hex y hy\u2080 with \u27e8m, hm\u27e9 ** case mk.intro.intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) n : \u2115 hmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a)) x : G hxH : x \u2208 \u2191H hnx : n \u2022 a < x hxn : x \u2264 (n + 1) \u2022 a hxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x y : G hxy : y < x hyH : y \u2208 H hy\u2080 : 0 < y m : \u2115 hm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a) \u22a2 False ** cases' lt_or_le m n with hmn hnm ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) g : G hg : g > 0 \u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) ** rcases existsUnique_add_zsmul_mem_Ico h\u2080 0 (g - a) with \u27e8m, \u27e8hm, hm'\u27e9, -\u27e9 ** case intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) g : G hg : g > 0 m : \u2124 hm : g - a \u2264 0 + m \u2022 a hm' : 0 + m \u2022 a < g - a + a \u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) ** simp only [zero_add, sub_le_iff_le_add, sub_add_cancel, \u2190 add_one_zsmul] at hm hm' ** case intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) g : G hg : g > 0 m : \u2124 hm : g \u2264 (m + 1) \u2022 a hm' : m \u2022 a < g \u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) ** lift m to \u2115 ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) g : G hg : g > 0 m : \u2124 hm : g \u2264 (m + 1) \u2022 a hm' : m \u2022 a < g \u22a2 0 \u2264 m ** rw [\u2190 Int.lt_add_one_iff, \u2190 zsmul_lt_zsmul_iff h\u2080, zero_zsmul] ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) g : G hg : g > 0 m : \u2124 hm : g \u2264 (m + 1) \u2022 a hm' : m \u2022 a < g \u22a2 0 < (m + 1) \u2022 a ** exact hg.trans_le hm ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) g : G hg : g > 0 m : \u2115 hm : g \u2264 (\u2191m + 1) \u2022 a hm' : \u2191m \u2022 a < g \u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) ** simp only [\u2190 Nat.cast_succ, coe_nat_zsmul] at hm hm' ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) g : G hg : g > 0 m : \u2115 hm : g \u2264 Nat.succ m \u2022 a hm' : m \u2022 a < g \u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) ** exact \u27e8m, hm', hm\u27e9 ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) \u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) ** rcases (bot_or_exists_ne_zero H).resolve_left hbot with \u27e8g, hgH, hg\u2080\u27e9 ** case intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) g : G hgH : g \u2208 H hg\u2080 : g \u2260 0 \u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) ** rcases hex |g| (abs_pos.2 hg\u2080) with \u27e8n, hn\u27e9 ** case intro.intro.intro G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) g : G hgH : g \u2208 H hg\u2080 : g \u2260 0 n : \u2115 hn : |g| \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) \u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) ** exact \u27e8n, _, (@abs_mem_iff (AddSubgroup G) G _ _).2 hgH, hn\u27e9 ** G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) \u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b ** rcases Nat.findX this with \u27e8n, \u27e8x, hxH, hnx, hxn\u27e9, hmin\u27e9 ** case mk.intro.intro.intro.intro.intro.intro.intro.intro.inl G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) n : \u2115 hmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a)) x : G hxH : x \u2208 \u2191H hnx : n \u2022 a < x hxn : x \u2264 (n + 1) \u2022 a hxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x y : G hxy : y < x hyH : y \u2208 H hy\u2080 : 0 < y m : \u2115 hm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a) hmn : m < n \u22a2 False ** exact hmin m hmn \u27e8y, hyH, hm\u27e9 ** case mk.intro.intro.intro.intro.intro.intro.intro.intro.inr G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) n : \u2115 hmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a)) x : G hxH : x \u2208 \u2191H hnx : n \u2022 a < x hxn : x \u2264 (n + 1) \u2022 a hxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x y : G hxy : y < x hyH : y \u2208 H hy\u2080 : 0 < y m : \u2115 hm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a) hnm : n \u2264 m \u22a2 False ** refine disjoint_left.1 hd (sub_mem hxH hyH) \u27e8sub_pos.2 hxy, sub_lt_iff_lt_add'.2 ?_\u27e9 ** case mk.intro.intro.intro.intro.intro.intro.intro.intro.inr G : Type u_1 inst\u271d\u00b9 : LinearOrderedAddCommGroup G inst\u271d : Archimedean G H : AddSubgroup G hbot : H \u2260 \u22a5 a : G h\u2080 : 0 < a hd : Disjoint (\u2191H) (Ioo 0 a) hex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) this : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) n : \u2115 hmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a)) x : G hxH : x \u2208 \u2191H hnx : n \u2022 a < x hxn : x \u2264 (n + 1) \u2022 a hxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x y : G hxy : y < x hyH : y \u2208 H hy\u2080 : 0 < y m : \u2115 hm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a) hnm : n \u2264 m \u22a2 x < y + a ** 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 _ ** Qed", + "informal": "" + }, + { + "formal": "Multiset.Nodup.toFinset_inj ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t l l' : Multiset \u03b1 hl : Nodup l hl' : Nodup l' h : toFinset l = toFinset l' \u22a2 l = l' ** simpa [\u2190 toFinset_eq hl, \u2190 toFinset_eq hl'] using h ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Field.charP_iff_model_fieldOfChar ** p : \u2115 K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K \u22a2 K \u22a8 Theory.fieldOfChar p \u2194 CharP K p ** simp only [Theory.fieldOfChar, Theory.model_union_iff,\n (show (Theory.field.Model K) by infer_instance), true_and] ** p : \u2115 K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K \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 CharP K p ** split_ifs with hp0 hp ** p : \u2115 K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K \u22a2 K \u22a8 Theory.field ** infer_instance ** case pos p : \u2115 K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K hp0 : p = 0 \u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K p ** subst hp0 ** case pos K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K \u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K 0 ** 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] ** case pos K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K \u22a2 CharZero K \u2194 CharP K 0 ** exact \u27e8fun _ => CharP.ofCharZero _, fun _ => CharP.charP_to_charZero K\u27e9 ** case pos p : \u2115 K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K hp0 : \u00acp = 0 hp : Nat.Prime p \u22a2 K \u22a8 {eqZero p} \u2194 CharP K p ** simp only [Theory.model_iff, Set.mem_singleton_iff, Sentence.Realize, forall_eq,\n realize_eqZero, \u2190 CharP.charP_iff_prime_eq_zero hp] ** case neg p : \u2115 K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K hp0 : \u00acp = 0 hp : \u00acNat.Prime p \u22a2 K \u22a8 {\u22a5} \u2194 CharP K p ** simp only [Theory.model_iff, Set.mem_singleton_iff, Sentence.Realize,\n forall_eq, Formula.realize_bot, false_iff] ** case neg p : \u2115 K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K hp0 : \u00acp = 0 hp : \u00acNat.Prime p \u22a2 \u00acCharP K p ** intro H ** case neg p : \u2115 K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : CompatibleRing K hp0 : \u00acp = 0 hp : \u00acNat.Prime p H : CharP K p \u22a2 False ** cases (CharP.char_is_prime_or_zero K p) <;> simp_all ** Qed", + "informal": "" + }, + { + "formal": "contDiff_clm_apply_iff ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u00b9\u00b9 : NormedAddCommGroup D inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup X inst\u271d\u00b2 : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f\u271d f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m n\u271d : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F inst\u271d\u00b9 : CompleteSpace \ud835\udd5c n : \u2115\u221e f : E \u2192 F \u2192L[\ud835\udd5c] G inst\u271d : FiniteDimensional \ud835\udd5c F \u22a2 ContDiff \ud835\udd5c n f \u2194 \u2200 (y : F), ContDiff \ud835\udd5c n fun x => \u2191(f x) y ** simp_rw [\u2190 contDiffOn_univ, contDiffOn_clm_apply] ** Qed", + "informal": "" + }, + { + "formal": "one_div_lt_of_neg ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedField \u03b1 a b c d : \u03b1 n : \u2124 ha : a < 0 hb : b < 0 \u22a2 1 / a < b \u2194 1 / b < a ** simpa using inv_lt_of_neg ha hb ** Qed", + "informal": "" + }, + { + "formal": "List.map_eq_foldr ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 f : \u03b1 \u2192 \u03b2 l : List \u03b1 \u22a2 map f l = foldr (fun a bs => f a :: bs) [] l ** induction l <;> simp [*] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.WidePullback.hom_eq_lift ** J : Type w C : Type u inst\u271d\u00b2 : Category.{v, u} C D : Type ?u.185654 inst\u271d\u00b9 : Category.{v\u2082, ?u.185654} D B : D objs : J \u2192 D arrows : (j : J) \u2192 objs j \u27f6 B inst\u271d : HasWidePullback B objs arrows X : D f : X \u27f6 B fs : (j : J) \u2192 X \u27f6 objs j w : \u2200 (j : J), fs j \u226b arrows j = f g : X \u27f6 widePullback B (fun j => objs j) arrows \u22a2 \u2200 (j : J), (fun j => g \u226b \u03c0 arrows j) j \u226b arrows j = g \u226b base arrows ** aesop_cat ** J : Type w C : Type u inst\u271d\u00b2 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9 : Category.{v\u2082, u_1} D B : D objs : J \u2192 D arrows : (j : J) \u2192 objs j \u27f6 B inst\u271d : HasWidePullback B objs arrows X : D f : X \u27f6 B fs : (j : J) \u2192 X \u27f6 objs j w : \u2200 (j : J), fs j \u226b arrows j = f g : X \u27f6 widePullback B (fun j => objs j) arrows \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) ** apply eq_lift_of_comp_eq ** case a J : Type w C : Type u inst\u271d\u00b2 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9 : Category.{v\u2082, u_1} D B : D objs : J \u2192 D arrows : (j : J) \u2192 objs j \u27f6 B inst\u271d : HasWidePullback B objs arrows X : D f : X \u27f6 B fs : (j : J) \u2192 X \u27f6 objs j w : \u2200 (j : J), fs j \u226b arrows j = f g : X \u27f6 widePullback B (fun j => objs j) arrows \u22a2 \u2200 (j : J), g \u226b \u03c0 arrows j = g \u226b \u03c0 arrows j case a J : Type w C : Type u inst\u271d\u00b2 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9 : Category.{v\u2082, u_1} D B : D objs : J \u2192 D arrows : (j : J) \u2192 objs j \u27f6 B inst\u271d : HasWidePullback B objs arrows X : D f : X \u27f6 B fs : (j : J) \u2192 X \u27f6 objs j w : \u2200 (j : J), fs j \u226b arrows j = f g : X \u27f6 widePullback B (fun j => objs j) arrows \u22a2 g \u226b base arrows = g \u226b base arrows ** aesop_cat ** case a J : Type w C : Type u inst\u271d\u00b2 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9 : Category.{v\u2082, u_1} D B : D objs : J \u2192 D arrows : (j : J) \u2192 objs j \u27f6 B inst\u271d : HasWidePullback B objs arrows X : D f : X \u27f6 B fs : (j : J) \u2192 X \u27f6 objs j w : \u2200 (j : J), fs j \u226b arrows j = f g : X \u27f6 widePullback B (fun j => objs j) arrows \u22a2 g \u226b base arrows = g \u226b base arrows ** rfl ** Qed", + "informal": "" + }, + { + "formal": "IsAntichain.image ** \u03b1 : Type u_1 \u03b2 : Type u_2 r r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop s t : Set \u03b1 a b : \u03b1 hs : IsAntichain r s f : \u03b1 \u2192 \u03b2 h : \u2200 \u2983a b : \u03b1\u2984, r' (f a) (f b) \u2192 r a b \u22a2 IsAntichain r' (f '' s) ** rintro _ \u27e8b, hb, rfl\u27e9 _ \u27e8c, hc, rfl\u27e9 hbc hr ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 r r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop r' : \u03b2 \u2192 \u03b2 \u2192 Prop s t : Set \u03b1 a b\u271d : \u03b1 hs : IsAntichain r s f : \u03b1 \u2192 \u03b2 h : \u2200 \u2983a b : \u03b1\u2984, r' (f a) (f b) \u2192 r a b b : \u03b1 hb : b \u2208 s c : \u03b1 hc : c \u2208 s hbc : f b \u2260 f c hr : r' (f b) (f c) \u22a2 False ** exact hs hb hc (ne_of_apply_ne _ hbc) (h hr) ** Qed", + "informal": "" + }, + { + "formal": "dist_triangle4_left ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d : PseudoMetricSpace \u03b1 x\u2081 y\u2081 x\u2082 y\u2082 : \u03b1 \u22a2 dist x\u2082 y\u2082 \u2264 dist x\u2081 y\u2081 + (dist x\u2081 x\u2082 + dist y\u2081 y\u2082) ** rw [add_left_comm, dist_comm x\u2081, \u2190 add_assoc] ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d : PseudoMetricSpace \u03b1 x\u2081 y\u2081 x\u2082 y\u2082 : \u03b1 \u22a2 dist x\u2082 y\u2082 \u2264 dist x\u2082 x\u2081 + dist x\u2081 y\u2081 + dist y\u2081 y\u2082 ** apply dist_triangle4 ** Qed", + "informal": "" + }, + { + "formal": "tendsto_of_uniformContinuous_subtype ** \u03b1 : Type ua \u03b2 : Type ub \u03b3 : Type uc \u03b4 : Type ud \u03b9 : Sort u_1 inst\u271d\u00b9 : UniformSpace \u03b1 inst\u271d : UniformSpace \u03b2 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 a : \u03b1 hf : UniformContinuous fun x => f \u2191x ha : s \u2208 \ud835\udcdd a \u22a2 Tendsto f (\ud835\udcdd a) (\ud835\udcdd (f a)) ** rw [(@map_nhds_subtype_coe_eq_nhds \u03b1 _ s a (mem_of_mem_nhds ha) ha).symm] ** \u03b1 : Type ua \u03b2 : Type ub \u03b3 : Type uc \u03b4 : Type ud \u03b9 : Sort u_1 inst\u271d\u00b9 : UniformSpace \u03b1 inst\u271d : UniformSpace \u03b2 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 a : \u03b1 hf : UniformContinuous fun x => f \u2191x ha : s \u2208 \ud835\udcdd a \u22a2 Tendsto f (map Subtype.val (\ud835\udcdd { val := a, property := (_ : a \u2208 s) })) (\ud835\udcdd (f a)) ** exact tendsto_map' hf.rst.imntinuous.continuousAt ** Qed", + "informal": "" + }, + { + "formal": "EuclideanSpace.inner_single_left ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 i : \u03b9 a : \ud835\udd5c v : EuclideanSpace \ud835\udd5c \u03b9 \u22a2 inner (single i a) v = \u2191(starRingEnd \ud835\udd5c) a * v i ** simp [apply_ite conj] ** Qed", + "informal": "" + }, + { + "formal": "divp_eq_one_iff_eq ** \u03b1 : Type u inst\u271d : Monoid \u03b1 a\u271d b c a : \u03b1 u : \u03b1\u02e3 \u22a2 a /\u209a u * \u2191u = 1 * \u2191u \u2194 a = \u2191u ** rw [divp_mul_cancel, one_mul] ** Qed", + "informal": "" + }, + { + "formal": "round_sub_int ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedRing \u03b1 inst\u271d : FloorRing \u03b1 x : \u03b1 y : \u2124 \u22a2 round (x - \u2191y) = round x - y ** rw [sub_eq_add_neg] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedRing \u03b1 inst\u271d : FloorRing \u03b1 x : \u03b1 y : \u2124 \u22a2 round (x + -\u2191y) = round x - y ** norm_cast ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedRing \u03b1 inst\u271d : FloorRing \u03b1 x : \u03b1 y : \u2124 \u22a2 round (x + \u2191(-y)) = round x - y ** rw [round_add_int, sub_eq_add_neg] ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_range_add ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 f : \u2115 \u2192 \u03b2 n m : \u2115 \u22a2 \u220f x in range (n + m), f x = (\u220f x in range n, f x) * \u220f x in range m, f (n + x) ** induction' m with m hm ** case zero \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 f : \u2115 \u2192 \u03b2 n : \u2115 \u22a2 \u220f x in range (n + Nat.zero), f x = (\u220f x in range n, f x) * \u220f x in range Nat.zero, f (n + x) ** simp ** case succ \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 f : \u2115 \u2192 \u03b2 n m : \u2115 hm : \u220f x in range (n + m), f x = (\u220f x in range n, f x) * \u220f x in range m, f (n + x) \u22a2 \u220f x in range (n + Nat.succ m), f x = (\u220f x in range n, f x) * \u220f x in range (Nat.succ m), f (n + x) ** erw [Nat.add_succ, prod_range_succ, prod_range_succ, hm, mul_assoc] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.toNNReal_top_mul ** \u03b1 : Type u_1 \u03b2 : Type u_2 a\u271d b c d : \u211d\u22650\u221e r p q : \u211d\u22650 a : \u211d\u22650\u221e \u22a2 ENNReal.toNNReal (\u22a4 * a) = 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "isLUB_Ico ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d\u00b3 : Preorder \u03b1 inst\u271d\u00b2 : Preorder \u03b2 s t : Set \u03b1 a\u271d b\u271d : \u03b1 inst\u271d\u00b9 : SemilatticeInf \u03b3 inst\u271d : DenselyOrdered \u03b3 a b : \u03b3 hab : a < b \u22a2 IsLUB (Ico a b) b ** simpa only [dual_Ioc] using isGLB_Ioc hab.dual ** Qed", + "informal": "" + }, + { + "formal": "Computable.map_decode_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u22a2 (Computable\u2082 fun a n => Option.map (f a) (decode n)) \u2194 Computable\u2082 f ** convert (bind_decode_iff (f := fun a => Option.some \u2218 f a)).trans option_some_iff ** case h.e'_1.h.e'_7.h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 x\u271d\u00b9 : \u03b1 x\u271d : \u2115 \u22a2 Option.map (f x\u271d\u00b9) (decode x\u271d) = Option.bind (decode x\u271d) (Option.some \u2218 f x\u271d\u00b9) ** apply Option.map_eq_bind ** Qed", + "informal": "" + }, + { + "formal": "Monotone.disjointed_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : GeneralizedBooleanAlgebra \u03b1 f : \u2115 \u2192 \u03b1 hf : Monotone f n : \u2115 \u22a2 disjointed f (n + 1) = f (n + 1) \\ f n ** rw [disjointed_succ, hf.partialSups_eq] ** Qed", + "informal": "" + }, + { + "formal": "Order.not_isSuccLimit_succ_of_not_isMax ** \u03b1 : Type u_1 inst\u271d\u00b9 : Preorder \u03b1 a : \u03b1 inst\u271d : SuccOrder \u03b1 ha : \u00acIsMax a \u22a2 \u00acIsSuccLimit (succ a) ** contrapose! ha ** \u03b1 : Type u_1 inst\u271d\u00b9 : Preorder \u03b1 a : \u03b1 inst\u271d : SuccOrder \u03b1 ha : IsSuccLimit (succ a) \u22a2 IsMax a ** exact ha.isMax ** Qed", + "informal": "" + }, + { + "formal": "Finset.sup'_map ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : SemilatticeSup \u03b1 s\u271d : Finset \u03b2 H : Finset.Nonempty s\u271d f\u271d : \u03b2 \u2192 \u03b1 s : Finset \u03b3 f : \u03b3 \u21aa \u03b2 g : \u03b2 \u2192 \u03b1 hs : Finset.Nonempty (map f s) hs' : optParam (Finset.Nonempty s) (_ : Finset.Nonempty s) \u22a2 sup' (map f s) hs g = sup' s hs' (g \u2218 \u2191f) ** rw [\u2190 WithBot.coe_eq_coe, coe_sup', sup_map, coe_sup'] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : SemilatticeSup \u03b1 s\u271d : Finset \u03b2 H : Finset.Nonempty s\u271d f\u271d : \u03b2 \u2192 \u03b1 s : Finset \u03b3 f : \u03b3 \u21aa \u03b2 g : \u03b2 \u2192 \u03b1 hs : Finset.Nonempty (map f s) hs' : optParam (Finset.Nonempty s) (_ : Finset.Nonempty s) \u22a2 sup s ((WithBot.some \u2218 g) \u2218 \u2191f) = sup s (WithBot.some \u2218 g \u2218 \u2191f) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Submodule.comap_unop_one ** \u03b9 : Sort u\u03b9 R : Type u inst\u271d\u00b2 : CommSemiring R A : Type v inst\u271d\u00b9 : Semiring A inst\u271d : Algebra R A S T : Set A M N P Q : Submodule R A m n : A \u22a2 comap (\u2191(LinearEquiv.symm (opLinearEquiv R))) 1 = 1 ** rw [\u2190 map_equiv_eq_comap_symm, map_op_one] ** Qed", + "informal": "" + }, + { + "formal": "Nat.succ_mul_choose_eq ** \u22a2 succ 0 * choose 0 0 = choose (succ 0) (succ 0) * succ 0 ** decide ** k : \u2115 \u22a2 succ 0 * choose 0 (k + 1) = choose (succ 0) (succ (k + 1)) * succ (k + 1) ** simp [choose] ** n : \u2115 \u22a2 succ (n + 1) * choose (n + 1) 0 = choose (succ (n + 1)) (succ 0) * succ 0 ** simp [choose, mul_succ, succ_eq_add_one, add_comm] ** n k : \u2115 \u22a2 succ (n + 1) * choose (n + 1) (k + 1) = choose (succ (n + 1)) (succ (k + 1)) * succ (k + 1) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Real.rpow_eq_nhds_of_pos ** p : \u211d \u00d7 \u211d hp_fst : 0 < p.1 \u22a2 (fun x => x.1 ^ x.2) =\u1da0[\ud835\udcdd p] fun x => rexp (log x.1 * x.2) ** suffices : \u2200\u1da0 x : \u211d \u00d7 \u211d in \ud835\udcdd p, 0 < x.1 ** p : \u211d \u00d7 \u211d hp_fst : 0 < p.1 this : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1 \u22a2 (fun x => x.1 ^ x.2) =\u1da0[\ud835\udcdd p] fun x => rexp (log x.1 * x.2) case this p : \u211d \u00d7 \u211d hp_fst : 0 < p.1 \u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1 ** exact\n this.mono fun x hx => by\n dsimp only\n rw [rpow_def_of_pos hx] ** case this p : \u211d \u00d7 \u211d hp_fst : 0 < p.1 \u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1 ** exact IsOpen.eventually_mem (isOpen_lt continuous_const continuous_fst) hp_fst ** p : \u211d \u00d7 \u211d hp_fst : 0 < p.1 this : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1 x : \u211d \u00d7 \u211d hx : 0 < x.1 \u22a2 (fun x => x.1 ^ x.2) x = (fun x => rexp (log x.1 * x.2)) x ** dsimp only ** p : \u211d \u00d7 \u211d hp_fst : 0 < p.1 this : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1 x : \u211d \u00d7 \u211d hx : 0 < x.1 \u22a2 x.1 ^ x.2 = rexp (log x.1 * x.2) ** rw [rpow_def_of_pos hx] ** Qed", + "informal": "" + }, + { + "formal": "Set.mulIndicator_comp_of_one ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b9 : One M inst\u271d : One N s t : Set \u03b1 f g\u271d : \u03b1 \u2192 M a : \u03b1 g : M \u2192 N hg : g 1 = 1 \u22a2 mulIndicator s (g \u2218 f) = g \u2218 mulIndicator s f ** funext ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b9 : One M inst\u271d : One N s t : Set \u03b1 f g\u271d : \u03b1 \u2192 M a : \u03b1 g : M \u2192 N hg : g 1 = 1 x\u271d : \u03b1 \u22a2 mulIndicator s (g \u2218 f) x\u271d = (g \u2218 mulIndicator s f) x\u271d ** simp only [mulIndicator] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b9 : One M inst\u271d : One N s t : Set \u03b1 f g\u271d : \u03b1 \u2192 M a : \u03b1 g : M \u2192 N hg : g 1 = 1 x\u271d : \u03b1 \u22a2 (if x\u271d \u2208 s then (g \u2218 f) x\u271d else 1) = (g \u2218 mulIndicator s f) x\u271d ** split_ifs <;> simp [*] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.exp_transpose ** \ud835\udd42 : Type u_1 m : Type u_2 n : Type u_3 p : Type u_4 n' : m \u2192 Type u_5 \ud835\udd38 : Type u_6 inst\u271d\u2077 : Fintype m inst\u271d\u2076 : DecidableEq m inst\u271d\u2075 : Field \ud835\udd42 inst\u271d\u2074 : CommRing \ud835\udd38 inst\u271d\u00b3 : TopologicalSpace \ud835\udd38 inst\u271d\u00b2 : TopologicalRing \ud835\udd38 inst\u271d\u00b9 : Algebra \ud835\udd42 \ud835\udd38 inst\u271d : T2Space \ud835\udd38 A : Matrix m m \ud835\udd38 \u22a2 exp \ud835\udd42 A\u1d40 = (exp \ud835\udd42 A)\u1d40 ** simp_rw [exp_eq_tsum, transpose_tsum, transpose_smul, transpose_pow] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.exists_ne_odd_degree_of_exists_odd_degree ** V : Type u G : SimpleGraph V inst\u271d\u00b9 : Fintype V inst\u271d : DecidableRel G.Adj v : V h : Odd (degree G v) \u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w) ** haveI := Classical.decEq V ** V : Type u G : SimpleGraph V inst\u271d\u00b9 : Fintype V inst\u271d : DecidableRel G.Adj v : V h : Odd (degree G v) this : DecidableEq V \u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w) ** rcases G.odd_card_odd_degree_vertices_ne v h with \u27e8k, hg\u27e9 ** case intro V : Type u G : SimpleGraph V inst\u271d\u00b9 : Fintype V inst\u271d : DecidableRel G.Adj v : V h : Odd (degree G v) this : DecidableEq V k : \u2115 hg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1 \u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w) ** have hg' : (filter (fun w : V => w \u2260 v \u2227 Odd (G.degree w)) univ).card > 0 := by\n rw [hg]\n apply Nat.succ_pos ** case intro V : Type u G : SimpleGraph V inst\u271d\u00b9 : Fintype V inst\u271d : DecidableRel G.Adj v : V h : Odd (degree G v) this : DecidableEq V k : \u2115 hg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1 hg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0 \u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w) ** rcases card_pos.mp hg' with \u27e8w, hw\u27e9 ** case intro.intro V : Type u G : SimpleGraph V inst\u271d\u00b9 : Fintype V inst\u271d : DecidableRel G.Adj v : V h : Odd (degree G v) this : DecidableEq V k : \u2115 hg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1 hg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0 w : V hw : w \u2208 filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ \u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w) ** simp only [true_and_iff, mem_filter, mem_univ, Ne.def] at hw ** case intro.intro V : Type u G : SimpleGraph V inst\u271d\u00b9 : Fintype V inst\u271d : DecidableRel G.Adj v : V h : Odd (degree G v) this : DecidableEq V k : \u2115 hg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1 hg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0 w : V hw : \u00acw = v \u2227 Odd (degree G w) \u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w) ** exact \u27e8w, hw\u27e9 ** V : Type u G : SimpleGraph V inst\u271d\u00b9 : Fintype V inst\u271d : DecidableRel G.Adj v : V h : Odd (degree G v) this : DecidableEq V k : \u2115 hg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1 \u22a2 card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0 ** rw [hg] ** V : Type u G : SimpleGraph V inst\u271d\u00b9 : Fintype V inst\u271d : DecidableRel G.Adj v : V h : Odd (degree G v) this : DecidableEq V k : \u2115 hg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1 \u22a2 2 * k + 1 > 0 ** apply Nat.succ_pos ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.MonoidalCategory.tensor_inv_hom_id' ** C\u271d : Type u \ud835\udc9e : Category.{v, u} C\u271d inst\u271d\u00b3 : MonoidalCategory C\u271d C : Type u inst\u271d\u00b2 : Category.{v, u} C inst\u271d\u00b9 : MonoidalCategory C U V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C f : V \u27f6 W inst\u271d : IsIso f g : X \u27f6 Y h : Y \u27f6 Z \u22a2 (g \u2297 inv f) \u226b (h \u2297 f) = (g \u2297 \ud835\udfd9 W) \u226b (h \u2297 \ud835\udfd9 W) ** rw [\u2190 tensor_comp, IsIso.inv_hom_id, comp_tensor_id] ** Qed", + "informal": "" + }, + { + "formal": "Asymptotics.isBigOWith_inv ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 G : Type u_5 E' : Type u_6 F' : Type u_7 G' : Type u_8 E'' : Type u_9 F'' : Type u_10 G'' : Type u_11 E''' : Type u_12 R : Type u_13 R' : Type u_14 \ud835\udd5c : Type u_15 \ud835\udd5c' : Type u_16 inst\u271d\u00b9\u00b3 : Norm E inst\u271d\u00b9\u00b2 : Norm F inst\u271d\u00b9\u00b9 : Norm G inst\u271d\u00b9\u2070 : SeminormedAddCommGroup E' inst\u271d\u2079 : SeminormedAddCommGroup F' inst\u271d\u2078 : SeminormedAddCommGroup G' inst\u271d\u2077 : NormedAddCommGroup E'' inst\u271d\u2076 : NormedAddCommGroup F'' inst\u271d\u2075 : NormedAddCommGroup G'' inst\u271d\u2074 : SeminormedRing R inst\u271d\u00b3 : SeminormedAddGroup E''' inst\u271d\u00b2 : SeminormedRing R' inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedField \ud835\udd5c' c c' c\u2081 c\u2082 : \u211d f : \u03b1 \u2192 E g : \u03b1 \u2192 F k : \u03b1 \u2192 G f' : \u03b1 \u2192 E' g' : \u03b1 \u2192 F' k' : \u03b1 \u2192 G' f'' : \u03b1 \u2192 E'' g'' : \u03b1 \u2192 F'' k'' : \u03b1 \u2192 G'' l l' : Filter \u03b1 hc : 0 < c \u22a2 IsBigOWith c\u207b\u00b9 l f g \u2194 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016 ** simp only [IsBigOWith_def, \u2190 div_eq_inv_mul, le_div_iff' hc] ** Qed", + "informal": "" + }, + { + "formal": "Set.mem_ordConnectedComponent_comm ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x y z : \u03b1 \u22a2 y \u2208 ordConnectedComponent s x \u2194 x \u2208 ordConnectedComponent s y ** rw [mem_ordConnectedComponent, mem_ordConnectedComponent, uIcc_comm] ** Qed", + "informal": "" + }, + { + "formal": "Dioph.sub_dioph ** \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g x y z : \u2115 o : y = x + z \u2228 y \u2264 z \u2227 x = 0 \u22a2 y - z = x ** rcases o with (ae | \u27e8yz, x0\u27e9) ** case inl \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g x y z : \u2115 ae : y = x + z \u22a2 y - z = x ** rw [ae, add_tsub_cancel_right] ** case inr.intro \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g x y z : \u2115 yz : y \u2264 z x0 : x = 0 \u22a2 y - z = x ** rw [x0, tsub_eq_zero_iff_le.mpr yz] ** \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g x y z : \u2115 \u22a2 y - z = x \u2192 y = x + z \u2228 y \u2264 z \u2227 x = 0 ** rintro rfl ** \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g y z : \u2115 \u22a2 y = y - z + z \u2228 y \u2264 z \u2227 y - z = 0 ** cases' le_total y z with yz zy ** case inl \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g y z : \u2115 yz : y \u2264 z \u22a2 y = y - z + z \u2228 y \u2264 z \u2227 y - z = 0 ** exact Or.inr \u27e8yz, tsub_eq_zero_iff_le.mpr yz\u27e9 ** case inr \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g y z : \u2115 zy : z \u2264 y \u22a2 y = y - z + z \u2228 y \u2264 z \u2227 y - z = 0 ** exact Or.inl (tsub_add_cancel_of_le zy).symm ** Qed", + "informal": "" + }, + { + "formal": "Filter.mem_coclosed_compact' ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s t : Set \u03b1 \u22a2 s \u2208 coclosedCompact \u03b1 \u2194 \u2203 t, IsClosed t \u2227 IsCompact t \u2227 s\u1d9c \u2286 t ** simp only [mem_coclosedCompact, compl_subset_comm] ** Qed", + "informal": "" + }, + { + "formal": "equicontinuousAt_finite ** \u03b9 : Type u_1 \u03ba : Type u_2 X : Type u_3 Y : Type u_4 Z : Type u_5 \u03b1 : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 \ud835\udcd5 : Type u_9 inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : TopologicalSpace Y inst\u271d\u2074 : TopologicalSpace Z inst\u271d\u00b3 : UniformSpace \u03b1 inst\u271d\u00b2 : UniformSpace \u03b2 inst\u271d\u00b9 : UniformSpace \u03b3 inst\u271d : Finite \u03b9 F : \u03b9 \u2192 X \u2192 \u03b1 x\u2080 : X \u22a2 EquicontinuousAt F x\u2080 \u2194 \u2200 (i : \u03b9), ContinuousAt (F i) x\u2080 ** simp [EquicontinuousAt, ContinuousAt, (nhds_basis_uniformity' (\ud835\udce4 \u03b1).basis_sets).tendsto_right_iff,\n UniformSpace.ball, @forall_swap _ \u03b9] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.mem_Ici ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrderTop \u03b1 a x : \u03b1 \u22a2 x \u2208 Ici a \u2194 a \u2264 x ** rw [Ici, \u2190 Finset.mem_def, Finset.mem_Ici] ** Qed", + "informal": "" + }, + { + "formal": "contDiffWithinAt_zero ** \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s \u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x \u2194 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u) ** constructor ** case mp \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s \u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x \u2192 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u) ** intro h ** case mp \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s h : ContDiffWithinAt \ud835\udd5c 0 f s x \u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u) ** obtain \u27e8u, H, p, hp\u27e9 := h 0 le_rfl ** case mp.intro.intro.intro \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u\u271d : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s h : ContDiffWithinAt \ud835\udd5c 0 f s x u : Set E H : u \u2208 \ud835\udcdd[insert x s] x p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hp : HasFTaylorSeriesUpToOn (\u21910) f p u \u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u) ** refine' \u27e8u, _, _\u27e9 ** case mp.intro.intro.intro.refine'_1 \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u\u271d : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s h : ContDiffWithinAt \ud835\udd5c 0 f s x u : Set E H : u \u2208 \ud835\udcdd[insert x s] x p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hp : HasFTaylorSeriesUpToOn (\u21910) f p u \u22a2 u \u2208 \ud835\udcdd[s] x ** simpa [hx] using H ** case mp.intro.intro.intro.refine'_2 \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u\u271d : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s h : ContDiffWithinAt \ud835\udd5c 0 f s x u : Set E H : u \u2208 \ud835\udcdd[insert x s] x p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hp : HasFTaylorSeriesUpToOn (\u21910) f p u \u22a2 ContinuousOn f (s \u2229 u) ** simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp ** case mp.intro.intro.intro.refine'_2 \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u\u271d : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s h : ContDiffWithinAt \ud835\udd5c 0 f s x u : Set E H : u \u2208 \ud835\udcdd[insert x s] x p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hp : ContinuousOn f u \u2227 \u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x \u22a2 ContinuousOn f (s \u2229 u) ** exact hp.1.mono (inter_subset_right s u) ** case mpr \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s \u22a2 (\u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)) \u2192 ContDiffWithinAt \ud835\udd5c 0 f s x ** rintro \u27e8u, H, hu\u27e9 ** case mpr.intro.intro \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u\u271d : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s u : Set E H : u \u2208 \ud835\udcdd[s] x hu : ContinuousOn f (s \u2229 u) \u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x ** rw [\u2190 contDiffWithinAt_inter' H] ** case mpr.intro.intro \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u\u271d : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s u : Set E H : u \u2208 \ud835\udcdd[s] x hu : ContinuousOn f (s \u2229 u) \u22a2 ContDiffWithinAt \ud835\udd5c 0 f (s \u2229 u) x ** have h' : x \u2208 s \u2229 u := \u27e8hx, mem_of_mem_nhdsWithin hx H\u27e9 ** case mpr.intro.intro \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u\u271d : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hx : x \u2208 s u : Set E H : u \u2208 \ud835\udcdd[s] x hu : ContinuousOn f (s \u2229 u) h' : x \u2208 s \u2229 u \u22a2 ContDiffWithinAt \ud835\udd5c 0 f (s \u2229 u) x ** exact (contDiffOn_zero.mpr hu).contDiffWithinAt h' ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.nmul_nadd_le\u2083' ** a b c d a' b' c' : Ordinal.{u} ha : a' \u2264 a hb : b' \u2264 b hc : c' \u2264 c \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 a \u2a33 (b \u2a33 c) \u266f a' \u2a33 (b' \u2a33 c) \u266f a' \u2a33 (b \u2a33 c') \u266f a \u2a33 (b' \u2a33 c') ** simp only [nmul_comm _ (_ \u2a33 _)] ** case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' \u2264 a hb : b' \u2264 b hc : c' \u2264 c \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' ** simp only [nadd_eq_add, NatOrdinal.toOrdinal_toNatOrdinal] ** case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' \u2264 a hb : b' \u2264 b hc : c' \u2264 c \u22a2 \u2191toOrdinal (\u2191toNatOrdinal (b \u2a33 c \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c \u2a33 a') + \u2191toNatOrdinal (b \u2a33 c' \u2a33 a') + \u2191toNatOrdinal (b' \u2a33 c' \u2a33 a)) = \u2191toOrdinal (\u2191toNatOrdinal (b \u2a33 c \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c' \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c \u2a33 a') + \u2191toNatOrdinal (b \u2a33 c' \u2a33 a')) ** abel_nf ** Qed", + "informal": "" + }, + { + "formal": "padicValNat.prime_pow ** p a b : \u2115 hp : Fact (Nat.Prime p) n : \u2115 \u22a2 padicValNat p (p ^ n) = n ** rw [padicValNat.pow _ (@Fact.out p.Prime).ne_zero, padicValNat_self, mul_one] ** Qed", + "informal": "" + }, + { + "formal": "FdRep.char_one ** k : Type u inst\u271d\u00b9 : Field k G : Type u inst\u271d : Monoid G V : FdRep k G \u22a2 character V 1 = \u2191(finrank k (CoeSort.coe V)) ** simp only [character, map_one, trace_one] ** Qed", + "informal": "" + }, + { + "formal": "lipschitzWith_iff_norm_div_le ** \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b2 : SeminormedGroup E inst\u271d\u00b9 : SeminormedGroup F inst\u271d : SeminormedGroup G s : Set E a a\u2081 a\u2082 b b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d f : E \u2192 F C : \u211d\u22650 \u22a2 LipschitzWith C f \u2194 \u2200 (x y : E), \u2016f x / f y\u2016 \u2264 \u2191C * \u2016x / y\u2016 ** simp only [lipschitzWith_iff_dist_le_mul, dist_eq_norm_div] ** Qed", + "informal": "" + }, + { + "formal": "IsPrimitiveRoot.pow_sub_one_eq ** M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R h\u03b6 : IsPrimitiveRoot \u03b6 k hk : 1 < k \u22a2 \u03b6 ^ Nat.pred k = -\u2211 i in range (Nat.pred k), \u03b6 ^ i ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Antitone.mulIndicator_eventuallyEq_iInter ** \u03b1 : Type u_1 \u03b2 : Type u_2 M : Type u_3 E : Type u_4 \u03b9 : Type u_5 inst\u271d\u00b9 : Preorder \u03b9 inst\u271d : One \u03b2 s : \u03b9 \u2192 Set \u03b1 hs : Antitone s f : \u03b1 \u2192 \u03b2 a : \u03b1 \u22a2 (fun i => mulIndicator (s i) f a) =\u1da0[atTop] fun x => mulIndicator (\u22c2 i, s i) f a ** classical exact hs.piecewise_eventually_eq_iInter f 1 a ** \u03b1 : Type u_1 \u03b2 : Type u_2 M : Type u_3 E : Type u_4 \u03b9 : Type u_5 inst\u271d\u00b9 : Preorder \u03b9 inst\u271d : One \u03b2 s : \u03b9 \u2192 Set \u03b1 hs : Antitone s f : \u03b1 \u2192 \u03b2 a : \u03b1 \u22a2 (fun i => mulIndicator (s i) f a) =\u1da0[atTop] fun x => mulIndicator (\u22c2 i, s i) f a ** exact hs.piecewise_eventually_eq_iInter f 1 a ** Qed", + "informal": "" + }, + { + "formal": "WithTop.add_coe_eq_top_iff ** \u03b1 : Type u \u03b2 : Type v inst\u271d : Add \u03b1 a b c d : WithTop \u03b1 x\u271d y\u271d : \u03b1 x : WithTop \u03b1 y : \u03b1 \u22a2 x + \u2191y = \u22a4 \u2194 x = \u22a4 ** induction x using WithTop.recTopCoe <;> simp [\u2190 coe_add] ** Qed", + "informal": "" + }, + { + "formal": "Nat.one_lt_cast ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : AddCommMonoidWithOne \u03b1 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 inst\u271d\u00b9 : ZeroLEOneClass \u03b1 inst\u271d : CharZero \u03b1 m n : \u2115 \u22a2 1 < \u2191n \u2194 1 < n ** rw [\u2190 cast_one, cast_lt] ** Qed", + "informal": "" + }, + { + "formal": "Set.Ici_disjoint_Iic ** \u03b9 : Sort u \u03b1 : Type v \u03b2 : Type w inst\u271d : Preorder \u03b1 a b c : \u03b1 \u22a2 Disjoint (Ici a) (Iic b) \u2194 \u00aca \u2264 b ** rw [Set.disjoint_iff_inter_eq_empty, Ici_inter_Iic, Icc_eq_empty_iff] ** Qed", + "informal": "" + }, + { + "formal": "FdRep.Iso.conj_\u03c1 ** k G : Type u inst\u271d\u00b9 : Field k inst\u271d : Monoid G V W : FdRep k G i : V \u2245 W g : G \u22a2 \u2191(\u03c1 W) g = \u2191(LinearEquiv.conj (isoToLinearEquiv i)) (\u2191(\u03c1 V) g) ** erw [FdRep.isoToLinearEquiv, \u2190 FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply] ** k G : Type u inst\u271d\u00b9 : Field k inst\u271d : Monoid G V W : FdRep k G i : V \u2245 W g : G \u22a2 \u2191(\u03c1 W) g = ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).inv \u226b \u2191(\u03c1 V) g \u226b ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom ** rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)] ** k G : Type u inst\u271d\u00b9 : Field k inst\u271d : Monoid G V W : FdRep k G i : V \u2245 W g : G \u22a2 ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom \u226b \u2191(\u03c1 W) g = \u2191(\u03c1 V) g \u226b ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom ** exact (i.hom.comm g).symm ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.Scheme.Pullback.lift_comp_\u03b9 ** C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J \u22a2 pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g ** rw [\u2190 pullback.condition_assoc, Category.assoc, p_comm] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J \u22a2 pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i = pullback.fst ** apply ((gluing \ud835\udcb0 f g).openCover.pullbackCover pullback.fst).hom_ext ** case h C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J \u22a2 \u2200 (x : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J), OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) x \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i = OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) x \u226b pullback.fst ** intro j ** case h C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) j \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i = OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) j \u226b pullback.fst ** dsimp only [OpenCover.pullbackCover] ** case h C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.fst \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i = pullback.fst \u226b pullback.fst ** trans pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b (gluing \ud835\udcb0 f g).\u03b9 _ ** C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.fst \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i = 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 ** rw [\u2190 show _ = fV \ud835\udcb0 f g j i \u226b _ from (gluing \ud835\udcb0 f g).glue_condition j i] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.fst \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i = 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 \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i ** simp_rw [\u2190 Category.assoc] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 (pullback.fst \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g)) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i = ((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) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i ** congr 1 ** case e_a C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.fst \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) = (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 ** rw [gluing_f, gluing_t] ** case e_a C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.fst \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) = (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i) \u226b pullback.fst ** apply pullback.hom_ext <;> simp_rw [Category.assoc] ** case e_a.h\u2080 C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.fst \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b pullback.fst = pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i \u226b pullback.fst \u226b pullback.fst ** rw [t_fst_fst, pullback.lift_fst, pullbackFst\u03b9ToV_snd] ** case e_a.h\u2080 C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.snd ** rfl ** case e_a.h\u2081 C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.fst \u226b pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b pullback.snd = pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i \u226b pullback.fst \u226b pullback.snd ** rw [t_fst_snd, pullback.lift_snd, pullbackFst\u03b9ToV_fst_assoc, pullback.condition_assoc] ** case e_a.h\u2081 C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \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 ** erw [Multicoequalizer.\u03c0_desc] ** case e_a.h\u2081 C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.snd \u226b pullback.snd = pullback.snd \u226b pullback.snd ** rfl ** C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \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 ** rw [pullback.condition, \u2190 Category.assoc] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \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.snd \u226b OpenCover.map (GlueData.openCover (gluing \ud835\udcb0 f g)) j ** congr 1 ** case e_a C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i = pullback.snd ** apply pullback.hom_ext ** case e_a.h\u2080 C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \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 ** simp only [pullbackFst\u03b9ToV_fst] ** case e_a.h\u2080 C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \u22a2 pullback.snd \u226b pullback.fst = pullback.snd \u226b pullback.fst ** rfl ** case e_a.h\u2081 C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y Z : Scheme \ud835\udcb0 : OpenCover X f : X \u27f6 Z g : Y \u27f6 Z inst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g s : PullbackCone f g i : \ud835\udcb0.J j : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J \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 ** simp only [pullbackFst\u03b9ToV_fst] ** Qed", + "informal": "" + }, + { + "formal": "TopologicalSpace.Opens.openEmbedding_obj_top ** X : TopCat U : Opens \u2191X \u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj \u22a4 = U ** ext1 ** case h X : TopCat U : Opens \u2191X \u22a2 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj \u22a4) = \u2191U ** exact Set.image_univ.trans Subtype.range_coe ** Qed", + "informal": "" + }, + { + "formal": "spectrum.pow_image_subset ** \ud835\udd5c : Type u A : Type v inst\u271d\u00b2 : Field \ud835\udd5c inst\u271d\u00b9 : Ring A inst\u271d : Algebra \ud835\udd5c A a : A n : \u2115 \u22a2 (fun x => x ^ n) '' \u03c3 a \u2286 \u03c3 (a ^ n) ** simpa only [eval_pow, eval_X, aeval_X_pow] using subset_polynomial_aeval a (X ^ n : \ud835\udd5c[X]) ** Qed", + "informal": "" + }, + { + "formal": "NonUnitalSubsemiring.mem_sSup_of_directedOn ** R : Type u S\u271d : Type v T : Type w inst\u271d\u00b3 : NonUnitalNonAssocSemiring R M : Subsemigroup R inst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d inst\u271d\u00b9 : NonUnitalNonAssocSemiring T F : Type u_1 inst\u271d : NonUnitalRingHomClass F R S\u271d S : Set (NonUnitalSubsemiring R) Sne : Set.Nonempty S hS : DirectedOn (fun x x_1 => x \u2264 x_1) S x : R \u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s ** haveI : Nonempty S := Sne.to_subtype ** R : Type u S\u271d : Type v T : Type w inst\u271d\u00b3 : NonUnitalNonAssocSemiring R M : Subsemigroup R inst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d inst\u271d\u00b9 : NonUnitalNonAssocSemiring T F : Type u_1 inst\u271d : NonUnitalRingHomClass F R S\u271d S : Set (NonUnitalSubsemiring R) Sne : Set.Nonempty S hS : DirectedOn (fun x x_1 => x \u2264 x_1) S x : R this : Nonempty \u2191S \u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s ** simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, Subtype.exists, exists_prop] ** Qed", + "informal": "" + }, + { + "formal": "Relation.ReflGen.mono ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d b c d : \u03b1 p : \u03b1 \u2192 \u03b1 \u2192 Prop hp : \u2200 (a b : \u03b1), r a b \u2192 p a b a : \u03b1 \u22a2 ReflGen p a a ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Interval.coe_dual ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Sort u_5 \u03ba : \u03b9 \u2192 Sort u_6 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : PartialOrder \u03b2 s\u271d t : Interval \u03b1 a b : \u03b1 s : Interval \u03b1 \u22a2 \u2191(\u2191dual s) = \u2191ofDual \u207b\u00b9' \u2191s ** cases s with\n| none => rfl\n| some s\u2080 => exact NonemptyInterval.coe_dual s\u2080 ** case none \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Sort u_5 \u03ba : \u03b9 \u2192 Sort u_6 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : PartialOrder \u03b2 s t : Interval \u03b1 a b : \u03b1 \u22a2 \u2191(\u2191dual none) = \u2191ofDual \u207b\u00b9' \u2191none ** rfl ** case some \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Sort u_5 \u03ba : \u03b9 \u2192 Sort u_6 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : PartialOrder \u03b2 s t : Interval \u03b1 a b : \u03b1 s\u2080 : NonemptyInterval \u03b1 \u22a2 \u2191(\u2191dual (some s\u2080)) = \u2191ofDual \u207b\u00b9' \u2191(some s\u2080) ** exact NonemptyInterval.coe_dual s\u2080 ** Qed", + "informal": "" + }, + { + "formal": "Dioph.div_dioph ** \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g this : 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 z x y : \u2115 \u22a2 y = 0 \u2227 z = 0 \u2228 z * y \u2264 x \u2227 x < (z + 1) * y \u2194 x / y = z ** refine Iff.trans ?_ eq_comm ** \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g this : 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 z x y : \u2115 \u22a2 y = 0 \u2227 z = 0 \u2228 z * y \u2264 x \u2227 x < (z + 1) * y \u2194 z = x / y ** 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 ** \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g this : 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 z x y : \u2115 y0 : y = 0 \u22a2 y = 0 \u2227 z = 0 \u2228 z * y \u2264 x \u2227 x < (z + 1) * y \u2194 z = x / y ** rw [y0, Nat.div_zero] ** \u03b1 \u03b2 : Type n : \u2115 f g : (\u03b1 \u2192 \u2115) \u2192 \u2115 df : DiophFn f dg : DiophFn g this : 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 z x y : \u2115 y0 : y = 0 \u22a2 0 = 0 \u2227 z = 0 \u2228 z * 0 \u2264 x \u2227 x < (z + 1) * 0 \u2194 z = 0 ** 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 ** Qed", + "informal": "" + }, + { + "formal": "NonUnitalSubring.mem_closure_iff ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q : R hq : q \u2208 \u2191(Subsemigroup.closure s) \u22a2 0 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** rw [zero_mul q] ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q : R hq : q \u2208 \u2191(Subsemigroup.closure s) \u22a2 0 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** apply AddSubgroup.zero_mem _ ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q : R hq : q \u2208 \u2191(Subsemigroup.closure s) p\u2081 p\u2082 : R ihp\u2081 : p\u2081 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ihp\u2082 : p\u2082 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 (p\u2081 + p\u2082) * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** rw [add_mul p\u2081 p\u2082 q] ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q : R hq : q \u2208 \u2191(Subsemigroup.closure s) p\u2081 p\u2082 : R ihp\u2081 : p\u2081 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ihp\u2082 : p\u2082 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 p\u2081 * q + p\u2082 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** apply AddSubgroup.add_mem _ ihp\u2081 ihp\u2082 ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d\u00b9 : R h : x\u271d\u00b9 \u2208 closure s x\u271d y : R hx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q : R hq : q \u2208 \u2191(Subsemigroup.closure s) x : R hx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 -x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** have f : -x * q = -(x * q) := by simp ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f\u271d : R \u2192\u2099+* S s : Set R x\u271d\u00b9 : R h : x\u271d\u00b9 \u2208 closure s x\u271d y : R hx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q : R hq : q \u2208 \u2191(Subsemigroup.closure s) x : R hx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) f : -x * q = -(x * q) \u22a2 -x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** rw [f] ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f\u271d : R \u2192\u2099+* S s : Set R x\u271d\u00b9 : R h : x\u271d\u00b9 \u2208 closure s x\u271d y : R hx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q : R hq : q \u2208 \u2191(Subsemigroup.closure s) x : R hx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) f : -x * q = -(x * q) \u22a2 -(x * q) \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** apply AddSubgroup.neg_mem _ hx ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d\u00b9 : R h : x\u271d\u00b9 \u2208 closure s x\u271d y : R hx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q : R hq : q \u2208 \u2191(Subsemigroup.closure s) x : R hx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 -x * q = -(x * q) ** simp ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 x * 0 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** rw [mul_zero x] ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 0 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** apply AddSubgroup.zero_mem _ ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q\u2081 q\u2082 : R ihq\u2081 : x * q\u2081 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ihq\u2082 : x * q\u2082 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 x * (q\u2081 + q\u2082) \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** rw [mul_add x q\u2081 q\u2082] ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) q\u2081 q\u2082 : R ihq\u2081 : x * q\u2081 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ihq\u2082 : x * q\u2082 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 x * q\u2081 + x * q\u2082 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** apply AddSubgroup.add_mem _ ihq\u2081 ihq\u2082 ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) z : R hz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 x * -z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** have f : x * -z = -(x * z) := by simp ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f\u271d : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) z : R hz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) f : x * -z = -(x * z) \u22a2 x * -z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** rw [f] ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f\u271d : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) z : R hz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) f : x * -z = -(x * z) \u22a2 -(x * z) \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) ** apply AddSubgroup.neg_mem _ hz ** F : Type w R : Type u S : Type v T : Type u_1 inst\u271d\u00b3 : NonUnitalNonAssocRing R inst\u271d\u00b2 : NonUnitalNonAssocRing S inst\u271d\u00b9 : NonUnitalNonAssocRing T inst\u271d : NonUnitalRingHomClass F R S g : S \u2192\u2099+* T f : R \u2192\u2099+* S s : Set R x\u271d : R h : x\u271d \u2208 closure s x y : R hx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) hy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) z : R hz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s) \u22a2 x * -z = -(x * z) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Metric.equicontinuousAt_iff_pair ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 F : \u03b9 \u2192 \u03b2 \u2192 \u03b1 x\u2080 : \u03b2 \u22a2 EquicontinuousAt F x\u2080 \u2194 \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 ** rw [equicontinuousAt_iff_pair] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 F : \u03b9 \u2192 \u03b2 \u2192 \u03b1 x\u2080 : \u03b2 \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 \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 ** constructor <;> intro H ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 F : \u03b9 \u2192 \u03b2 \u2192 \u03b1 x\u2080 : \u03b2 H : \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 \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 ** intro \u03b5 h\u03b5 ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 F : \u03b9 \u2192 \u03b2 \u2192 \u03b1 x\u2080 : \u03b2 H : \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 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 \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 ** exact H _ (dist_mem_uniformity h\u03b5) ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 F : \u03b9 \u2192 \u03b2 \u2192 \u03b1 x\u2080 : \u03b2 H : \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 \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 ** intro U hU ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 F : \u03b9 \u2192 \u03b2 \u2192 \u03b1 x\u2080 : \u03b2 H : \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 U : Set (\u03b1 \u00d7 \u03b1) hU : U \u2208 \ud835\udce4 \u03b1 \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 ** rcases mem_uniformity_dist.mp hU with \u27e8\u03b5, h\u03b5, h\u03b5U\u27e9 ** case mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 F : \u03b9 \u2192 \u03b2 \u2192 \u03b1 x\u2080 : \u03b2 H : \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 U : Set (\u03b1 \u00d7 \u03b1) hU : U \u2208 \ud835\udce4 \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 h\u03b5U : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 U \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 ** refine' Exists.imp (fun V => And.imp_right fun h => _) (H _ h\u03b5) ** case mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 F : \u03b9 \u2192 \u03b2 \u2192 \u03b1 x\u2080 : \u03b2 H : \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 U : Set (\u03b1 \u00d7 \u03b1) hU : U \u2208 \ud835\udce4 \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 h\u03b5U : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 U V : Set \u03b2 h : \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 \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 ** exact fun x hx x' hx' i => h\u03b5U (h _ hx _ hx' i) ** Qed", + "informal": "" + }, + { + "formal": "Matrix.transpose_smul ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 m' : o \u2192 Type u_5 n' : o \u2192 Type u_6 R\u271d : Type u_7 S : Type u_8 \u03b1 : Type v \u03b2 : Type w \u03b3 : Type u_9 R : Type u_10 inst\u271d : SMul R \u03b1 c : R M : Matrix m n \u03b1 \u22a2 (c \u2022 M)\u1d40 = c \u2022 M\u1d40 ** ext ** case a.h l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 m' : o \u2192 Type u_5 n' : o \u2192 Type u_6 R\u271d : Type u_7 S : Type u_8 \u03b1 : Type v \u03b2 : Type w \u03b3 : Type u_9 R : Type u_10 inst\u271d : SMul R \u03b1 c : R M : Matrix m n \u03b1 i\u271d : n x\u271d : m \u22a2 (c \u2022 M)\u1d40 i\u271d x\u271d = (c \u2022 M\u1d40) i\u271d x\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "List.eq_of_perm_of_sorted ** \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081 l\u2082 : List \u03b1 p : l\u2081 ~ l\u2082 s\u2081 : Sorted r l\u2081 s\u2082 : Sorted r l\u2082 \u22a2 l\u2081 = l\u2082 ** induction' s\u2081 with a l\u2081 h\u2081 s\u2081 IH generalizing l\u2082 ** case nil \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081 l\u2082\u271d : List \u03b1 p\u271d : l\u2081 ~ l\u2082\u271d s\u2082\u271d : Sorted r l\u2082\u271d l\u2082 : List \u03b1 p : [] ~ l\u2082 s\u2082 : Sorted r l\u2082 \u22a2 [] = l\u2082 ** exact p.nil_eq ** case cons \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081\u271d l\u2082\u271d : List \u03b1 p\u271d : l\u2081\u271d ~ l\u2082\u271d s\u2082\u271d : Sorted r l\u2082\u271d a : \u03b1 l\u2081 : List \u03b1 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a' s\u2081 : Pairwise r l\u2081 IH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082 l\u2082 : List \u03b1 p : a :: l\u2081 ~ l\u2082 s\u2082 : Sorted r l\u2082 \u22a2 a :: l\u2081 = l\u2082 ** have : a \u2208 l\u2082 := p.subset (mem_cons_self _ _) ** case cons \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081\u271d l\u2082\u271d : List \u03b1 p\u271d : l\u2081\u271d ~ l\u2082\u271d s\u2082\u271d : Sorted r l\u2082\u271d a : \u03b1 l\u2081 : List \u03b1 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a' s\u2081 : Pairwise r l\u2081 IH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082 l\u2082 : List \u03b1 p : a :: l\u2081 ~ l\u2082 s\u2082 : Sorted r l\u2082 this : a \u2208 l\u2082 \u22a2 a :: l\u2081 = l\u2082 ** rcases mem_split this with \u27e8u\u2082, v\u2082, rfl\u27e9 ** case cons.intro.intro \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081\u271d l\u2082 : List \u03b1 p\u271d : l\u2081\u271d ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 l\u2081 : List \u03b1 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a' s\u2081 : Pairwise r l\u2081 IH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082 u\u2082 v\u2082 : List \u03b1 p : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this : a \u2208 u\u2082 ++ a :: v\u2082 \u22a2 a :: l\u2081 = u\u2082 ++ a :: v\u2082 ** have p' := (perm_cons a).1 (p.trans perm_middle) ** case cons.intro.intro \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081\u271d l\u2082 : List \u03b1 p\u271d : l\u2081\u271d ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 l\u2081 : List \u03b1 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a' s\u2081 : Pairwise r l\u2081 IH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082 u\u2082 v\u2082 : List \u03b1 p : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this : a \u2208 u\u2082 ++ a :: v\u2082 p' : l\u2081 ~ u\u2082 ++ v\u2082 \u22a2 a :: l\u2081 = u\u2082 ++ a :: v\u2082 ** obtain rfl := IH p' (s\u2082.sublist <| by simp) ** case cons.intro.intro \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081 l\u2082 : List \u03b1 p\u271d : l\u2081 ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 u\u2082 v\u2082 : List \u03b1 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this : a \u2208 u\u2082 ++ a :: v\u2082 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a' s\u2081 : Pairwise r (u\u2082 ++ v\u2082) IH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082 p : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082 p' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082 \u22a2 a :: (u\u2082 ++ v\u2082) = u\u2082 ++ a :: v\u2082 ** change a :: u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082) ** case cons.intro.intro \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081 l\u2082 : List \u03b1 p\u271d : l\u2081 ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 u\u2082 v\u2082 : List \u03b1 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this : a \u2208 u\u2082 ++ a :: v\u2082 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a' s\u2081 : Pairwise r (u\u2082 ++ v\u2082) IH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082 p : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082 p' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082 \u22a2 a :: u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082) ** rw [\u2190 append_assoc] ** case cons.intro.intro \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081 l\u2082 : List \u03b1 p\u271d : l\u2081 ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 u\u2082 v\u2082 : List \u03b1 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this : a \u2208 u\u2082 ++ a :: v\u2082 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a' s\u2081 : Pairwise r (u\u2082 ++ v\u2082) IH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082 p : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082 p' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082 \u22a2 a :: u\u2082 ++ v\u2082 = u\u2082 ++ [a] ++ v\u2082 ** congr ** case cons.intro.intro.e_a \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081 l\u2082 : List \u03b1 p\u271d : l\u2081 ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 u\u2082 v\u2082 : List \u03b1 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this : a \u2208 u\u2082 ++ a :: v\u2082 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a' s\u2081 : Pairwise r (u\u2082 ++ v\u2082) IH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082 p : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082 p' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082 \u22a2 a :: u\u2082 = u\u2082 ++ [a] ** 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])) ** case cons.intro.intro.e_a \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081 l\u2082 : List \u03b1 p\u271d : l\u2081 ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 u\u2082 v\u2082 : List \u03b1 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this\u271d : a \u2208 u\u2082 ++ a :: v\u2082 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a' s\u2081 : Pairwise r (u\u2082 ++ v\u2082) IH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082 p : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082 p' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082 this : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a \u22a2 a :: u\u2082 = u\u2082 ++ [a] ** 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] ** \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081\u271d l\u2082 : List \u03b1 p\u271d : l\u2081\u271d ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 l\u2081 : List \u03b1 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a' s\u2081 : Pairwise r l\u2081 IH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082 u\u2082 v\u2082 : List \u03b1 p : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this : a \u2208 u\u2082 ++ a :: v\u2082 p' : l\u2081 ~ u\u2082 ++ v\u2082 \u22a2 u\u2082 ++ v\u2082 <+ u\u2082 ++ a :: v\u2082 ** simp ** \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 inst\u271d : IsAntisymm \u03b1 r l\u2081 l\u2082 : List \u03b1 p\u271d : l\u2081 ~ l\u2082 s\u2082\u271d : Sorted r l\u2082 a : \u03b1 u\u2082 v\u2082 : List \u03b1 s\u2082 : Sorted r (u\u2082 ++ a :: v\u2082) this : a \u2208 u\u2082 ++ a :: v\u2082 h\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a' s\u2081 : Pairwise r (u\u2082 ++ v\u2082) IH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082 p : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082 p' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082 x : \u03b1 m : x \u2208 u\u2082 \u22a2 x \u2208 u\u2082 ++ v\u2082 ** simp [m] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.toReal_mono' ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 h : a \u2264 b ht : b = \u22a4 \u2192 a = \u22a4 \u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b ** rcases eq_or_ne a \u221e with rfl | ha ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 b c d : \u211d\u22650\u221e r p q : \u211d\u22650 h : \u22a4 \u2264 b ht : b = \u22a4 \u2192 \u22a4 = \u22a4 \u22a2 ENNReal.toReal \u22a4 \u2264 ENNReal.toReal b ** exact toReal_nonneg ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 h : a \u2264 b ht : b = \u22a4 \u2192 a = \u22a4 ha : a \u2260 \u22a4 \u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b ** exact toReal_mono (mt ht ha) h ** Qed", + "informal": "" + }, + { + "formal": "frontier_inter_open_inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 inst\u271d : TopologicalSpace \u03b4 s t : Set \u03b1 ht : IsOpen t \u22a2 frontier (s \u2229 t) \u2229 t = frontier s \u2229 t ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.lift_down ** \u03b1\u271d \u03b2\u271d : Type u a : Cardinal.{u} b : Cardinal.{max u v} \u03b1 : Type u \u03b2 : Type (max u v) \u22a2 #\u03b2 \u2264 lift.{v, u} #\u03b1 \u2192 \u2203 a', lift.{v, u} a' = #\u03b2 ** rw [\u2190 lift_id #\u03b2, \u2190 lift_umax, \u2190 lift_umax.{u, v}, lift_mk_le.{v}] ** \u03b1\u271d \u03b2\u271d : Type u a : Cardinal.{u} b : Cardinal.{max u v} \u03b1 : Type u \u03b2 : Type (max u v) \u22a2 Nonempty (\u03b2 \u21aa \u03b1) \u2192 \u2203 a', lift.{max u v, u} a' = lift.{max u v, max u v} #\u03b2 ** 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 ** Qed", + "informal": "" + }, + { + "formal": "AntilipschitzWith.comp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : PseudoEMetricSpace \u03b1 inst\u271d\u00b9 : PseudoEMetricSpace \u03b2 inst\u271d : PseudoEMetricSpace \u03b3 K : \u211d\u22650 f\u271d : \u03b1 \u2192 \u03b2 Kg : \u211d\u22650 g : \u03b2 \u2192 \u03b3 hg : AntilipschitzWith Kg g Kf : \u211d\u22650 f : \u03b1 \u2192 \u03b2 hf : AntilipschitzWith Kf f x y : \u03b1 \u22a2 \u2191Kf * (\u2191Kg * edist (g (f x)) (g (f y))) = \u2191(Kf * Kg) * edist ((g \u2218 f) x) ((g \u2218 f) y) ** rw [ENNReal.coe_mul, mul_assoc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : PseudoEMetricSpace \u03b1 inst\u271d\u00b9 : PseudoEMetricSpace \u03b2 inst\u271d : PseudoEMetricSpace \u03b3 K : \u211d\u22650 f\u271d : \u03b1 \u2192 \u03b2 Kg : \u211d\u22650 g : \u03b2 \u2192 \u03b3 hg : AntilipschitzWith Kg g Kf : \u211d\u22650 f : \u03b1 \u2192 \u03b2 hf : AntilipschitzWith Kf f x y : \u03b1 \u22a2 \u2191Kf * (\u2191Kg * edist (g (f x)) (g (f y))) = \u2191Kf * (\u2191Kg * edist ((g \u2218 f) x) ((g \u2218 f) y)) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Perm.zpow_apply_comm ** \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 \u03c3 : Perm \u03b1 m n : \u2124 x : \u03b1 \u22a2 \u2191(\u03c3 ^ m) (\u2191(\u03c3 ^ n) x) = \u2191(\u03c3 ^ n) (\u2191(\u03c3 ^ m) x) ** rw [\u2190 Equiv.Perm.mul_apply, \u2190 Equiv.Perm.mul_apply, zpow_mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.nextCoeff_X_sub_C ** R : Type u S : Type v a b c\u271d d : R n m : \u2115 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : Nontrivial R inst\u271d : Ring S c : S \u22a2 nextCoeff (X - \u2191C c) = -c ** rw [sub_eq_add_neg, \u2190 map_neg C c, nextCoeff_X_add_C] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.nat_cast_sub ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 m n : \u2115 \u22a2 \u2191(m - n) = \u2191m - \u2191n ** rw [\u2190 coe_nat, Nat.cast_tsub, coe_sub, coe_nat, coe_nat] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.prod.triangle ** C : Type u inst\u271d\u00b2 : Category.{v, u} C X\u271d Y\u271d : C inst\u271d\u00b9 : HasTerminal C inst\u271d : HasBinaryProducts C X Y : C \u22a2 (associator X (\u22a4_ C) Y).hom \u226b map (\ud835\udfd9 X) (leftUnitor Y).hom = map (rightUnitor X).hom (\ud835\udfd9 Y) ** ext <;> simp ** Qed", + "informal": "" + }, + { + "formal": "List.prod_map_hom ** \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G\u271d : Type u_7 R : Type u_8 inst\u271d\u00b3 : Monoid M inst\u271d\u00b2 : Monoid N inst\u271d\u00b9 : Monoid P l l\u2081 l\u2082 : List M a : M L : List \u03b9 f : \u03b9 \u2192 M G : Type u_9 inst\u271d : MonoidHomClass G M N g : G \u22a2 prod (map (\u2191g \u2218 f) L) = \u2191g (prod (map f L)) ** rw [\u2190 prod_hom, map_map] ** Qed", + "informal": "" + }, + { + "formal": "CompositionAsSet.blocks_sum ** n : \u2115 c : CompositionAsSet n \u22a2 sum (blocks c) = n ** have : c.blocks.take c.length = c.blocks := take_all_of_le (by simp [blocks]) ** n : \u2115 c : CompositionAsSet n this : take (length c) (blocks c) = blocks c \u22a2 sum (blocks c) = n ** rw [\u2190 this, c.blocks_partial_sum c.length_lt_card_boundaries, c.boundary_length] ** n : \u2115 c : CompositionAsSet n this : take (length c) (blocks c) = blocks c \u22a2 \u2191(Fin.last n) = n ** rfl ** n : \u2115 c : CompositionAsSet n \u22a2 List.length (blocks c) \u2264 length c ** simp [blocks] ** Qed", + "informal": "" + }, + { + "formal": "JoinedIn_of_segment_subset ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E\u271d : Type u_3 inst\u271d\u2079 : AddCommGroup E\u271d inst\u271d\u2078 : Module \u211d E\u271d inst\u271d\u2077 : TopologicalSpace E\u271d inst\u271d\u2076 : TopologicalAddGroup E\u271d inst\u271d\u2075 : ContinuousSMul \u211d E\u271d E : Type u_4 inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : TopologicalSpace E inst\u271d\u00b9 : ContinuousAdd E inst\u271d : ContinuousSMul \u211d E x y : E s : Set E h : [x-[\u211d]y] \u2286 s \u22a2 JoinedIn s x y ** have A : Continuous (fun t \u21a6 (1 - t) \u2022 x + t \u2022 y : \u211d \u2192 E) := by continuity ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E\u271d : Type u_3 inst\u271d\u2079 : AddCommGroup E\u271d inst\u271d\u2078 : Module \u211d E\u271d inst\u271d\u2077 : TopologicalSpace E\u271d inst\u271d\u2076 : TopologicalAddGroup E\u271d inst\u271d\u2075 : ContinuousSMul \u211d E\u271d E : Type u_4 inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : TopologicalSpace E inst\u271d\u00b9 : ContinuousAdd E inst\u271d : ContinuousSMul \u211d E x y : E s : Set E h : [x-[\u211d]y] \u2286 s A : Continuous fun t => (1 - t) \u2022 x + t \u2022 y \u22a2 JoinedIn s x y ** apply JoinedIn.ofLine A.continuousOn (by simp) (by simp) ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E\u271d : Type u_3 inst\u271d\u2079 : AddCommGroup E\u271d inst\u271d\u2078 : Module \u211d E\u271d inst\u271d\u2077 : TopologicalSpace E\u271d inst\u271d\u2076 : TopologicalAddGroup E\u271d inst\u271d\u2075 : ContinuousSMul \u211d E\u271d E : Type u_4 inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : TopologicalSpace E inst\u271d\u00b9 : ContinuousAdd E inst\u271d : ContinuousSMul \u211d E x y : E s : Set E h : [x-[\u211d]y] \u2286 s A : Continuous fun t => (1 - t) \u2022 x + t \u2022 y \u22a2 (fun t => (1 - t) \u2022 x + t \u2022 y) '' unitInterval \u2286 s ** convert h ** case h.e'_3 \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E\u271d : Type u_3 inst\u271d\u2079 : AddCommGroup E\u271d inst\u271d\u2078 : Module \u211d E\u271d inst\u271d\u2077 : TopologicalSpace E\u271d inst\u271d\u2076 : TopologicalAddGroup E\u271d inst\u271d\u2075 : ContinuousSMul \u211d E\u271d E : Type u_4 inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : TopologicalSpace E inst\u271d\u00b9 : ContinuousAdd E inst\u271d : ContinuousSMul \u211d E x y : E s : Set E h : [x-[\u211d]y] \u2286 s A : Continuous fun t => (1 - t) \u2022 x + t \u2022 y \u22a2 (fun t => (1 - t) \u2022 x + t \u2022 y) '' unitInterval = [x-[\u211d]y] ** rw [segment_eq_image \u211d x y] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E\u271d : Type u_3 inst\u271d\u2079 : AddCommGroup E\u271d inst\u271d\u2078 : Module \u211d E\u271d inst\u271d\u2077 : TopologicalSpace E\u271d inst\u271d\u2076 : TopologicalAddGroup E\u271d inst\u271d\u2075 : ContinuousSMul \u211d E\u271d E : Type u_4 inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : TopologicalSpace E inst\u271d\u00b9 : ContinuousAdd E inst\u271d : ContinuousSMul \u211d E x y : E s : Set E h : [x-[\u211d]y] \u2286 s \u22a2 Continuous fun t => (1 - t) \u2022 x + t \u2022 y ** continuity ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E\u271d : Type u_3 inst\u271d\u2079 : AddCommGroup E\u271d inst\u271d\u2078 : Module \u211d E\u271d inst\u271d\u2077 : TopologicalSpace E\u271d inst\u271d\u2076 : TopologicalAddGroup E\u271d inst\u271d\u2075 : ContinuousSMul \u211d E\u271d E : Type u_4 inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : TopologicalSpace E inst\u271d\u00b9 : ContinuousAdd E inst\u271d : ContinuousSMul \u211d E x y : E s : Set E h : [x-[\u211d]y] \u2286 s A : Continuous fun t => (1 - t) \u2022 x + t \u2022 y \u22a2 (1 - 0) \u2022 x + 0 \u2022 y = x ** simp ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E\u271d : Type u_3 inst\u271d\u2079 : AddCommGroup E\u271d inst\u271d\u2078 : Module \u211d E\u271d inst\u271d\u2077 : TopologicalSpace E\u271d inst\u271d\u2076 : TopologicalAddGroup E\u271d inst\u271d\u2075 : ContinuousSMul \u211d E\u271d E : Type u_4 inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : TopologicalSpace E inst\u271d\u00b9 : ContinuousAdd E inst\u271d : ContinuousSMul \u211d E x y : E s : Set E h : [x-[\u211d]y] \u2286 s A : Continuous fun t => (1 - t) \u2022 x + t \u2022 y \u22a2 (1 - 1) \u2022 x + 1 \u2022 y = y ** simp ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.Monic.comp ** R : Type u S : Type v T : Type w a b : R n : \u2115 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R p q : R[X] hp : Monic p hq : Monic q h : natDegree q \u2260 0 \u22a2 Monic (Polynomial.comp p q) ** rw [Monic.def, leadingCoeff_comp h, Monic.def.1 hp, Monic.def.1 hq, one_pow, one_mul] ** Qed", + "informal": "" + }, + { + "formal": "InnerProductGeometry.norm_div_sin_angle_sub_of_inner_eq_zero ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x y : V h : inner x y = 0 h0 : x = 0 \u2228 y \u2260 0 \u22a2 \u2016y\u2016 / Real.sin (angle x (x - y)) = \u2016x - y\u2016 ** rw [\u2190 neg_eq_zero, \u2190 inner_neg_right] at h ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x y : V h : inner x (-y) = 0 h0 : x = 0 \u2228 y \u2260 0 \u22a2 \u2016y\u2016 / Real.sin (angle x (x - y)) = \u2016x - y\u2016 ** rw [\u2190 neg_ne_zero] at h0 ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x y : V h : inner x (-y) = 0 h0 : x = 0 \u2228 -y \u2260 0 \u22a2 \u2016y\u2016 / Real.sin (angle x (x - y)) = \u2016x - y\u2016 ** rw [sub_eq_add_neg, \u2190 norm_neg, norm_div_sin_angle_add_of_inner_eq_zero h h0] ** Qed", + "informal": "" + }, + { + "formal": "SmoothBumpFunction.nhds_basis_tsupport ** E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E H : Type uH inst\u271d\u2074 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : ChartedSpace H M inst\u271d\u00b9 : SmoothManifoldWithCorners I M c : M f : SmoothBumpFunction I c x : M inst\u271d : T2Space M \u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f => tsupport \u2191f ** 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 _ ** E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E H : Type uH inst\u271d\u2074 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : ChartedSpace H M inst\u271d\u00b9 : SmoothManifoldWithCorners I M c : M f : SmoothBumpFunction I c x : M inst\u271d : T2Space M this : HasBasis (\ud835\udcdd c) (fun x => True) fun f => \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I) \u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f => tsupport \u2191f ** refine' this.to_hasBasis' (fun f _ => \u27e8f, trivial, f.tsupport_subset_symm_image_closedBall\u27e9)\n fun f _ => f.tsupport_mem_nhds ** E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E H : Type uH inst\u271d\u2074 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : ChartedSpace H M inst\u271d\u00b9 : SmoothManifoldWithCorners I M c : M f : SmoothBumpFunction I c x : M inst\u271d : T2Space M \u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f => \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I) ** rw [\u2190 map_extChartAt_symm_nhdsWithin_range I c] ** E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E H : Type uH inst\u271d\u2074 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : ChartedSpace H M inst\u271d\u00b9 : SmoothManifoldWithCorners I M c : M f : SmoothBumpFunction I c x : M inst\u271d : T2Space M \u22a2 HasBasis (map (\u2191(LocalEquiv.symm (extChartAt I c))) (\ud835\udcdd[range \u2191I] \u2191(extChartAt I c) c)) (fun x => True) fun f => \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I) ** exact nhdsWithin_range_basis.map _ ** Qed", + "informal": "" + }, + { + "formal": "Colex.sum_two_pow_lt_iff_lt ** \u03b1 : Type u_1 A B : Finset \u2115 z : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i \u22a2 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i \u2194 toColex A < toColex B ** 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 ** \u03b1 : Type u_1 A B : Finset \u2115 z : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i h : \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i h\u2081 : A = B \u2228 B < A \u22a2 A = B \u2192 False ** rintro rfl ** \u03b1 : Type u_1 A : Finset \u2115 z : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i h : \u2211 i in A, 2 ^ i < \u2211 i in A, 2 ^ i h\u2081 : A = A \u2228 A < A \u22a2 False ** apply irrefl _ h ** \u03b1 : Type u_1 A B : Finset \u2115 \u22a2 \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i ** intro A B ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 \u22a2 toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i ** rw [\u2190 sdiff_lt_sdiff_iff_lt, Colex.lt_def] ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 \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 ** rintro \u27e8k, z, kA, kB\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A \u22a2 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i ** rw [\u2190 sdiff_union_inter A B] ** case intro.intro.intro \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A \u22a2 \u2211 i in A \\ B \u222a A \u2229 B, 2 ^ i < \u2211 i in B, 2 ^ i ** conv_rhs => rw [\u2190 sdiff_union_inter B A] ** case intro.intro.intro \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A \u22a2 \u2211 i in A \\ B \u222a A \u2229 B, 2 ^ i < \u2211 i in B \\ A \u222a B \u2229 A, 2 ^ i ** rw [sum_union (disjoint_sdiff_inter _ _), sum_union (disjoint_sdiff_inter _ _), inter_comm,\n add_lt_add_iff_right] ** case intro.intro.intro \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A \u22a2 \u2211 x in A \\ B, 2 ^ x < \u2211 x in B \\ A, 2 ^ x ** apply lt_of_lt_of_le (@Nat.sum_two_pow_lt k (A \\ B) _) ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A \u22a2 \u2200 {x : \u2115}, x \u2208 A \\ B \u2192 x < k ** intro x hx ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A x : \u2115 hx : x \u2208 A \\ B \u22a2 x < k ** apply lt_of_le_of_ne (le_of_not_lt _) ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A x : \u2115 hx : x \u2208 A \\ B \u22a2 \u00ack < x ** intro kx ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A x : \u2115 hx : x \u2208 A \\ B kx : k < x \u22a2 False ** have := (z kx).1 hx ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A x : \u2115 hx : x \u2208 A \\ B kx : k < x this : x \u2208 B \\ A \u22a2 False ** rw [mem_sdiff] at this hx ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A x : \u2115 hx : x \u2208 A \u2227 \u00acx \u2208 B kx : k < x this : x \u2208 B \u2227 \u00acx \u2208 A \u22a2 False ** exact hx.2 this.1 ** case intro.intro.intro \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A \u22a2 2 ^ k \u2264 \u2211 x in B \\ A, 2 ^ x ** apply single_le_sum (fun _ _ => Nat.zero_le _) kB ** \u03b1 : Type u_1 A\u271d B\u271d A B : Finset \u2115 k : \u2115 z : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A) kA : \u00ack \u2208 A \\ B kB : k \u2208 B \\ A x : \u2115 hx : x \u2208 A \\ B \u22a2 x \u2260 k ** apply ne_of_mem_of_not_mem hx kA ** Qed", + "informal": "" + }, + { + "formal": "MvPowerSeries.map_X ** \u03c3 : Type u_1 R : Type u_2 S : Type u_3 T : Type u_4 inst\u271d\u00b2 : Semiring R inst\u271d\u00b9 : Semiring S inst\u271d : Semiring T f : R \u2192+* S g : S \u2192+* T s : \u03c3 \u22a2 \u2191(map \u03c3 f) (X s) = X s ** simp [MvPowerSeries.X] ** Qed", + "informal": "" + }, + { + "formal": "Primrec.dom_bool\u2082 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 f : Bool \u2192 Bool \u2192 \u03b1 x\u271d : Bool \u00d7 Bool a b : Bool \u22a2 (bif (a, b).1 then f true (a, b).2 else f false (a, b).2) = f (a, b).1 (a, b).2 ** cases a <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "Zsqrtd.sqLe_cancel ** d\u271d : \u2124 c d x y z w : \u2115 zw : SqLe y d x c h : SqLe (x + z) c (y + w) d \u22a2 SqLe z c w d ** apply le_of_not_gt ** case a d\u271d : \u2124 c d x y z w : \u2115 zw : SqLe y d x c h : SqLe (x + z) c (y + w) d \u22a2 \u00acc * z * z > d * w * w ** intro l ** case a d\u271d : \u2124 c d x y z w : \u2115 zw : SqLe y d x c h : SqLe (x + z) c (y + w) d l : c * z * z > d * w * w \u22a2 False ** refine' not_le_of_gt _ h ** case a d\u271d : \u2124 c d x y z w : \u2115 zw : SqLe y d x c h : SqLe (x + z) c (y + w) d l : c * z * z > d * w * w \u22a2 c * (x + z) * (x + z) > d * (y + w) * (y + w) ** simp only [SqLe, mul_add, mul_comm, mul_left_comm, add_assoc, gt_iff_lt] ** case a d\u271d : \u2124 c d x y z w : \u2115 zw : SqLe y d x c h : SqLe (x + z) c (y + w) d l : c * z * z > d * w * w \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))) ** have hm := sqLe_add_mixed zw (le_of_lt l) ** case a d\u271d : \u2124 c d x y z w : \u2115 zw : SqLe y d x c h : SqLe (x + z) c (y + w) d l : c * z * z > d * w * w hm : d * (y * w) \u2264 c * (x * z) \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))) ** simp only [SqLe, mul_assoc, gt_iff_lt] at l zw ** case a d\u271d : \u2124 c d x y z w : \u2115 h : SqLe (x + z) c (y + w) d hm : d * (y * w) \u2264 c * (x * z) l : d * (w * w) < c * (z * z) zw : d * (y * y) \u2264 c * (x * x) \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))) ** 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)) _) ** Qed", + "informal": "" + }, + { + "formal": "LocalEquiv.trans_refl_restr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 e : LocalEquiv \u03b1 \u03b2 e' : LocalEquiv \u03b2 \u03b3 s : Set \u03b2 \u22a2 (LocalEquiv.trans e (LocalEquiv.restr (LocalEquiv.refl \u03b2) s)).source = (LocalEquiv.restr e (\u2191e \u207b\u00b9' s)).source ** simp [trans_source] ** Qed", + "informal": "" + }, + { + "formal": "Commute.div_mul_div_comm ** G : Type u_1 inst\u271d : DivisionMonoid G a b c d : G hbd : Commute b d hbc : Commute b\u207b\u00b9 c \u22a2 a / b * (c / d) = a * c / (b * d) ** simp_rw [div_eq_mul_inv, mul_inv_rev, hbd.inv_inv.symm.eq, hbc.mul_mul_mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "Std.HashMap.Imp.Buckets.WF.mk' ** \u03b1 : Type u_1 \u03b2 : Type u_2 n : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 h : 0 < n \u22a2 WF (Buckets.mk n) ** refine \u27e8fun _ h => ?_, fun i h => ?_\u27e9 ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 n : Nat inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 h\u271d : 0 < n inst\u271d\u00b9 : LawfulHashable \u03b1 inst\u271d : PartialEquivBEq \u03b1 x\u271d : AssocList \u03b1 \u03b2 h : x\u271d \u2208 (Buckets.mk n).val.data \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d) ** simp [Buckets.mk, empty', mkArray, List.mem_replicate] at h ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 n : Nat inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 h\u271d : 0 < n inst\u271d\u00b9 : LawfulHashable \u03b1 inst\u271d : PartialEquivBEq \u03b1 x\u271d : AssocList \u03b1 \u03b2 h : \u00acn = 0 \u2227 x\u271d = AssocList.nil \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d) ** simp [h, List.Pairwise.nil] ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 n : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 h\u271d : 0 < n i : Nat h : i < Array.size (Buckets.mk n).val \u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.mk n).val) = i) (Buckets.mk n).val[i] ** simp [Buckets.mk, empty', mkArray, Array.getElem_eq_data_get, AssocList.All] ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P \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 ** refine' \u27e8fun h => _, fun h => _\u27e9 ** case refine'_1 V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : Collinear \u211d {p\u2081, p\u2082, p\u2083} \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 ** replace h := h.wbtw_or_wbtw_or_wbtw ** case refine'_1 V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : 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 \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 ** by_cases h\u2081\u2082 : p\u2081 = p\u2082 ** case neg V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : 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 h\u2081\u2082 : \u00acp\u2081 = p\u2082 \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 ** by_cases h\u2083\u2082 : p\u2083 = p\u2082 ** case neg V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : 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 h\u2081\u2082 : \u00acp\u2081 = p\u2082 h\u2083\u2082 : \u00acp\u2083 = p\u2082 \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 ** rw [or_iff_right h\u2081\u2082, or_iff_right h\u2083\u2082] ** case neg V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : 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 h\u2081\u2082 : \u00acp\u2081 = p\u2082 h\u2083\u2082 : \u00acp\u2083 = p\u2082 \u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 ** rcases h with (h | h | h) ** case pos V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : 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 h\u2081\u2082 : p\u2081 = p\u2082 \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 ** exact Or.inl h\u2081\u2082 ** case pos V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : 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 h\u2081\u2082 : \u00acp\u2081 = p\u2082 h\u2083\u2082 : p\u2083 = p\u2082 \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 ** exact Or.inr (Or.inl h\u2083\u2082) ** case neg.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h\u2081\u2082 : \u00acp\u2081 = p\u2082 h\u2083\u2082 : \u00acp\u2083 = p\u2082 h : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 ** exact Or.inr (angle_eq_pi_iff_sbtw.2 \u27e8h, Ne.symm h\u2081\u2082, Ne.symm h\u2083\u2082\u27e9) ** case neg.inr.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h\u2081\u2082 : \u00acp\u2081 = p\u2082 h\u2083\u2082 : \u00acp\u2083 = p\u2082 h : Wbtw \u211d p\u2082 p\u2083 p\u2081 \u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 ** exact Or.inl (h.angle\u2083\u2081\u2082_eq_zero_of_ne h\u2083\u2082) ** case neg.inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h\u2081\u2082 : \u00acp\u2081 = p\u2082 h\u2083\u2082 : \u00acp\u2083 = p\u2082 h : Wbtw \u211d p\u2083 p\u2081 p\u2082 \u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 ** exact Or.inl (h.angle\u2082\u2083\u2081_eq_zero_of_ne h\u2081\u2082) ** case refine'_2 V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : 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 \u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} ** rcases h with (rfl | rfl | h | h) ** case refine'_2.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2083 : P \u22a2 Collinear \u211d {p\u2081, p\u2081, p\u2083} ** simpa using collinear_pair \u211d p\u2081 p\u2083 ** case refine'_2.inr.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2083 : P \u22a2 Collinear \u211d {p\u2081, p\u2083, p\u2083} ** simpa using collinear_pair \u211d p\u2081 p\u2083 ** case refine'_2.inr.inr.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : \u2220 p\u2081 p\u2082 p\u2083 = 0 \u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} ** rw [angle_eq_zero_iff_ne_and_wbtw] at h ** case refine'_2.inr.inr.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : 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 \u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} ** rcases h with (\u27e8-, h\u27e9 | \u27e8-, h\u27e9) ** case refine'_2.inr.inr.inl.inl.intro V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : Wbtw \u211d p\u2082 p\u2081 p\u2083 \u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} ** rw [Set.insert_comm] ** case refine'_2.inr.inr.inl.inl.intro V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : Wbtw \u211d p\u2082 p\u2081 p\u2083 \u22a2 Collinear \u211d {p\u2082, p\u2081, p\u2083} ** exact h.collinear ** case refine'_2.inr.inr.inl.inr.intro V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : Wbtw \u211d p\u2082 p\u2083 p\u2081 \u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} ** rw [Set.insert_comm, Set.pair_comm] ** case refine'_2.inr.inr.inl.inr.intro V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : Wbtw \u211d p\u2082 p\u2083 p\u2081 \u22a2 Collinear \u211d {p\u2082, p\u2083, p\u2081} ** exact h.collinear ** case refine'_2.inr.inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 \u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} ** rw [angle_eq_pi_iff_sbtw] at h ** case refine'_2.inr.inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P p\u2081 p\u2082 p\u2083 : P h : Sbtw \u211d p\u2081 p\u2082 p\u2083 \u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} ** exact h.wbtw.collinear ** Qed", + "informal": "" + }, + { + "formal": "inv_mem_nhds_one ** \u03b1 : Type u \u03b2 : Type v G : Type w H : Type x inst\u271d\u00b3 : TopologicalSpace G inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalGroup G inst\u271d : TopologicalSpace \u03b1 f : \u03b1 \u2192 G s : Set \u03b1 x : \u03b1 S : Set G hS : S \u2208 \ud835\udcdd 1 \u22a2 S\u207b\u00b9 \u2208 \ud835\udcdd 1 ** rwa [\u2190 nhds_one_symm'] at hS ** Qed", + "informal": "" + }, + { + "formal": "HasFTaylorSeriesUpToOn.prod ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s \u22a2 HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => ContinuousMultilinearMap.prod (p y k) (q y k)) s ** set L := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun _ : Fin m => E) F G ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s L : (m : \u2115) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G \u22a2 HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => ContinuousMultilinearMap.prod (p y k) (q y k)) s ** constructor ** case zero_eq \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s L : (m : \u2115) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G \u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x) ** intro x hx ** case zero_eq \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x\u271d x\u2080 : E c : F b : E \u00d7 F \u2192 G m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s L : (m : \u2115) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G x : E hx : x \u2208 s \u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x) ** rw [\u2190 hf.zero_eq x hx, \u2190 hg.zero_eq x hx] ** case zero_eq \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x\u271d x\u2080 : E c : F b : E \u00d7 F \u2192 G m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s L : (m : \u2115) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G x : E hx : x \u2208 s \u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (ContinuousMultilinearMap.uncurry0 (p x 0), ContinuousMultilinearMap.uncurry0 (q x 0)) ** rfl ** case fderivWithin \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s L : (m : \u2115) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G \u22a2 \u2200 (m : \u2115), \u2191m < n \u2192 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s x ** intro m hm x hx ** case fderivWithin \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x\u271d x\u2080 : E c : F b : E \u00d7 F \u2192 G m\u271d n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s L : (m : \u2115) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G m : \u2115 hm : \u2191m < n x : E hx : x \u2208 s \u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s x ** convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x\n ((hf.fderivWithin m hm x hx).prod (hg.fderivWithin m hm x hx)) ** case cont \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s L : (m : \u2115) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G \u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s ** intro m hm ** case cont \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u2079 : NormedAddCommGroup D inst\u271d\u2078 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g\u271d : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m\u271d n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F hf : HasFTaylorSeriesUpToOn n f p s g : E \u2192 G q : E \u2192 FormalMultilinearSeries \ud835\udd5c E G hg : HasFTaylorSeriesUpToOn n g q s L : (m : \u2115) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G m : \u2115 hm : \u2191m \u2264 n \u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s ** exact (L m).continuous.comp_continuousOn ((hf.rst.imnt m hm).prod (hg.cont m hm)) ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.support_esymm'' ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R \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 ** rw [esymm_eq_sum_monomial] ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R \u22a2 support (\u2211 t in powersetCard n univ, \u2191(monomial (\u2211 i in t, fun\u2080 | i => 1)) 1) = Finset.biUnion (powersetCard n univ) fun t => (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support ** simp only [\u2190 single_eq_monomial] ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R \u22a2 support (\u2211 x in powersetCard n univ, fun\u2080 | \u2211 i in x, fun\u2080 | i => 1 => 1) = Finset.biUnion (powersetCard n univ) fun t => (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support ** refine' Finsupp.support_sum_eq_biUnion (powersetCard n (univ : Finset \u03c3)) _ ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R \u22a2 \u2200 (i\u2081 i\u2082 : Finset \u03c3), 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 ** intro s t hst ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t \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 ** rw [Finset.disjoint_left, Finsupp.support_single_ne_zero _ one_ne_zero] ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t \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 ** rw [Finsupp.support_single_ne_zero _ one_ne_zero] ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t \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} ** simp only [one_ne_zero, mem_singleton, Finsupp.mem_support_iff] ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t \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 ** rintro a h rfl ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 \u22a2 False ** have := congr_arg Finsupp.support h ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support \u22a2 False ** rw [Finsupp.support_sum_eq_biUnion, Finsupp.support_sum_eq_biUnion] at this ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support \u22a2 False case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support \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 case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support \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 ** 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] ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support hsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x} \u22a2 False case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support \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 case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support \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 ** have hs := biUnion_congr (of_eq_true (eq_self s)) (hsingle s) ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support hsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x} hs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a} \u22a2 False case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support \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 case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support \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 ** have ht := biUnion_congr (of_eq_true (eq_self t)) (hsingle t) ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support hsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x} hs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a} ht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a} \u22a2 False case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support \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 case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support \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 ** rw [hs, ht] at this ** case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support \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 case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support \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 ** all_goals intro x y; simp [Finsupp.support_single_disjoint] ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support \u22a2 \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x} ** intros _ x _ ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support s\u271d : Finset \u03c3 x : \u03c3 a\u271d : x \u2208 s\u271d \u22a2 (fun\u2080 | x => 1).support = {x} ** rw [Finsupp.support_single_ne_zero x one_ne_zero] ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support this : (Finset.biUnion t fun a => {a}) = Finset.biUnion s fun a => {a} hsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x} hs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a} ht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a} \u22a2 False ** simp only [biUnion_singleton_eq_self] at this ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support this\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support hsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x} hs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a} ht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a} this : t = s \u22a2 False ** exact absurd this hst.symm ** case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support \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 ** intro x y ** case h \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2075 : CommSemiring R inst\u271d\u2074 : CommSemiring S inst\u271d\u00b3 : Fintype \u03c3 inst\u271d\u00b2 : Fintype \u03c4 n : \u2115 inst\u271d\u00b9 : DecidableEq \u03c3 inst\u271d : Nontrivial R s t : Finset \u03c3 hst : s \u2260 t h : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1 this : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support x y : \u03c3 \u22a2 x \u2260 y \u2192 Disjoint (fun\u2080 | x => 1).support (fun\u2080 | y => 1).support ** simp [Finsupp.support_single_disjoint] ** Qed", + "informal": "" + }, + { + "formal": "Finset.image_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b2 f\u271d g : \u03b1 \u2192 \u03b2 s : Finset \u03b1 t : Finset \u03b2 a\u271d : \u03b1 b c : \u03b2 f : \u03b1 \u2192 \u03b2 a : \u03b1 x : \u03b2 \u22a2 x \u2208 image f {a} \u2194 x \u2208 {f a} ** simpa only [mem_image, exists_prop, mem_singleton, exists_eq_left] using eq_comm ** Qed", + "informal": "" + }, + { + "formal": "IsDedekindDomain.HeightOneSpectrum.IntValuation.le_max_iff_min_le ** R : Type u_1 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R K : Type u_2 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v : HeightOneSpectrum R a b c : \u2115 \u22a2 \u2191ofAdd (-\u2191c) \u2264 max (\u2191ofAdd (-\u2191a)) (\u2191ofAdd (-\u2191b)) \u2194 min a b \u2264 c ** 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] ** Qed", + "informal": "" + }, + { + "formal": "coprime_sq_sub_sq_sum_of_odd_odd ** m n : \u2124 h : Int.gcd m n = 1 hm : m % 2 = 1 hn : n % 2 = 1 \u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227 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 ** cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2 ** case intro m n : \u2124 h : Int.gcd m n = 1 hm : m % 2 = 1 hn : n % 2 = 1 m0 : \u2124 hm2 : m - 1 = m0 * 2 \u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227 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 ** cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2 ** case intro.intro m n : \u2124 h : Int.gcd m n = 1 hm : m % 2 = 1 hn : n % 2 = 1 m0 : \u2124 hm2 : m - 1 = m0 * 2 n0 : \u2124 hn2 : n - 1 = n0 * 2 \u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227 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 ** rw [sub_eq_iff_eq_add] at hm2 hn2 ** case intro.intro m n : \u2124 h : Int.gcd m n = 1 hm : m % 2 = 1 hn : n % 2 = 1 m0 : \u2124 hm2 : m = m0 * 2 + 1 n0 : \u2124 hn2 : n = n0 * 2 + 1 \u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227 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 ** subst m ** case intro.intro n : \u2124 hn : n % 2 = 1 m0 n0 : \u2124 hn2 : n = n0 * 2 + 1 h : Int.gcd (m0 * 2 + 1) n = 1 hm : (m0 * 2 + 1) % 2 = 1 \u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + n ^ 2 \u2227 2 \u2223 (m0 * 2 + 1) ^ 2 - n ^ 2 \u2227 ((m0 * 2 + 1) ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227 Int.gcd (((m0 * 2 + 1) ^ 2 - n ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + n ^ 2) / 2) = 1 ** subst n ** case intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 \u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227 2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227 ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1 ** have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by\n ring ** case intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) \u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227 2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227 ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1 ** have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring ** case intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) \u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227 2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227 ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1 ** 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 ** case intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227 2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227 ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1 ** refine' \u27e8\u27e8_, h1\u27e9, \u27e8_, h2\u27e9, h3, _\u27e9 ** case intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u22a2 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1 ** have h20 : (2 : \u2124) \u2260 0 := by decide ** case intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 \u22a2 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1 ** rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20] ** case intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 \u22a2 Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 ** by_contra h4 ** case intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 \u22a2 False ** obtain \u27e8p, hp, hp1, hp2\u27e9 := Nat.Prime.not_coprime_iff_dvd.mp h4 ** case intro.intro.intro.intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) hp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) \u22a2 False ** apply hp.not_dvd_one ** case intro.intro.intro.intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) hp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) \u22a2 p \u2223 1 ** rw [\u2190 h] ** case intro.intro.intro.intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) hp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) \u22a2 p \u2223 Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) ** rw [\u2190 Int.coe_nat_dvd_left] at hp1 hp2 ** case intro.intro.intro.intro.intro m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) hp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 \u22a2 p \u2223 Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) ** apply Nat.dvd_gcd ** m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 \u22a2 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) ** ring ** m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) \u22a2 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) ** ring ** m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) \u22a2 ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 ** rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right] ** case H m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) \u22a2 2 \u2260 0 ** exact by decide ** m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) \u22a2 2 \u2260 0 ** decide ** m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u22a2 2 \u2260 0 ** decide ** case intro.intro.intro.intro.intro.a m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) hp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 \u22a2 p \u2223 Int.natAbs (m0 * 2 + 1) ** apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp ** case intro.intro.intro.intro.intro.a.h m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) hp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 \u22a2 \u2191p \u2223 (m0 * 2 + 1) ^ 2 ** convert dvd_add hp1 hp2 ** case h.e'_4 m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) hp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 \u22a2 (m0 * 2 + 1) ^ 2 = 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) + (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) ** ring ** case intro.intro.intro.intro.intro.a m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) hp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 \u22a2 p \u2223 Int.natAbs (n0 * 2 + 1) ** apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp ** case intro.intro.intro.intro.intro.a.h m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) hp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 \u22a2 \u2191p \u2223 (n0 * 2 + 1) ^ 2 ** convert dvd_sub hp2 hp1 ** case h.e'_4 m0 n0 : \u2124 hm : (m0 * 2 + 1) % 2 = 1 hn : (n0 * 2 + 1) % 2 = 1 h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1 h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 h20 : 2 \u2260 0 h4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1 p : \u2115 hp : Nat.Prime p hp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) hp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 \u22a2 (n0 * 2 + 1) ^ 2 = 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 - 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) ** ring ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.aleph0_le_cof ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} \u22a2 \u2135\u2080 \u2264 cof o \u2194 IsLimit o ** rcases zero_or_succ_or_limit o with (rfl | \u27e8o, rfl\u27e9 | l) ** case inl \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop \u22a2 \u2135\u2080 \u2264 cof 0 \u2194 IsLimit 0 ** simp [not_zero_isLimit, Cardinal.aleph0_ne_zero] ** case inr.inl.intro \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} \u22a2 \u2135\u2080 \u2264 cof (succ o) \u2194 IsLimit (succ o) ** simp [not_succ_isLimit, Cardinal.one_lt_aleph0] ** case inr.inr \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o \u22a2 \u2135\u2080 \u2264 cof o \u2194 IsLimit o ** simp [l] ** case inr.inr \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o \u22a2 \u2135\u2080 \u2264 cof o ** refine' le_of_not_lt fun h => _ ** case inr.inr \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 \u22a2 False ** cases' Cardinal.lt_aleph0.1 h with n e ** case inr.inr.intro \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 n : \u2115 e : cof o = \u2191n \u22a2 False ** have := cof_cof o ** case inr.inr.intro \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 n : \u2115 e : cof o = \u2191n this : cof (ord (cof o)) = cof o \u22a2 False ** rw [e, ord_nat] at this ** case inr.inr.intro \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 n : \u2115 e : cof o = \u2191n this : cof \u2191n = \u2191n \u22a2 False ** cases n ** case inr.inr.intro.zero \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 e : cof o = \u2191Nat.zero this : cof \u2191Nat.zero = \u2191Nat.zero \u22a2 False ** simp at e ** case inr.inr.intro.zero \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 this : cof \u2191Nat.zero = \u2191Nat.zero e : o = 0 \u22a2 False ** simp [e, not_zero_isLimit] at l ** case inr.inr.intro.succ \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 n\u271d : \u2115 e : cof o = \u2191(Nat.succ n\u271d) this : cof \u2191(Nat.succ n\u271d) = \u2191(Nat.succ n\u271d) \u22a2 False ** rw [nat_cast_succ, cof_succ] at this ** case inr.inr.intro.succ \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 n\u271d : \u2115 e : cof o = \u2191(Nat.succ n\u271d) this : 1 = \u2191(Nat.succ n\u271d) \u22a2 False ** rw [\u2190 this, cof_eq_one_iff_is_succ] at e ** case inr.inr.intro.succ \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < \u2135\u2080 n\u271d : \u2115 e : \u2203 a, o = succ a this : 1 = \u2191(Nat.succ n\u271d) \u22a2 False ** rcases e with \u27e8a, rfl\u27e9 ** case inr.inr.intro.succ.intro \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop n\u271d : \u2115 this : 1 = \u2191(Nat.succ n\u271d) a : Ordinal.{u_2} l : IsLimit (succ a) h : cof (succ a) < \u2135\u2080 \u22a2 False ** exact not_succ_isLimit _ l ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Language.Formula.realize_iExs ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M \u22a2 Formula.Realize (Formula.iExs f \u03c6) v \u2194 \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a) ** let e := Classical.choice (Classical.choose_spec (Finite.exists_equiv_fin \u03b3)) ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) \u22a2 Formula.Realize (Formula.iExs f \u03c6) v \u2194 \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a) ** rw [Formula.iExs] ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) \u22a2 Formula.Realize (let e := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))); exs (relabel (fun a => Sum.map id (\u2191e) (f a)) \u03c6)) v \u2194 \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a) ** simp only [Nat.add_zero, realize_exs, realize_relabel, Function.comp,\n castAdd_zero, castIso_refl, OrderIso.refl_apply, Sum.elim_map, id_eq] ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) \u22a2 (\u2203 xs, Realize \u03c6 (fun x => Sum.elim (fun x => v x) (fun x => xs (Fin.cast (_ : Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x))) (f x)) fun x => xs (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x)) \u2194 \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a) ** rw [\u2190 not_iff_not, not_exists, not_exists] ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) \u22a2 (\u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M), \u00acRealize \u03c6 (fun x_1 => Sum.elim (fun x => v x) (fun x_2 => x (Fin.cast (_ : Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x_2))) (f x_1)) fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194 \u2200 (x : \u03b3 \u2192 M), \u00acFormula.Realize \u03c6 fun a => Sum.elim v x (f a) ** refine Equiv.forall_congr ?_ ?_ ** case refine_1 L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) \u22a2 (Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M) \u2243 (\u03b3 \u2192 M) ** 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 ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) x\u271d : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M \u22a2 (fun v => v \u2218 \u2191e.symm) ((fun v => v \u2218 \u2191e) x\u271d) = x\u271d ** simp [Function.comp] ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) x\u271d : \u03b3 \u2192 M \u22a2 (fun v => v \u2218 \u2191e) ((fun v => v \u2218 \u2191e.symm) x\u271d) = x\u271d ** simp [Function.comp] ** case refine_2 L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) \u22a2 \u2200 {x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M}, (\u00acRealize \u03c6 (fun x_1 => Sum.elim (fun x => v x) (fun x_2 => x (Fin.cast (_ : Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x_2))) (f x_1)) fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194 \u00acFormula.Realize \u03c6 fun a => Sum.elim v (\u2191{ toFun := fun v => v \u2218 \u2191e, invFun := fun v => v \u2218 \u2191e.symm, left_inv := (_ : \u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M), (fun x_1 => x (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm x_1))) = x), right_inv := (_ : \u2200 (x : \u03b3 \u2192 M), (fun x_1 => x (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x_1))) = x) } x) (f a) ** intro x ** case refine_2 L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M \u22a2 (\u00acRealize \u03c6 (fun x_1 => Sum.elim (fun x => v x) (fun x_2 => x (Fin.cast (_ : Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x_2))) (f x_1)) fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194 \u00acFormula.Realize \u03c6 fun a => Sum.elim v (\u2191{ toFun := fun v => v \u2218 \u2191e, invFun := fun v => v \u2218 \u2191e.symm, left_inv := (_ : \u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M), (fun x_1 => x (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm x_1))) = x), right_inv := (_ : \u2200 (x : \u03b3 \u2192 M), (fun x_1 => x (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x_1))) = x) } x) (f a) ** rw [Formula.Realize, iff_iff_eq] ** case refine_2 L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M \u22a2 (\u00acRealize \u03c6 (fun x_1 => Sum.elim (fun x => v x) (fun x_2 => x (Fin.cast (_ : Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x_2))) (f x_1)) fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) = \u00acRealize \u03c6 (fun a => Sum.elim v (\u2191{ toFun := fun v => v \u2218 \u2191e, invFun := fun v => v \u2218 \u2191e.symm, left_inv := (_ : \u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M), (fun x_1 => x (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm x_1))) = x), right_inv := (_ : \u2200 (x : \u03b3 \u2192 M), (fun x_1 => x (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x_1))) = x) } x) (f a)) default ** congr ** case refine_2.e_a.e__xs L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M \u22a2 (fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) = default ** funext i ** case refine_2.e_a.e__xs.h L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b3 : Structure L M inst\u271d\u00b2 : Structure L N inst\u271d\u00b9 : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L inst\u271d : Finite \u03b3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 \u03c6 : Formula L \u03b1 v : \u03b2 \u2192 M e : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))) x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M i : Fin 0 \u22a2 x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) i) = default i ** exact i.elim0 ** Qed", + "informal": "" + }, + { + "formal": "Monoid.CoprodI.lift_word_prod_nontrivial_of_head_card ** \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j \u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1 ** obtain \u27e8h, hn1, hnh\u27e9 := Cardinal.three_le hcard 1 w.head\u207b\u00b9 ** case intro.intro \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j h : H i hn1 : h \u2260 1 hnh : h \u2260 (NeWord.head w)\u207b\u00b9 \u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1 ** have hnot1 : h * w.head \u2260 1 := by\n rw [\u2190 div_inv_eq_mul]\n exact div_ne_one_of_ne hnh ** case intro.intro \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j h : H i hn1 : h \u2260 1 hnh : h \u2260 (NeWord.head w)\u207b\u00b9 hnot1 : h * NeWord.head w \u2260 1 \u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1 ** 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)) ** case intro.intro \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j h : H i hn1 : h \u2260 1 hnh : h \u2260 (NeWord.head w)\u207b\u00b9 hnot1 : h * NeWord.head w \u2260 1 w' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1)) \u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1 ** have hw' : lift f w'.prod \u2260 1 :=\n lift_word_prod_nontrivial_of_head_eq_last f X hXnonempty hXdisj hpp w' ** case intro.intro \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j h : H i hn1 : h \u2260 1 hnh : h \u2260 (NeWord.head w)\u207b\u00b9 hnot1 : h * NeWord.head w \u2260 1 w' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1)) hw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1 \u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1 ** intro heq1 ** case intro.intro \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j h : H i hn1 : h \u2260 1 hnh : h \u2260 (NeWord.head w)\u207b\u00b9 hnot1 : h * NeWord.head w \u2260 1 w' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1)) hw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1 heq1 : \u2191(\u2191lift f) (NeWord.prod w) = 1 \u22a2 False ** apply hw' ** case intro.intro \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j h : H i hn1 : h \u2260 1 hnh : h \u2260 (NeWord.head w)\u207b\u00b9 hnot1 : h * NeWord.head w \u2260 1 w' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1)) hw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1 heq1 : \u2191(\u2191lift f) (NeWord.prod w) = 1 \u22a2 \u2191(\u2191lift f) (NeWord.prod w') = 1 ** simp [heq1] ** \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j h : H i hn1 : h \u2260 1 hnh : h \u2260 (NeWord.head w)\u207b\u00b9 \u22a2 h * NeWord.head w \u2260 1 ** rw [\u2190 div_inv_eq_mul] ** \u03b9 : Type u_1 M : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i) N : Type u_3 inst\u271d\u00b3 : Monoid N hnontriv : Nontrivial \u03b9 G : Type u_4 inst\u271d\u00b2 : Group G H : \u03b9 \u2192 Type u_5 inst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i) f : (i : \u03b9) \u2192 H i \u2192* G hcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i) \u03b1 : Type u_6 inst\u271d : MulAction G \u03b1 X : \u03b9 \u2192 Set \u03b1 hXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i) hXdisj : Pairwise fun i j => Disjoint (X i) (X j) hpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i i j : \u03b9 w : NeWord H i j hcard : 3 \u2264 #(H i) hheadtail : i \u2260 j h : H i hn1 : h \u2260 1 hnh : h \u2260 (NeWord.head w)\u207b\u00b9 \u22a2 h / (NeWord.head w)\u207b\u00b9 \u2260 1 ** exact div_ne_one_of_ne hnh ** Qed", + "informal": "" + }, + { + "formal": "Finset.sup'_lt_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s : Finset \u03b9 H : Finset.Nonempty s f : \u03b9 \u2192 \u03b1 a : \u03b1 \u22a2 sup' s H f < a \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i < a ** rw [\u2190 WithBot.coe_lt_coe, coe_sup', Finset.sup_lt_iff (WithBot.bot_lt_coe a)] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s : Finset \u03b9 H : Finset.Nonempty s f : \u03b9 \u2192 \u03b1 a : \u03b1 \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 ** exact ball_congr (fun _ _ => WithBot.coe_lt_coe) ** Qed", + "informal": "" + }, + { + "formal": "TopCat.snd_openEmbedding_of_left_openEmbedding ** J : Type v inst\u271d : SmallCategory J X\u271d Y\u271d Z : TopCat X Y S : TopCat f : X \u27f6 S H : OpenEmbedding \u2191f g : Y \u27f6 S \u22a2 OpenEmbedding \u2191pullback.snd ** 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)) ** case h.e'_5.h J : Type v inst\u271d : SmallCategory J X\u271d Y\u271d Z : TopCat X Y S : TopCat f : X \u27f6 S H : OpenEmbedding \u2191f g : Y \u27f6 S e_2\u271d : (forget TopCat).obj Y = \u2191Y \u22a2 \u2191pullback.snd = \u2191(homeoOfIso (asIso pullback.snd)) \u2218 \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)) ** erw [\u2190 coe_comp] ** case h.e'_5.h J : Type v inst\u271d : SmallCategory J X\u271d Y\u271d Z : TopCat X Y S : TopCat f : X \u27f6 S H : OpenEmbedding \u2191f g : Y \u27f6 S e_2\u271d : (forget TopCat).obj Y = \u2191Y \u22a2 \u2191pullback.snd = \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) ** simp ** J : Type v inst\u271d : SmallCategory J X\u271d Y\u271d Z : TopCat X Y S : TopCat f : X \u27f6 S H : OpenEmbedding \u2191f g : Y \u27f6 S \u22a2 g \u226b \ud835\udfd9 S = \ud835\udfd9 Y \u226b g ** simp ** Qed", + "informal": "" + }, + { + "formal": "ZMod.coe_valMinAbs ** k n : \u2115 h\u271d : k = n + 1 x : ZMod (namedPattern k (n + 1) h\u271d) \u22a2 \u2191(valMinAbs x) = x ** rw [valMinAbs_def_pos] ** k n : \u2115 h\u271d : k = n + 1 x : ZMod (namedPattern k (n + 1) h\u271d) \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 ** split_ifs ** case pos k n : \u2115 h\u271d\u00b9 : k = n + 1 x : ZMod (namedPattern k (n + 1) h\u271d\u00b9) h\u271d : val x \u2264 namedPattern k (n + 1) h\u271d\u00b9 / 2 \u22a2 \u2191\u2191(val x) = x ** rw [Int.cast_ofNat, nat_cast_zmod_val] ** case neg k n : \u2115 h\u271d\u00b9 : k = n + 1 x : ZMod (namedPattern k (n + 1) h\u271d\u00b9) h\u271d : \u00acval x \u2264 namedPattern k (n + 1) h\u271d\u00b9 / 2 \u22a2 \u2191(\u2191(val x) - \u2191(namedPattern k (n + 1) h\u271d\u00b9)) = x ** rw [Int.cast_sub, Int.cast_ofNat, nat_cast_zmod_val, Int.cast_ofNat, nat_cast_self, sub_zero] ** Qed", + "informal": "" + }, + { + "formal": "NonUnitalAlgHom.coe_injective ** R : Type u A : Type v B : Type w C : Type w\u2081 inst\u271d\u2076 : Monoid R inst\u271d\u2075 : NonUnitalNonAssocSemiring A inst\u271d\u2074 : DistribMulAction R A inst\u271d\u00b3 : NonUnitalNonAssocSemiring B inst\u271d\u00b2 : DistribMulAction R B inst\u271d\u00b9 : NonUnitalNonAssocSemiring C inst\u271d : DistribMulAction R C \u22a2 Function.Injective FunLike.coe ** rintro \u27e8\u27e8\u27e8f, _\u27e9, _\u27e9, _\u27e9 \u27e8\u27e8\u27e8g, _\u27e9, _\u27e9, _\u27e9 h ** case mk.mk.mk.mk.mk.mk R : Type u A : Type v B : Type w C : Type w\u2081 inst\u271d\u2076 : Monoid R inst\u271d\u2075 : NonUnitalNonAssocSemiring A inst\u271d\u2074 : DistribMulAction R A inst\u271d\u00b3 : NonUnitalNonAssocSemiring B inst\u271d\u00b2 : DistribMulAction R B inst\u271d\u00b9 : NonUnitalNonAssocSemiring C inst\u271d : DistribMulAction R C f : A \u2192 B map_smul'\u271d\u00b9 : \u2200 (m : R) (x : A), f (m \u2022 x) = m \u2022 f x map_zero'\u271d\u00b9 : MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } 0 = 0 map_add'\u271d\u00b9 : \u2200 (x y : A), MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } (x + y) = MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } x + MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } y map_mul'\u271d\u00b9 : \u2200 (x y : A), MulActionHom.toFun { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 }.toMulActionHom (x * y) = MulActionHom.toFun { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 }.toMulActionHom x * MulActionHom.toFun { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 }.toMulActionHom y g : A \u2192 B map_smul'\u271d : \u2200 (m : R) (x : A), g (m \u2022 x) = m \u2022 g x map_zero'\u271d : MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } 0 = 0 map_add'\u271d : \u2200 (x y : A), MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } (x + y) = MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } x + MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } y map_mul'\u271d : \u2200 (x y : A), MulActionHom.toFun { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }.toMulActionHom (x * y) = MulActionHom.toFun { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }.toMulActionHom x * MulActionHom.toFun { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }.toMulActionHom y h : \u2191{ toDistribMulActionHom := { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 }, map_mul' := map_mul'\u271d\u00b9 } = \u2191{ toDistribMulActionHom := { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }, map_mul' := map_mul'\u271d } \u22a2 { toDistribMulActionHom := { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 }, map_mul' := map_mul'\u271d\u00b9 } = { toDistribMulActionHom := { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }, map_mul' := map_mul'\u271d } ** congr ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.preserves_mono_of_preservesLimit ** C : Type u\u2081 D : Type u\u2082 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} C inst\u271d\u00b2 : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y : C f : X \u27f6 Y inst\u271d\u00b9 : PreservesLimit (cospan f f) F inst\u271d : Mono f \u22a2 Mono (F.map f) ** have := isLimitPullbackConeMapOfIsLimit F _ (PullbackCone.isLimitMkIdId f) ** C : Type u\u2081 D : Type u\u2082 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} C inst\u271d\u00b2 : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y : C f : X \u27f6 Y inst\u271d\u00b9 : PreservesLimit (cospan f f) F inst\u271d : Mono f this : let_fun this := (_ : F.map (\ud835\udfd9 X) \u226b F.map f = F.map (\ud835\udfd9 X) \u226b F.map f); IsLimit (PullbackCone.mk (F.map (\ud835\udfd9 X)) (F.map (\ud835\udfd9 X)) this) \u22a2 Mono (F.map f) ** simp_rw [F.map_id] at this ** C : Type u\u2081 D : Type u\u2082 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} C inst\u271d\u00b2 : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y : C f : X \u27f6 Y inst\u271d\u00b9 : PreservesLimit (cospan f f) F inst\u271d : Mono f this : 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)) \u22a2 Mono (F.map f) ** apply PullbackCone.mono_of_isLimitMkIdId _ this ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.martingale_of_condexp_sub_eq_zero_nat ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 \u22a2 Martingale f \ud835\udca2 \u03bc ** 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 ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 i : \u2115 \u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i] ** rw [\u2190 neg_sub] ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 i : \u2115 \u22a2 0 \u2264\u1d50[\u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i] ** refine' (EventuallyEq.trans _ (condexp_neg _).symm).le ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 i : \u2115 \u22a2 0 =\u1d50[\u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] ** filter_upwards [hf i] with x hx ** case h \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 i : \u2115 x : \u03a9 hx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = OfNat.ofNat 0 x \u22a2 OfNat.ofNat 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x ** simpa only [Pi.zero_apply, Pi.neg_apply, zero_eq_neg] ** Qed", + "informal": "" + }, + { + "formal": "WithSeminorms.continuous_seminorm ** \ud835\udd5c : Type u_1 \ud835\udd5c\u2082 : Type u_2 \ud835\udd5d : Type u_3 \ud835\udd5d\u2082 : Type u_4 E : Type u_5 F : Type u_6 G : Type u_7 \u03b9 : Type u_8 \u03b9' : Type u_9 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : AddCommGroup E inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Nonempty \u03b9 t : TopologicalSpace E p : SeminormFamily \ud835\udd5c E \u03b9 hp : WithSeminorms p i : \u03b9 \u22a2 Continuous \u2191(p i) ** have := hp.topologicalAddGroup ** \ud835\udd5c : Type u_1 \ud835\udd5c\u2082 : Type u_2 \ud835\udd5d : Type u_3 \ud835\udd5d\u2082 : Type u_4 E : Type u_5 F : Type u_6 G : Type u_7 \u03b9 : Type u_8 \u03b9' : Type u_9 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : AddCommGroup E inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Nonempty \u03b9 t : TopologicalSpace E p : SeminormFamily \ud835\udd5c E \u03b9 hp : WithSeminorms p i : \u03b9 this : TopologicalAddGroup E \u22a2 Continuous \u2191(p i) ** exact continuous_iInf_dom (@continuous_norm _ (p i).toSeminormedAddGroup) ** Qed", + "informal": "" + }, + { + "formal": "Zspan.fract_fract ** E : Type u_1 \u03b9 : Type u_2 K : Type u_3 inst\u271d\u2074 : NormedLinearOrderedField K inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace K E b : Basis \u03b9 K E inst\u271d\u00b9 : FloorRing K inst\u271d : Fintype \u03b9 m : E x\u271d : \u03b9 \u22a2 \u2191(\u2191b.repr (fract b (fract b m))) x\u271d = \u2191(\u2191b.repr (fract b m)) x\u271d ** classical simp only [repr_fract_apply, Int.fract_fract] ** E : Type u_1 \u03b9 : Type u_2 K : Type u_3 inst\u271d\u2074 : NormedLinearOrderedField K inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace K E b : Basis \u03b9 K E inst\u271d\u00b9 : FloorRing K inst\u271d : Fintype \u03b9 m : E x\u271d : \u03b9 \u22a2 \u2191(\u2191b.repr (fract b (fract b m))) x\u271d = \u2191(\u2191b.repr (fract b m)) x\u271d ** simp only [repr_fract_apply, Int.fract_fract] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.mem_roots_sub_C' ** R : Type u S : Type v T : Type w a\u271d b : R n : \u2115 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R p\u271d q p : R[X] a x : R \u22a2 x \u2208 roots (p - \u2191C a) \u2194 p \u2260 \u2191C a \u2227 eval x p = a ** rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C] ** Qed", + "informal": "" + }, + { + "formal": "LocalizedModule.r.isEquiv ** R : Type u inst\u271d\u00b2 : CommSemiring R S : Submonoid R M : Type v inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M x\u271d : M \u00d7 { x // x \u2208 S } m : M s : { x // x \u2208 S } \u22a2 1 \u2022 (m, s).2 \u2022 (m, s).1 = 1 \u2022 (m, s).2 \u2022 (m, s).1 ** rw [one_smul] ** R : Type u inst\u271d\u00b2 : CommSemiring R S : Submonoid R M : Type v inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M x\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : M \u00d7 { x // x \u2208 S } m1 : M s1 : { x // x \u2208 S } m2 : M s2 : { x // x \u2208 S } x\u271d\u00b9 : r S M (m1, s1) (m2, s2) m3 : M s3 : { x // x \u2208 S } x\u271d : r S M (m2, s2) (m3, s3) u1 : { x // x \u2208 S } hu1 : u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1 = u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1 u2 : { x // x \u2208 S } hu2 : u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1 = u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1 \u22a2 r S M (m1, s1) (m3, s3) ** use u1 * u2 * s2 ** case h R : Type u inst\u271d\u00b2 : CommSemiring R S : Submonoid R M : Type v inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M x\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : M \u00d7 { x // x \u2208 S } m1 : M s1 : { x // x \u2208 S } m2 : M s2 : { x // x \u2208 S } x\u271d\u00b9 : r S M (m1, s1) (m2, s2) m3 : M s3 : { x // x \u2208 S } x\u271d : r S M (m2, s2) (m3, s3) u1 : { x // x \u2208 S } hu1 : u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1 = u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1 u2 : { x // x \u2208 S } hu2 : u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1 = u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1 hu1' : (fun x x_1 => x \u2022 x_1) (u2 * s3) (u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1) = (fun x x_1 => x \u2022 x_1) (u2 * s3) (u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1) hu2' : (fun x x_1 => x \u2022 x_1) (u1 * s1) (u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1) = (fun x x_1 => x \u2022 x_1) (u1 * s1) (u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1) \u22a2 (u1 * u2 * s2) \u2022 (m3, s3).2 \u2022 (m1, s1).1 = (u1 * u2 * s2) \u2022 (m1, s1).2 \u2022 (m3, s3).1 ** simp only [\u2190 mul_smul, smul_assoc, mul_assoc, mul_comm, mul_left_comm] at hu1' hu2' \u22a2 ** case h R : Type u inst\u271d\u00b2 : CommSemiring R S : Submonoid R M : Type v inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M x\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : M \u00d7 { x // x \u2208 S } m1 : M s1 : { x // x \u2208 S } m2 : M s2 : { x // x \u2208 S } x\u271d\u00b9 : r S M (m1, s1) (m2, s2) m3 : M s3 : { x // x \u2208 S } x\u271d : r S M (m2, s2) (m3, s3) u1 : { x // x \u2208 S } hu1 : u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1 = u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1 u2 : { x // x \u2208 S } hu2 : u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1 = u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1 hu1' : (s1 * (s3 * (u1 * u2))) \u2022 m2 = (s2 * (s3 * (u1 * u2))) \u2022 m1 hu2' : (s1 * (s2 * (u1 * u2))) \u2022 m3 = (s1 * (s3 * (u1 * u2))) \u2022 m2 \u22a2 (s2 * (s3 * (u1 * u2))) \u2022 m1 = (s1 * (s2 * (u1 * u2))) \u2022 m3 ** rw [hu2', hu1'] ** Qed", + "informal": "" + }, + { + "formal": "Associates.mem_factors_of_dvd ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d : UniqueFactorizationMonoid \u03b1 dec : DecidableEq \u03b1 dec' : DecidableEq (Associates \u03b1) a p : \u03b1 ha0 : a \u2260 0 hp : Irreducible p hd : p \u2223 a \u22a2 Associates.mk p \u2208 factors (Associates.mk a) ** rw [factors_mk _ ha0] ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d : UniqueFactorizationMonoid \u03b1 dec : DecidableEq \u03b1 dec' : DecidableEq (Associates \u03b1) a p : \u03b1 ha0 : a \u2260 0 hp : Irreducible p hd : p \u2223 a \u22a2 Associates.mk p \u2208 \u2191(factors' a) ** exact mem_factorSet_some.mpr (mem_factors'_of_dvd ha0 hp hd) ** Qed", + "informal": "" + }, + { + "formal": "Real.exists_extension_norm_eq ** E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** 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 ** case intro.intro E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d g : E \u2192\u2097[\u211d] \u211d g_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x g_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016 \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** 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 ** E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d c : \u211d hc : 0 < c x : E \u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (c \u2022 x) = c * (fun x => \u2016f\u2016 * \u2016x\u2016) x ** simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm] ** E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d x y : E \u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (x + y) \u2264 (fun x => \u2016f\u2016 * \u2016x\u2016) x + (fun x => \u2016f\u2016 * \u2016x\u2016) y ** rw [\u2190 left_distrib] ** E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d x y : E \u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (x + y) \u2264 \u2016f\u2016 * (\u2016x\u2016 + \u2016y\u2016) ** exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@norm_nonneg _ _ f) ** case intro.intro E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d g : E \u2192\u2097[\u211d] \u211d g_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x g_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016 g' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016) \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** refine' \u27e8g', g_eq, _\u27e9 ** case intro.intro E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d g : E \u2192\u2097[\u211d] \u211d g_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x g_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016 g' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016) \u22a2 \u2016g'\u2016 = \u2016f\u2016 ** apply le_antisymm (g.mkContinuous_norm_le (norm_nonneg f) _) ** case intro.intro E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d g : E \u2192\u2097[\u211d] \u211d g_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x g_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016 g' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016) \u22a2 \u2016f\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 ** refine' f.op_norm_le_bound (norm_nonneg _) fun x => _ ** case intro.intro E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d g : E \u2192\u2097[\u211d] \u211d g_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x g_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016 g' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016) x : { x // x \u2208 p } \u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016 ** dsimp at g_eq ** case intro.intro E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d g : E \u2192\u2097[\u211d] \u211d g_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x g_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016 g' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016) x : { x // x \u2208 p } \u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016 ** rw [\u2190 g_eq] ** case intro.intro E : Type u_1 inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \u211d E p : Subspace \u211d E f : { x // x \u2208 p } \u2192L[\u211d] \u211d g : E \u2192\u2097[\u211d] \u211d g_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x g_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016 g' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016) x : { x // x \u2208 p } \u22a2 \u2016\u2191g \u2191x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016 ** apply g'.le_op_norm ** Qed", + "informal": "" + }, + { + "formal": "DirectSum.coe_of_mul_apply_of_not_le ** \u03b9 : Type u_1 \u03c3 : Type u_2 S : Type u_3 R : Type u_4 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : SetLike \u03c3 R inst\u271d\u00b2 : AddSubmonoidClass \u03c3 R A : \u03b9 \u2192 \u03c3 inst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9 inst\u271d : SetLike.GradedMonoid A i : \u03b9 r : { x // x \u2208 A i } r' : \u2a01 (i : \u03b9), { x // x \u2208 A i } n : \u03b9 h : \u00aci \u2264 n \u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = 0 ** rw [coe_mul_apply_eq_dfinsupp_sum] ** \u03b9 : Type u_1 \u03c3 : Type u_2 S : Type u_3 R : Type u_4 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : SetLike \u03c3 R inst\u271d\u00b2 : AddSubmonoidClass \u03c3 R A : \u03b9 \u2192 \u03c3 inst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9 inst\u271d : SetLike.GradedMonoid A i : \u03b9 r : { x // x \u2208 A i } r' : \u2a01 (i : \u03b9), { x // x \u2208 A i } n : \u03b9 h : \u00aci \u2264 n \u22a2 (DFinsupp.sum (\u2191(of (fun i => { x // x \u2208 A i }) i) r) fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0) = 0 ** apply (DFinsupp.sum_single_index _).trans ** \u03b9 : Type u_1 \u03c3 : Type u_2 S : Type u_3 R : Type u_4 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : SetLike \u03c3 R inst\u271d\u00b2 : AddSubmonoidClass \u03c3 R A : \u03b9 \u2192 \u03c3 inst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9 inst\u271d : SetLike.GradedMonoid A i : \u03b9 r : { x // x \u2208 A i } r' : \u2a01 (i : \u03b9), { x // x \u2208 A i } n : \u03b9 h : \u00aci \u2264 n \u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = 0 \u03b9 : Type u_1 \u03c3 : Type u_2 S : Type u_3 R : Type u_4 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : SetLike \u03c3 R inst\u271d\u00b2 : AddSubmonoidClass \u03c3 R A : \u03b9 \u2192 \u03c3 inst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9 inst\u271d : SetLike.GradedMonoid A i : \u03b9 r : { x // x \u2208 A i } r' : \u2a01 (i : \u03b9), { x // x \u2208 A i } n : \u03b9 h : \u00aci \u2264 n \u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0 ** swap ** \u03b9 : Type u_1 \u03c3 : Type u_2 S : Type u_3 R : Type u_4 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : SetLike \u03c3 R inst\u271d\u00b2 : AddSubmonoidClass \u03c3 R A : \u03b9 \u2192 \u03c3 inst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9 inst\u271d : SetLike.GradedMonoid A i : \u03b9 r : { x // x \u2208 A i } r' : \u2a01 (i : \u03b9), { x // x \u2208 A i } n : \u03b9 h : \u00aci \u2264 n \u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0 ** simp_rw [ZeroMemClass.coe_zero, zero_mul, ite_self] ** \u03b9 : Type u_1 \u03c3 : Type u_2 S : Type u_3 R : Type u_4 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : SetLike \u03c3 R inst\u271d\u00b2 : AddSubmonoidClass \u03c3 R A : \u03b9 \u2192 \u03c3 inst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9 inst\u271d : SetLike.GradedMonoid A i : \u03b9 r : { x // x \u2208 A i } r' : \u2a01 (i : \u03b9), { x // x \u2208 A i } n : \u03b9 h : \u00aci \u2264 n \u22a2 (DFinsupp.sum r' fun j rj => 0) = 0 ** exact DFinsupp.sum_zero ** \u03b9 : Type u_1 \u03c3 : Type u_2 S : Type u_3 R : Type u_4 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : SetLike \u03c3 R inst\u271d\u00b2 : AddSubmonoidClass \u03c3 R A : \u03b9 \u2192 \u03c3 inst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9 inst\u271d : SetLike.GradedMonoid A i : \u03b9 r : { x // x \u2208 A i } r' : \u2a01 (i : \u03b9), { x // x \u2208 A i } n : \u03b9 h : \u00aci \u2264 n \u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = 0 ** rw [DFinsupp.sum, Finset.sum_ite_of_false _ _ fun x _ H => _, Finset.sum_const_zero] ** \u03b9 : Type u_1 \u03c3 : Type u_2 S : Type u_3 R : Type u_4 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : SetLike \u03c3 R inst\u271d\u00b2 : AddSubmonoidClass \u03c3 R A : \u03b9 \u2192 \u03c3 inst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9 inst\u271d : SetLike.GradedMonoid A i : \u03b9 r : { x // x \u2208 A i } r' : \u2a01 (i : \u03b9), { x // x \u2208 A i } n : \u03b9 h : \u00aci \u2264 n \u22a2 \u2200 (x : \u03b9), x \u2208 DFinsupp.support r' \u2192 i + x = n \u2192 False ** exact fun x _ H => h ((self_le_add_right i x).trans_eq H) ** Qed", + "informal": "" + }, + { + "formal": "eq_mpr_bijective ** \u03b1 \u03b2 : Sort u_1 h : \u03b1 = \u03b2 \u22a2 Bijective (Eq.mpr h) ** cases h ** case refl \u03b1 : Sort u_1 \u22a2 Bijective (Eq.mpr (_ : \u03b1 = \u03b1)) ** refine \u27e8fun _ _ \u21a6 id, fun x \u21a6 \u27e8x, rfl\u27e9\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.LocallyCoverDense.pushforward_cover_iff_cover_pullback ** C : Type u_1 inst\u271d\u2074 : Category.{u_3, u_1} C D : Type u_2 inst\u271d\u00b3 : Category.{u_4, u_2} D G : C \u2964 D J : GrothendieckTopology C K : GrothendieckTopology D A : Type v inst\u271d\u00b2 : Category.{u, v} A inst\u271d\u00b9 : Full G inst\u271d : Faithful G Hld : LocallyCoverDense K G X : C S : Sieve X \u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) \u2194 \u2203 T, Sieve.functorPullback G \u2191T = S ** constructor ** case mp C : Type u_1 inst\u271d\u2074 : Category.{u_3, u_1} C D : Type u_2 inst\u271d\u00b3 : Category.{u_4, u_2} D G : C \u2964 D J : GrothendieckTopology C K : GrothendieckTopology D A : Type v inst\u271d\u00b2 : Category.{u, v} A inst\u271d\u00b9 : Full G inst\u271d : Faithful G Hld : LocallyCoverDense K G X : C S : Sieve X \u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) \u2192 \u2203 T, Sieve.functorPullback G \u2191T = S ** intro hS ** case mp C : Type u_1 inst\u271d\u2074 : Category.{u_3, u_1} C D : Type u_2 inst\u271d\u00b3 : Category.{u_4, u_2} D G : C \u2964 D J : GrothendieckTopology C K : GrothendieckTopology D A : Type v inst\u271d\u00b2 : Category.{u, v} A inst\u271d\u00b9 : Full G inst\u271d : Faithful G Hld : LocallyCoverDense K G X : C S : Sieve X hS : GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) \u22a2 \u2203 T, Sieve.functorPullback G \u2191T = S ** exact \u27e8\u27e8_, hS\u27e9, (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S\u27e9 ** case mpr C : Type u_1 inst\u271d\u2074 : Category.{u_3, u_1} C D : Type u_2 inst\u271d\u00b3 : Category.{u_4, u_2} D G : C \u2964 D J : GrothendieckTopology C K : GrothendieckTopology D A : Type v inst\u271d\u00b2 : Category.{u, v} A inst\u271d\u00b9 : Full G inst\u271d : Faithful G Hld : LocallyCoverDense K G X : C S : Sieve X \u22a2 (\u2203 T, Sieve.functorPullback G \u2191T = S) \u2192 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) ** rintro \u27e8T, rfl\u27e9 ** case mpr.intro C : Type u_1 inst\u271d\u2074 : Category.{u_3, u_1} C D : Type u_2 inst\u271d\u00b3 : Category.{u_4, u_2} D G : C \u2964 D J : GrothendieckTopology C K : GrothendieckTopology D A : Type v inst\u271d\u00b2 : Category.{u, v} A inst\u271d\u00b9 : Full G inst\u271d : Faithful G Hld : LocallyCoverDense K G X : C T : \u2191(GrothendieckTopology.sieves K (G.obj X)) \u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G (Sieve.functorPullback G \u2191T)) ** exact Hld T ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.C_mul_comp ** R : Type u S : Type v T : Type w \u03b9 : Type y a b : R m n : \u2115 inst\u271d : Semiring R p q r : R[X] \u22a2 comp (\u2191C a * p) r = \u2191C a * comp p r ** 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] ** case h_add R : Type u S : Type v T : Type w \u03b9 : Type y a b : R m n : \u2115 inst\u271d : Semiring R p\u271d q\u271d r p q : R[X] hp : comp (\u2191C a * p) r = \u2191C a * comp p r hq : comp (\u2191C a * q) r = \u2191C a * comp q r \u22a2 comp (\u2191C a * (p + q)) r = \u2191C a * comp (p + q) r ** simp [hp, hq, mul_add] ** case h_monomial R : Type u S : Type v T : Type w \u03b9 : Type y a b\u271d : R m n\u271d : \u2115 inst\u271d : Semiring R p q r : R[X] n : \u2115 b : R \u22a2 comp (\u2191C a * \u2191(monomial n) b) r = \u2191C a * comp (\u2191(monomial n) b) r ** simp [mul_assoc] ** Qed", + "informal": "" + }, + { + "formal": "EMetric.subset_countable_closure_of_compact ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 inst\u271d : PseudoEMetricSpace \u03b1 x y z : \u03b1 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s\u271d t s : Set \u03b1 hs : IsCompact s \u22a2 \u2203 t, t \u2286 s \u2227 Set.Countable t \u2227 s \u2286 closure t ** refine' subset_countable_closure_of_almost_dense_set s fun \u03b5 h\u03b5 => _ ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 inst\u271d : PseudoEMetricSpace \u03b1 x y z : \u03b1 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s\u271d t s : Set \u03b1 hs : IsCompact s \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 \u22a2 \u2203 t, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, closedBall x \u03b5 ** rcases totallyBounded_iff'.1 hs.totallyBounded \u03b5 h\u03b5 with \u27e8t, -, htf, hst\u27e9 ** case intro.intro.intro \u03b1 : Type u \u03b2 : Type v X : Type u_1 inst\u271d : PseudoEMetricSpace \u03b1 x y z : \u03b1 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s\u271d t\u271d s : Set \u03b1 hs : IsCompact s \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 t : Set \u03b1 htf : Set.Finite t hst : s \u2286 \u22c3 y \u2208 t, ball y \u03b5 \u22a2 \u2203 t, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, closedBall x \u03b5 ** exact \u27e8t, htf.countable, hst.trans <| iUnion\u2082_mono fun _ _ => ball_subset_closedBall\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.exists_degree_lt ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : Fintype \u03c3 f : MvPolynomial \u03c3 R n : \u2115 h : totalDegree f < n * Fintype.card \u03c3 d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support f \u22a2 \u2203 i, \u2191d i < n ** contrapose! h ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : Fintype \u03c3 f : MvPolynomial \u03c3 R n : \u2115 d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support f h : \u2200 (i : \u03c3), n \u2264 \u2191d i \u22a2 n * Fintype.card \u03c3 \u2264 totalDegree f ** 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 ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : Fintype \u03c3 f : MvPolynomial \u03c3 R n : \u2115 d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support f h : \u2200 (i : \u03c3), n \u2264 \u2191d i \u22a2 n * Fintype.card \u03c3 = \u2211 _s : \u03c3, n ** rw [Finset.sum_const, Nat.nsmul_eq_mul, mul_comm, Finset.card_univ] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : Fintype \u03c3 f : MvPolynomial \u03c3 R n : \u2115 d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support f h : \u2200 (i : \u03c3), n \u2264 \u2191d i \u22a2 \u2211 s : \u03c3, \u2191d s \u2264 sum d fun x e => e ** rw [Finsupp.sum_fintype] ** case h R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : Fintype \u03c3 f : MvPolynomial \u03c3 R n : \u2115 d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support f h : \u2200 (i : \u03c3), n \u2264 \u2191d i \u22a2 \u03c3 \u2192 0 = 0 ** intros ** case h R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : Fintype \u03c3 f : MvPolynomial \u03c3 R n : \u2115 d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support f h : \u2200 (i : \u03c3), n \u2264 \u2191d i i\u271d : \u03c3 \u22a2 0 = 0 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Commute.geom_sum\u2082_mul ** \u03b1 : Type u inst\u271d : Ring \u03b1 x y : \u03b1 h : Commute x y n : \u2115 \u22a2 (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n ** have := (h.sub_left (Commute.refl y)).geom_sum\u2082_mul_add n ** \u03b1 : Type u inst\u271d : Ring \u03b1 x y : \u03b1 h : Commute x y n : \u2115 this : (\u2211 i in range n, (x - y + y) ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = (x - y + y) ^ n \u22a2 (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n ** rw [sub_add_cancel] at this ** \u03b1 : Type u inst\u271d : Ring \u03b1 x y : \u03b1 h : Commute x y n : \u2115 this : (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = x ^ n \u22a2 (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n ** rw [\u2190 this, add_sub_cancel] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.cliqueFree_completeMultipartiteGraph ** \u03b1 : Type u_1 G H : SimpleGraph \u03b1 m n : \u2115 s : Finset \u03b1 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 V : \u03b9 \u2192 Type u_3 hc : Fintype.card \u03b9 < n \u22a2 CliqueFree (completeMultipartiteGraph V) n ** rw [cliqueFree_iff, isEmpty_iff] ** \u03b1 : Type u_1 G H : SimpleGraph \u03b1 m n : \u2115 s : Finset \u03b1 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 V : \u03b9 \u2192 Type u_3 hc : Fintype.card \u03b9 < n \u22a2 \u22a4 \u21aag completeMultipartiteGraph V \u2192 False ** intro f ** \u03b1 : Type u_1 G H : SimpleGraph \u03b1 m n : \u2115 s : Finset \u03b1 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 V : \u03b9 \u2192 Type u_3 hc : Fintype.card \u03b9 < n f : \u22a4 \u21aag completeMultipartiteGraph V \u22a2 False ** obtain \u27e8v, w, hn, he\u27e9 := exists_ne_map_eq_of_card_lt (Sigma.fst \u2218 f) (by simp [hc]) ** case intro.intro.intro \u03b1 : Type u_1 G H : SimpleGraph \u03b1 m n : \u2115 s : Finset \u03b1 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 V : \u03b9 \u2192 Type u_3 hc : Fintype.card \u03b9 < n f : \u22a4 \u21aag completeMultipartiteGraph V v w : Fin n hn : v \u2260 w he : (Sigma.fst \u2218 \u2191f) v = (Sigma.fst \u2218 \u2191f) w \u22a2 False ** rw [\u2190 top_adj, \u2190 f.map_adj_iff, comap_Adj, top_adj] at hn ** case intro.intro.intro \u03b1 : Type u_1 G H : SimpleGraph \u03b1 m n : \u2115 s : Finset \u03b1 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 V : \u03b9 \u2192 Type u_3 hc : Fintype.card \u03b9 < n f : \u22a4 \u21aag completeMultipartiteGraph V v w : Fin n hn : (\u2191f v).fst \u2260 (\u2191f w).fst he : (Sigma.fst \u2218 \u2191f) v = (Sigma.fst \u2218 \u2191f) w \u22a2 False ** exact absurd he hn ** \u03b1 : Type u_1 G H : SimpleGraph \u03b1 m n : \u2115 s : Finset \u03b1 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 V : \u03b9 \u2192 Type u_3 hc : Fintype.card \u03b9 < n f : \u22a4 \u21aag completeMultipartiteGraph V \u22a2 Fintype.card \u03b9 < Fintype.card (Fin n) ** simp [hc] ** Qed", + "informal": "" + }, + { + "formal": "List.one_le_prod_of_one_le ** \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : Monoid N inst\u271d\u00b2 : Monoid P l\u271d l\u2081 l\u2082 : List M a : M inst\u271d\u00b9 : Preorder M inst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 l : List M hl\u2081 : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x \u22a2 1 \u2264 prod l ** induction' l with hd tl ih ** case cons \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : Monoid N inst\u271d\u00b2 : Monoid P l\u271d l\u2081 l\u2082 : List M a : M inst\u271d\u00b9 : Preorder M inst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 l : List M hl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x hd : M tl : List M ih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 1 \u2264 prod tl hl\u2081 : \u2200 (x : M), x \u2208 hd :: tl \u2192 1 \u2264 x \u22a2 1 \u2264 prod (hd :: tl) ** rw [prod_cons] ** case cons \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : Monoid N inst\u271d\u00b2 : Monoid P l\u271d l\u2081 l\u2082 : List M a : M inst\u271d\u00b9 : Preorder M inst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 l : List M hl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x hd : M tl : List M ih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 1 \u2264 prod tl hl\u2081 : \u2200 (x : M), x \u2208 hd :: tl \u2192 1 \u2264 x \u22a2 1 \u2264 hd * prod tl ** 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)) ** case nil \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : Monoid N inst\u271d\u00b2 : Monoid P l\u271d l\u2081 l\u2082 : List M a : M inst\u271d\u00b9 : Preorder M inst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 l : List M hl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x hl\u2081 : \u2200 (x : M), x \u2208 [] \u2192 1 \u2264 x \u22a2 1 \u2264 prod [] ** rfl ** Qed", + "informal": "" + }, + { + "formal": "associated_zero_iff_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MonoidWithZero \u03b1 a : \u03b1 h : a ~\u1d64 0 \u22a2 a = 0 ** let \u27e8u, h\u27e9 := h.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MonoidWithZero \u03b1 a : \u03b1 h\u271d : a ~\u1d64 0 u : \u03b1\u02e3 h : 0 * \u2191u = a \u22a2 a = 0 ** simpa using h.symm ** Qed", + "informal": "" + }, + { + "formal": "SemiconjBy.neg_left ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w R : Type x inst\u271d\u00b9 : Mul R inst\u271d : HasDistribNeg R a x y : R h : SemiconjBy a x y \u22a2 SemiconjBy (-a) x y ** simp only [SemiconjBy, h.eq, neg_mul, mul_neg] ** Qed", + "informal": "" + }, + { + "formal": "InnerProductSpace.continuousLinearMapOfBilin_apply ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : CompleteSpace E B : E \u2192L\u22c6[\ud835\udd5c] E \u2192L[\ud835\udd5c] \ud835\udd5c v w : E \u22a2 inner (\u2191B\u266f v) w = \u2191(\u2191B v) w ** rw [continuousLinearMapOfBilin, coe_comp', ContinuousLinearEquiv.coe_coe,\n LinearIsometryEquiv.coe_toContinuousLinearEquiv, Function.comp_apply, toDual_symm_apply] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.edgeDensity_add_edgeDensity_compl ** \ud835\udd5c : Type u_1 \u03b9 : Type u_2 \u03ba : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 G : SimpleGraph \u03b1 inst\u271d\u00b9 : DecidableRel G.Adj s s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 hs : Finset.Nonempty s ht : Finset.Nonempty t h : Disjoint s t \u22a2 edgeDensity G s t + edgeDensity G\u1d9c s t = 1 ** rw [edgeDensity_def, edgeDensity_def, div_add_div_same, div_eq_one_iff_eq] ** \ud835\udd5c : Type u_1 \u03b9 : Type u_2 \u03ba : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 G : SimpleGraph \u03b1 inst\u271d\u00b9 : DecidableRel G.Adj s s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 hs : Finset.Nonempty s ht : Finset.Nonempty t h : Disjoint s t \u22a2 \u2191(card (interedges G s t)) + \u2191(card (interedges G\u1d9c s t)) = \u2191(card s) * \u2191(card t) ** exact_mod_cast card_interedges_add_card_interedges_compl _ h ** \ud835\udd5c : Type u_1 \u03b9 : Type u_2 \u03ba : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 G : SimpleGraph \u03b1 inst\u271d\u00b9 : DecidableRel G.Adj s s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 hs : Finset.Nonempty s ht : Finset.Nonempty t h : Disjoint s t \u22a2 \u2191(card s) * \u2191(card t) \u2260 0 ** apply mul_ne_zero <;> exact_mod_cast Nat.pos_iff_ne_zero.1 (Nonempty.card_pos \u2039_\u203a) ** Qed", + "informal": "" + }, + { + "formal": "Submodule.fst_map_fst ** R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid M\u2082 inst\u271d\u00b9 : Module R M inst\u271d : Module R M\u2082 p : Submodule R M q : Submodule R M\u2082 \u22a2 map (LinearMap.fst R M M\u2082) (fst R M M\u2082) = \u22a4 ** rw [eq_top_iff] ** R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid M\u2082 inst\u271d\u00b9 : Module R M inst\u271d : Module R M\u2082 p : Submodule R M q : Submodule R M\u2082 \u22a2 \u22a4 \u2264 map (LinearMap.fst R M M\u2082) (fst R M M\u2082) ** rintro x - ** R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid M\u2082 inst\u271d\u00b9 : Module R M inst\u271d : Module R M\u2082 p : Submodule R M q : Submodule R M\u2082 x : M \u22a2 x \u2208 map (LinearMap.fst R M M\u2082) (fst R M M\u2082) ** simp only [fst, comap_bot, mem_map, mem_ker, snd_apply, fst_apply,\n Prod.exists, exists_eq_left, exists_eq] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Bicategory.inv_whiskerLeft ** B : Type u inst\u271d\u00b9 : Bicategory B a b c d e : B f : a \u27f6 b g h : b \u27f6 c \u03b7 : g \u27f6 h inst\u271d : IsIso \u03b7 \u22a2 inv (f \u25c1 \u03b7) = f \u25c1 inv \u03b7 ** apply IsIso.inv_eq_of_hom_inv_id ** case hom_inv_id B : Type u inst\u271d\u00b9 : Bicategory B a b c d e : B f : a \u27f6 b g h : b \u27f6 c \u03b7 : g \u27f6 h inst\u271d : IsIso \u03b7 \u22a2 f \u25c1 \u03b7 \u226b f \u25c1 inv \u03b7 = \ud835\udfd9 (f \u226b g) ** simp only [\u2190 whiskerLeft_comp, whiskerLeft_id, IsIso.hom_inv_id] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.MonoidalCategory.tensor_id_comp_id_tensor ** C\u271d : Type u \ud835\udc9e : Category.{v, u} C\u271d inst\u271d\u00b2 : MonoidalCategory C\u271d C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : MonoidalCategory C U V W X Y Z : C f : W \u27f6 X g : Y \u27f6 Z \u22a2 (g \u2297 \ud835\udfd9 W) \u226b (\ud835\udfd9 Z \u2297 f) = g \u2297 f ** rw [\u2190 tensor_comp] ** C\u271d : Type u \ud835\udc9e : Category.{v, u} C\u271d inst\u271d\u00b2 : MonoidalCategory C\u271d C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : MonoidalCategory C U V W X Y Z : C f : W \u27f6 X g : Y \u27f6 Z \u22a2 g \u226b \ud835\udfd9 Z \u2297 \ud835\udfd9 W \u226b f = g \u2297 f ** simp ** Qed", + "informal": "" + }, + { + "formal": "Ideal.IsPrime.homogeneousCore ** \u03b9 : Type u_1 \u03c3 : Type u_2 A : Type u_3 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9 inst\u271d\u00b2 : SetLike \u03c3 A inst\u271d\u00b9 : AddSubmonoidClass \u03c3 A \ud835\udc9c : \u03b9 \u2192 \u03c3 inst\u271d : GradedRing \ud835\udc9c I : Ideal A h : IsPrime I \u22a2 IsPrime (HomogeneousIdeal.toIdeal (Ideal.homogeneousCore \ud835\udc9c I)) ** apply (Ideal.homogeneousCore \ud835\udc9c I).is_homogeneous'.isPrime_of_homogeneous_mem_or_mem ** case homogeneous_mem_or_mem \u03b9 : Type u_1 \u03c3 : Type u_2 A : Type u_3 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9 inst\u271d\u00b2 : SetLike \u03c3 A inst\u271d\u00b9 : AddSubmonoidClass \u03c3 A \ud835\udc9c : \u03b9 \u2192 \u03c3 inst\u271d : GradedRing \ud835\udc9c I : Ideal A h : IsPrime I \u22a2 \u2200 {x y : A}, Homogeneous \ud835\udc9c x \u2192 Homogeneous \ud835\udc9c y \u2192 x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2192 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule ** rintro x y hx hy hxy ** case homogeneous_mem_or_mem \u03b9 : Type u_1 \u03c3 : Type u_2 A : Type u_3 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9 inst\u271d\u00b2 : SetLike \u03c3 A inst\u271d\u00b9 : AddSubmonoidClass \u03c3 A \ud835\udc9c : \u03b9 \u2192 \u03c3 inst\u271d : GradedRing \ud835\udc9c I : Ideal A h : IsPrime I x y : A hx : Homogeneous \ud835\udc9c x hy : Homogeneous \ud835\udc9c y hxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u22a2 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule ** have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le \ud835\udc9c I hxy) ** case homogeneous_mem_or_mem \u03b9 : Type u_1 \u03c3 : Type u_2 A : Type u_3 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9 inst\u271d\u00b2 : SetLike \u03c3 A inst\u271d\u00b9 : AddSubmonoidClass \u03c3 A \ud835\udc9c : \u03b9 \u2192 \u03c3 inst\u271d : GradedRing \ud835\udc9c I : Ideal A h : IsPrime I x y : A hx : Homogeneous \ud835\udc9c x hy : Homogeneous \ud835\udc9c y hxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule H : x \u2208 I \u2228 y \u2208 I \u22a2 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule ** refine' H.imp _ _ ** case I_ne_top \u03b9 : Type u_1 \u03c3 : Type u_2 A : Type u_3 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9 inst\u271d\u00b2 : SetLike \u03c3 A inst\u271d\u00b9 : AddSubmonoidClass \u03c3 A \ud835\udc9c : \u03b9 \u2192 \u03c3 inst\u271d : GradedRing \ud835\udc9c I : Ideal A h : IsPrime I \u22a2 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2260 \u22a4 ** exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le \ud835\udc9c I) ** case homogeneous_mem_or_mem.refine'_1 \u03b9 : Type u_1 \u03c3 : Type u_2 A : Type u_3 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9 inst\u271d\u00b2 : SetLike \u03c3 A inst\u271d\u00b9 : AddSubmonoidClass \u03c3 A \ud835\udc9c : \u03b9 \u2192 \u03c3 inst\u271d : GradedRing \ud835\udc9c I : Ideal A h : IsPrime I x y : A hx : Homogeneous \ud835\udc9c x hy : Homogeneous \ud835\udc9c y hxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule H : x \u2208 I \u2228 y \u2208 I \u22a2 x \u2208 I \u2192 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule ** exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hx ** case homogeneous_mem_or_mem.refine'_2 \u03b9 : Type u_1 \u03c3 : Type u_2 A : Type u_3 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9 inst\u271d\u00b2 : SetLike \u03c3 A inst\u271d\u00b9 : AddSubmonoidClass \u03c3 A \ud835\udc9c : \u03b9 \u2192 \u03c3 inst\u271d : GradedRing \ud835\udc9c I : Ideal A h : IsPrime I x y : A hx : Homogeneous \ud835\udc9c x hy : Homogeneous \ud835\udc9c y hxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule H : x \u2208 I \u2228 y \u2208 I \u22a2 y \u2208 I \u2192 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule ** exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy ** Qed", + "informal": "" + }, + { + "formal": "PNat.div_add_mod' ** m k : \u2115+ \u22a2 div m k * \u2191k + \u2191(mod m k) = \u2191m ** rw [mul_comm] ** m k : \u2115+ \u22a2 \u2191k * div m k + \u2191(mod m k) = \u2191m ** exact div_add_mod _ _ ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.mul_normal ** \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N \u22a2 \u2191(H \u2294 N) = \u2191H * \u2191N ** rw [sup_eq_closure_mul] ** \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N \u22a2 \u2191(closure (\u2191H * \u2191N)) = \u2191H * \u2191N ** refine Set.Subset.antisymm (fun x hx => ?_) subset_closure ** \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x : G hx : x \u2208 \u2191(closure (\u2191H * \u2191N)) \u22a2 x \u2208 \u2191H * \u2191N ** refine closure_induction'' (p := fun x => x \u2208 (H : Set G) * (N : Set G)) hx ?_ ?_ ?_ ?_ ** case refine_1 \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x : G hx : x \u2208 \u2191(closure (\u2191H * \u2191N)) \u22a2 \u2200 (x : G), x \u2208 \u2191H * \u2191N \u2192 (fun x => x \u2208 \u2191H * \u2191N) x ** rintro _ \u27e8x, y, hx, hy, rfl\u27e9 ** case refine_1.intro.intro.intro.intro \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x\u271d : G hx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N)) x y : G hx : x \u2208 \u2191H hy : y \u2208 \u2191N \u22a2 (fun x x_1 => x * x_1) x y \u2208 \u2191H * \u2191N ** exact mul_mem_mul hx hy ** case refine_2 \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x : G hx : x \u2208 \u2191(closure (\u2191H * \u2191N)) \u22a2 \u2200 (x : G), x \u2208 \u2191H * \u2191N \u2192 (fun x => x \u2208 \u2191H * \u2191N) x\u207b\u00b9 ** rintro _ \u27e8x, y, hx, hy, rfl\u27e9 ** case refine_2.intro.intro.intro.intro \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x\u271d : G hx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N)) x y : G hx : x \u2208 \u2191H hy : y \u2208 \u2191N \u22a2 ((fun x x_1 => x * x_1) x y)\u207b\u00b9 \u2208 \u2191H * \u2191N ** 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) ** case refine_3 \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x : G hx : x \u2208 \u2191(closure (\u2191H * \u2191N)) \u22a2 (fun x => x \u2208 \u2191H * \u2191N) 1 ** exact \u27e81, 1, one_mem _, one_mem _, mul_one 1\u27e9 ** case refine_4 \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x : G hx : x \u2208 \u2191(closure (\u2191H * \u2191N)) \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) ** rintro _ _ \u27e8x, y, hx, hy, rfl\u27e9 \u27e8x', y', hx', hy', rfl\u27e9 ** case refine_4.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x\u271d : G hx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N)) x y : G hx : x \u2208 \u2191H hy : y \u2208 \u2191N x' y' : G hx' : x' \u2208 \u2191H hy' : y' \u2208 \u2191N \u22a2 (fun x x_1 => x * x_1) x y * (fun x x_1 => x * x_1) x' y' \u2208 \u2191H * \u2191N ** refine \u27e8x * x', x'\u207b\u00b9 * y * x' * y', mul_mem hx hx', mul_mem ?_ hy', ?_\u27e9 ** case refine_4.intro.intro.intro.intro.intro.intro.intro.intro.refine_1 \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x\u271d : G hx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N)) x y : G hx : x \u2208 \u2191H hy : y \u2208 \u2191N x' y' : G hx' : x' \u2208 \u2191H hy' : y' \u2208 \u2191N \u22a2 x'\u207b\u00b9 * y * x' \u2208 N ** simpa using hN.conj_mem _ hy x'\u207b\u00b9 ** case refine_4.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 \u03b1 : Type u_1 G : Type u_2 A : Type u_3 S : Type u_4 inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G H N : Subgroup G hN : Normal N x\u271d : G hx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N)) x y : G hx : x \u2208 \u2191H hy : y \u2208 \u2191N x' y' : G hx' : x' \u2208 \u2191H hy' : y' \u2208 \u2191N \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' ** simp only [mul_assoc, mul_inv_cancel_left] ** Qed", + "informal": "" + }, + { + "formal": "Filter.sup_neBot ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f\u271d g\u271d : Filter \u03b1 s t : Set \u03b1 f g : Filter \u03b1 \u22a2 NeBot (f \u2294 g) \u2194 NeBot f \u2228 NeBot g ** simp only [neBot_iff, not_and_or, Ne.def, sup_eq_bot_iff] ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_disjUnion ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 h : Disjoint s\u2081 s\u2082 \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 ** refine' Eq.trans _ (fold_disjUnion h) ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 h : Disjoint s\u2081 s\u2082 \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) ** rw [one_mul] ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 h : Disjoint s\u2081 s\u2082 \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) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_stalk_iso ** C : Type u_1 inst\u271d\u2076 : Category.{u_2, u_1} C inst\u271d\u2075 : HasLimits C inst\u271d\u2074 : HasColimits C inst\u271d\u00b3 : ConcreteCategory C inst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C) inst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C) inst\u271d : PreservesFilteredColimits (CategoryTheory.forget C) X Y : SheafedSpace C f : X \u27f6 Y hf : OpenEmbedding \u2191f.base H : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x) U : Opens \u2191\u2191X.toPresheafedSpace \u22a2 IsIso (f.c.app (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f.base)).obj U))) ** 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) ** C : Type u_1 inst\u271d\u2076 : Category.{u_2, u_1} C inst\u271d\u2075 : HasLimits C inst\u271d\u2074 : HasColimits C inst\u271d\u00b3 : ConcreteCategory C inst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C) inst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C) inst\u271d : PreservesFilteredColimits (CategoryTheory.forget C) X Y : SheafedSpace C f : X \u27f6 Y hf : OpenEmbedding \u2191f.base H : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x) U : Opens \u2191\u2191X.toPresheafedSpace h : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace) [inst : \u2200 (x : { x // x \u2208 U }), IsIso ((TopCat.Presheaf.stalkFunctor C \u2191x).map (let_fun this := { val := f.c }; this).val)], IsIso ((let_fun this := { val := f.c }; this).val.app (op U)) \u22a2 IsIso (f.c.app (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f.base)).obj U))) ** refine @h _ ?_ ** C : Type u_1 inst\u271d\u2076 : Category.{u_2, u_1} C inst\u271d\u2075 : HasLimits C inst\u271d\u2074 : HasColimits C inst\u271d\u00b3 : ConcreteCategory C inst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C) inst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C) inst\u271d : PreservesFilteredColimits (CategoryTheory.forget C) X Y : SheafedSpace C f : X \u27f6 Y hf : OpenEmbedding \u2191f.base H : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x) U : Opens \u2191\u2191X.toPresheafedSpace h : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace) [inst : \u2200 (x : { x // x \u2208 U }), IsIso ((TopCat.Presheaf.stalkFunctor C \u2191x).map (let_fun this := { val := f.c }; this).val)], IsIso ((let_fun this := { val := f.c }; this).val.app (op U)) \u22a2 \u2200 (x : { x // x \u2208 (IsOpenMap.functor (_ : IsOpenMap \u2191f.base)).obj U }), IsIso ((TopCat.Presheaf.stalkFunctor C \u2191x).map (let_fun this := { val := f.c }; this).val) ** rintro \u27e8_, y, hy, rfl\u27e9 ** case mk.intro.intro C : Type u_1 inst\u271d\u2076 : Category.{u_2, u_1} C inst\u271d\u2075 : HasLimits C inst\u271d\u2074 : HasColimits C inst\u271d\u00b3 : ConcreteCategory C inst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C) inst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C) inst\u271d : PreservesFilteredColimits (CategoryTheory.forget C) X Y : SheafedSpace C f : X \u27f6 Y hf : OpenEmbedding \u2191f.base H : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x) U : Opens \u2191\u2191X.toPresheafedSpace h : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace) [inst : \u2200 (x : { x // x \u2208 U }), IsIso ((TopCat.Presheaf.stalkFunctor C \u2191x).map (let_fun this := { val := f.c }; this).val)], IsIso ((let_fun this := { val := f.c }; this).val.app (op U)) y : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace hy : y \u2208 \u2191U \u22a2 IsIso ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map (let_fun this := { val := f.c }; this).val) ** specialize H y ** case mk.intro.intro C : Type u_1 inst\u271d\u2076 : Category.{u_2, u_1} C inst\u271d\u2075 : HasLimits C inst\u271d\u2074 : HasColimits C inst\u271d\u00b3 : ConcreteCategory C inst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C) inst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C) inst\u271d : PreservesFilteredColimits (CategoryTheory.forget C) X Y : SheafedSpace C f : X \u27f6 Y hf : OpenEmbedding \u2191f.base U : Opens \u2191\u2191X.toPresheafedSpace h : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace) [inst : \u2200 (x : { x // x \u2208 U }), IsIso ((TopCat.Presheaf.stalkFunctor C \u2191x).map (let_fun this := { val := f.c }; this).val)], IsIso ((let_fun this := { val := f.c }; this).val.app (op U)) y : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace hy : y \u2208 \u2191U H : IsIso (PresheafedSpace.stalkMap f y) \u22a2 IsIso ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map (let_fun this := { val := f.c }; this).val) ** delta PresheafedSpace.stalkMap at H ** case mk.intro.intro C : Type u_1 inst\u271d\u2076 : Category.{u_2, u_1} C inst\u271d\u2075 : HasLimits C inst\u271d\u2074 : HasColimits C inst\u271d\u00b3 : ConcreteCategory C inst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C) inst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C) inst\u271d : PreservesFilteredColimits (CategoryTheory.forget C) X Y : SheafedSpace C f : X \u27f6 Y hf : OpenEmbedding \u2191f.base U : Opens \u2191\u2191X.toPresheafedSpace h : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace) [inst : \u2200 (x : { x // x \u2208 U }), IsIso ((TopCat.Presheaf.stalkFunctor C \u2191x).map (let_fun this := { val := f.c }; this).val)], IsIso ((let_fun this := { val := f.c }; this).val.app (op U)) y : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace hy : y \u2208 \u2191U H : IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y) \u22a2 IsIso ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map (let_fun this := { val := f.c }; this).val) ** haveI H' :=\n TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding C hf X.presheaf y ** case mk.intro.intro C : Type u_1 inst\u271d\u2076 : Category.{u_2, u_1} C inst\u271d\u2075 : HasLimits C inst\u271d\u2074 : HasColimits C inst\u271d\u00b3 : ConcreteCategory C inst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C) inst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C) inst\u271d : PreservesFilteredColimits (CategoryTheory.forget C) X Y : SheafedSpace C f : X \u27f6 Y hf : OpenEmbedding \u2191f.base U : Opens \u2191\u2191X.toPresheafedSpace h : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace) [inst : \u2200 (x : { x // x \u2208 U }), IsIso ((TopCat.Presheaf.stalkFunctor C \u2191x).map (let_fun this := { val := f.c }; this).val)], IsIso ((let_fun this := { val := f.c }; this).val.app (op U)) y : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace hy : y \u2208 \u2191U H : IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y) H' : IsIso (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y) \u22a2 IsIso ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map (let_fun this := { val := f.c }; this).val) ** have := @IsIso.comp_isIso _ _ _ _ _ _ _ H (@IsIso.inv_isIso _ _ _ _ _ H') ** case mk.intro.intro C : Type u_1 inst\u271d\u2076 : Category.{u_2, u_1} C inst\u271d\u2075 : HasLimits C inst\u271d\u2074 : HasColimits C inst\u271d\u00b3 : ConcreteCategory C inst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C) inst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C) inst\u271d : PreservesFilteredColimits (CategoryTheory.forget C) X Y : SheafedSpace C f : X \u27f6 Y hf : OpenEmbedding \u2191f.base U : Opens \u2191\u2191X.toPresheafedSpace h : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace) [inst : \u2200 (x : { x // x \u2208 U }), IsIso ((TopCat.Presheaf.stalkFunctor C \u2191x).map (let_fun this := { val := f.c }; this).val)], IsIso ((let_fun this := { val := f.c }; this).val.app (op U)) y : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace hy : y \u2208 \u2191U H : IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y) H' : IsIso (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y) this : IsIso (((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y) \u226b inv (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)) \u22a2 IsIso ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map (let_fun this := { val := f.c }; this).val) ** rwa [Category.assoc, IsIso.hom_inv_id, Category.comp_id] at this ** Qed", + "informal": "" + }, + { + "formal": "List.filterMap_eq_map ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 \u22a2 filterMap (some \u2218 f) = map f ** funext l ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 l : List \u03b1 \u22a2 filterMap (some \u2218 f) l = map f l ** induction l <;> simp [*] ** Qed", + "informal": "" + }, + { + "formal": "Part.fix_def ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom \u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x ** let p := fun i : \u2115 => (Fix.approx f i x).Dom ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom \u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x ** have : p (Nat.find h') := Nat.find_spec h' ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom this : p (Nat.find h') \u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x ** generalize hk : Nat.find h' = k ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom this : p (Nat.find h') k : \u2115 hk : Nat.find h' = k \u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x ** replace hk : Nat.find h' = k + (@Upto.zero p).val := hk ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom this : p (Nat.find h') k : \u2115 hk : Nat.find h' = k + \u2191Upto.zero \u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x ** rw [hk] at this ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom k : \u2115 this : p (k + \u2191Upto.zero) hk : Nat.find h' = k + \u2191Upto.zero \u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x ** revert hk ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom k : \u2115 this : p (k + \u2191Upto.zero) \u22a2 Nat.find h' = k + \u2191Upto.zero \u2192 Part.fix f x = Fix.approx f (Nat.succ k) x ** dsimp [Part.fix] ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom k : \u2115 this : p (k + \u2191Upto.zero) \u22a2 Nat.find h' = k + \u2191Upto.zero \u2192 (assert (\u2203 i, (Fix.approx f i x).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x) = Fix.approx f (Nat.succ k) x ** rw [assert_pos h'] ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom k : \u2115 this : p (k + \u2191Upto.zero) \u22a2 Nat.find h' = k + \u2191Upto.zero \u2192 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x = Fix.approx f (Nat.succ k) x ** revert this ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom k : \u2115 \u22a2 p (k + \u2191Upto.zero) \u2192 Nat.find h' = k + \u2191Upto.zero \u2192 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x = Fix.approx f (Nat.succ k) x ** generalize Upto.zero = z ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom k : \u2115 z : Upto p \u22a2 p (k + \u2191z) \u2192 Nat.find h' = k + \u2191z \u2192 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x = Fix.approx f (Nat.succ k) x ** intro _this hk ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom k : \u2115 z : Upto p _this : p (k + \u2191z) hk : Nat.find h' = k + \u2191z \u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x = Fix.approx f (Nat.succ k) x ** 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 _ ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom k : \u2115 z : Upto p _this : p (k + \u2191z) hk : Nat.find h' = k + \u2191z \u22a2 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ k) x' ** 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] ** case zero \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom z : Upto p _this : p (Nat.zero + \u2191z) hk : Nat.find h' = Nat.zero + \u2191z \u22a2 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ Nat.zero) x' ** intro x' ** case zero \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom z : Upto p _this : p (Nat.zero + \u2191z) hk : Nat.find h' = Nat.zero + \u2191z x' : \u03b1 \u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ Nat.zero) x' ** rw [Fix.approx, WellFounded.fix_eq, fixAux] ** case zero \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom z : Upto p _this : p (Nat.zero + \u2191z) hk : Nat.find h' = Nat.zero + \u2191z x' : \u03b1 \u22a2 f (fun x_1 => assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h) x_1) x' = f (Fix.approx f Nat.zero) x' ** congr ** case zero.e_a \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom z : Upto p _this : p (Nat.zero + \u2191z) hk : Nat.find h' = Nat.zero + \u2191z x' : \u03b1 \u22a2 (fun x_1 => assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h) x_1) = Fix.approx f Nat.zero ** ext x: 1 ** case zero.e_a.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x\u271d : \u03b1 h' : \u2203 i, (Fix.approx f i x\u271d).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom z : Upto p _this : p (Nat.zero + \u2191z) hk : Nat.find h' = Nat.zero + \u2191z x' x : \u03b1 \u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x\u271d).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x => (Fix.approx f x x\u271d).Dom)) (fixAux f) (Upto.succ z h) x) = Fix.approx f Nat.zero x ** rw [assert_neg] ** case zero.e_a.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x\u271d : \u03b1 h' : \u2203 i, (Fix.approx f i x\u271d).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom z : Upto p _this : p (Nat.zero + \u2191z) hk : Nat.find h' = Nat.zero + \u2191z x' x : \u03b1 \u22a2 none = Fix.approx f Nat.zero x case zero.e_a.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x\u271d : \u03b1 h' : \u2203 i, (Fix.approx f i x\u271d).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom z : Upto p _this : p (Nat.zero + \u2191z) hk : Nat.find h' = Nat.zero + \u2191z x' x : \u03b1 \u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom ** rfl ** case zero.e_a.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x\u271d : \u03b1 h' : \u2203 i, (Fix.approx f i x\u271d).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom z : Upto p _this : p (Nat.zero + \u2191z) hk : Nat.find h' = Nat.zero + \u2191z x' x : \u03b1 \u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom ** rw [Nat.zero_add] at _this ** case zero.e_a.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x\u271d : \u03b1 h' : \u2203 i, (Fix.approx f i x\u271d).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom z : Upto p _this : p \u2191z hk : Nat.find h' = Nat.zero + \u2191z x' x : \u03b1 \u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom ** simpa only [not_not, Coe] ** case succ \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z \u22a2 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ (Nat.succ n)) x' ** intro x' ** case succ \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' : \u03b1 \u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ (Nat.succ n)) x' ** rw [Fix.approx, WellFounded.fix_eq, fixAux] ** case succ \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' : \u03b1 \u22a2 f (fun x_1 => assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h) x_1) x' = f (Fix.approx f (n + 1)) x' ** congr ** case succ.e_a \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' : \u03b1 \u22a2 (fun x_1 => assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h) x_1) = Fix.approx f (n + 1) ** ext : 1 ** case succ.e_a.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' x\u271d : \u03b1 \u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) = Fix.approx f (n + 1) x\u271d ** 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 ** case succ.e_a.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' x\u271d : \u03b1 hh : \u00ac(Fix.approx f (\u2191z) x).Dom \u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) = Fix.approx f (n + 1) x\u271d ** rw [succ_add_eq_succ_add] at _this hk ** case succ.e_a.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (n + Nat.succ \u2191z) hk : Nat.find h' = n + Nat.succ \u2191z x' x\u271d : \u03b1 hh : \u00ac(Fix.approx f (\u2191z) x).Dom \u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h => WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) = Fix.approx f (n + 1) x\u271d ** rw [assert_pos hh, n_ih (Upto.succ z hh) _this hk] ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' x\u271d : \u03b1 \u22a2 \u00ac(Fix.approx f (\u2191z) x).Dom ** apply Nat.find_min h' ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' x\u271d : \u03b1 \u22a2 \u2191z < Nat.find h' ** rw [hk, Nat.succ_add, \u2190 Nat.add_succ] ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' x\u271d : \u03b1 \u22a2 \u2191z < n + Nat.succ \u2191z ** apply Nat.lt_of_succ_le ** case h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 f : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a) x : \u03b1 h' : \u2203 i, (Fix.approx f i x).Dom p : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom n : \u2115 n_ih : \u2200 (z : Upto p), p (n + \u2191z) \u2192 Nat.find h' = n + \u2191z \u2192 \u2200 (x' : \u03b1), WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' = Fix.approx f (Nat.succ n) x' z : Upto p _this : p (Nat.succ n + \u2191z) hk : Nat.find h' = Nat.succ n + \u2191z x' x\u271d : \u03b1 \u22a2 Nat.succ \u2191z \u2264 n + Nat.succ \u2191z ** apply Nat.le_add_left ** Qed", + "informal": "" + }, + { + "formal": "Rat.normalize_eq ** num : Int den : Nat den_nz : den \u2260 0 \u22a2 normalize num den = mk' (num / \u2191(Nat.gcd (Int.natAbs num) den)) (den / Nat.gcd (Int.natAbs num) den) ** simp only [normalize, maybeNormalize_eq,\n Int.div_eq_ediv_of_dvd (Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))] ** Qed", + "informal": "" + }, + { + "formal": "PrimeSpectrum.isClosed_range_comap_of_surjective ** R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S S' : Type u_1 inst\u271d : CommRing S' f : R \u2192+* S hf : Surjective \u2191f \u22a2 IsClosed (Set.range \u2191(comap f)) ** rw [range_comap_of_surjective _ f hf] ** R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S S' : Type u_1 inst\u271d : CommRing S' f : R \u2192+* S hf : Surjective \u2191f \u22a2 IsClosed (zeroLocus \u2191(ker f)) ** exact isClosed_zeroLocus _ ** Qed", + "informal": "" + }, + { + "formal": "toIocMod_add_left ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c : \u03b1 n : \u2124 a b : \u03b1 \u22a2 toIocMod hp a (p + b) = toIocMod hp a b ** rw [add_comm, toIocMod_add_right] ** Qed", + "informal": "" + }, + { + "formal": "Complex.sin_sq_add_cos_sq ** x y : \u2102 \u22a2 sin x ^ 2 + cos x ^ 2 = cosh (x * I) ^ 2 - sinh (x * I) ^ 2 ** rw [cosh_mul_I, sinh_mul_I, mul_pow, I_sq, mul_neg_one, sub_neg_eq_add, add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Isometry.diam_image ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : Isometry f s : Set \u03b1 \u22a2 Metric.diam (f '' s) = Metric.diam s ** rw [Metric.diam, Metric.diam, hf.ediam_image] ** Qed", + "informal": "" + }, + { + "formal": "Ideal.MvPolynomial.quotient_mk_comp_C_isIntegral_of_jacobson' ** n : \u2115 R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsJacobson R P : Ideal (MvPolynomial (Fin n) R) hP : IsMaximal P \u22a2 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R \u29f8 P)) ** induction' n with n IH ** case zero n : \u2115 R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsJacobson R P\u271d : Ideal (MvPolynomial (Fin n) R) hP\u271d : IsMaximal P\u271d P : Ideal (MvPolynomial (Fin Nat.zero) R) hP : IsMaximal P \u22a2 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin Nat.zero) R \u29f8 P)) ** apply RingHom.isIntegral_of_surjective ** case zero.hf n : \u2115 R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsJacobson R P\u271d : Ideal (MvPolynomial (Fin n) R) hP\u271d : IsMaximal P\u271d P : Ideal (MvPolynomial (Fin Nat.zero) R) hP : IsMaximal P \u22a2 Function.Surjective \u2191(algebraMap R (MvPolynomial (Fin Nat.zero) R \u29f8 P)) ** apply Function.Surjective.comp Quotient.mk_surjective ** case zero.hf n : \u2115 R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsJacobson R P\u271d : Ideal (MvPolynomial (Fin n) R) hP\u271d : IsMaximal P\u271d P : Ideal (MvPolynomial (Fin Nat.zero) R) hP : IsMaximal P \u22a2 Function.Surjective fun x => \u2191(algebraMap R (MvPolynomial (Fin Nat.zero) R)) x ** exact C_surjective (Fin 0) ** case succ n\u271d : \u2115 R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsJacobson R P\u271d : Ideal (MvPolynomial (Fin n\u271d) R) hP\u271d : IsMaximal P\u271d n : \u2115 IH : \u2200 (P : Ideal (MvPolynomial (Fin n) R)), IsMaximal P \u2192 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R \u29f8 P)) P : Ideal (MvPolynomial (Fin (Nat.succ n)) R) hP : IsMaximal P \u22a2 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin (Nat.succ n)) R \u29f8 P)) ** apply aux_IH IH (finSuccEquiv R n).symm P hP ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.rTensor_comp_map ** R : Type u_1 inst\u271d\u00b9\u2074 : CommSemiring R R' : Type u_2 inst\u271d\u00b9\u00b3 : Monoid R' R'' : Type u_3 inst\u271d\u00b9\u00b2 : Semiring R'' M : Type u_4 N : Type u_5 P : Type u_6 Q : Type u_7 S : Type u_8 inst\u271d\u00b9\u00b9 : AddCommMonoid M inst\u271d\u00b9\u2070 : AddCommMonoid N inst\u271d\u2079 : AddCommMonoid P inst\u271d\u2078 : AddCommMonoid Q inst\u271d\u2077 : AddCommMonoid S inst\u271d\u2076 : Module R M inst\u271d\u2075 : Module R N inst\u271d\u2074 : Module R P inst\u271d\u00b3 : Module R Q inst\u271d\u00b2 : Module R S inst\u271d\u00b9 : DistribMulAction R' M inst\u271d : Module R'' M g\u271d : P \u2192\u2097[R] Q f\u271d : N \u2192\u2097[R] P f' : P \u2192\u2097[R] S f : M \u2192\u2097[R] P g : N \u2192\u2097[R] Q \u22a2 comp (rTensor Q f') (map f g) = map (comp f' f) g ** simp only [lTensor, rTensor, \u2190 map_comp, id_comp, comp_id] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.condexp_bot ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c F inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 inst\u271d : IsProbabilityMeasure \u03bc f : \u03b1 \u2192 F' \u22a2 \u03bc[f|\u22a5] = fun x => \u222b (x : \u03b1), f x \u2202\u03bc ** refine' (condexp_bot' f).trans _ ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c F inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 inst\u271d : IsProbabilityMeasure \u03bc f : \u03b1 \u2192 F' \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 ** rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul] ** Qed", + "informal": "" + }, + { + "formal": "Int.nneg_mul_add_sq_of_abs_le_one ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 \u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n ** 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 ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n \u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n ** 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 ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n hnx' : n < 0 \u2192 x + \u2191n \u2264 0 \u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n ** rw [\u2190 mul_add, mul_nonneg_iff] ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n hnx' : n < 0 \u2192 x + \u2191n \u2264 0 \u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0 ** rcases lt_trichotomy n 0 with (h | rfl | h) ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hn : 0 < n \u22a2 0 \u2264 x + \u2191n ** have := _root_.add_le_add (neg_le_of_abs_le hx) (cast_one_le_of_pos hn) ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hn : 0 < n this : -1 + 1 \u2264 x + \u2191n \u22a2 0 \u2264 x + \u2191n ** rwa [add_left_neg] at this ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n hn : n < 0 \u22a2 x + \u2191n \u2264 0 ** have := _root_.add_le_add (le_of_abs_le hx) (cast_le_neg_one_of_neg hn) ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n hn : n < 0 this : x + \u2191n \u2264 1 + -1 \u22a2 x + \u2191n \u2264 0 ** rwa [add_right_neg] at this ** case inl F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n hnx' : n < 0 \u2192 x + \u2191n \u2264 0 h : n < 0 \u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0 ** exact Or.inr \u27e8by exact_mod_cast h.le, hnx' h\u27e9 ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n hnx' : n < 0 \u2192 x + \u2191n \u2264 0 h : n < 0 \u22a2 \u2191n \u2264 0 ** exact_mod_cast h.le ** case inr.inl F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < 0 \u2192 0 \u2264 x + \u21910 hnx' : 0 < 0 \u2192 x + \u21910 \u2264 0 \u22a2 0 \u2264 \u21910 \u2227 0 \u2264 x + \u21910 \u2228 \u21910 \u2264 0 \u2227 x + \u21910 \u2264 0 ** simp [le_total 0 x] ** case inr.inr F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n hnx' : n < 0 \u2192 x + \u2191n \u2264 0 h : 0 < n \u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0 ** exact Or.inl \u27e8by exact_mod_cast h.le, hnx h\u27e9 ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : LinearOrderedRing \u03b1 a b n : \u2124 x : \u03b1 hx : |x| \u2264 1 hnx : 0 < n \u2192 0 \u2264 x + \u2191n hnx' : n < 0 \u2192 x + \u2191n \u2264 0 h : 0 < n \u22a2 0 \u2264 \u2191n ** exact_mod_cast h.le ** Qed", + "informal": "" + }, + { + "formal": "NormedAddGroupHom.coe_injective ** V : Type u_1 V\u2081 : Type u_2 V\u2082 : Type u_3 V\u2083 : Type u_4 inst\u271d\u00b3 : SeminormedAddCommGroup V inst\u271d\u00b2 : SeminormedAddCommGroup V\u2081 inst\u271d\u00b9 : SeminormedAddCommGroup V\u2082 inst\u271d : SeminormedAddCommGroup V\u2083 f g : NormedAddGroupHom V\u2081 V\u2082 \u22a2 Function.Injective toFun ** apply coe_inj ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.polar_eq_iInter ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : NormedCommRing \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F B : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c s : Set E \u22a2 polar B s = \u22c2 x \u2208 s, {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1} ** ext ** case h \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : NormedCommRing \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F B : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c s : Set E x\u271d : F \u22a2 x\u271d \u2208 polar B s \u2194 x\u271d \u2208 \u22c2 x \u2208 s, {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1} ** simp only [polar_mem_iff, Set.mem_iInter, Set.mem_setOf_eq] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.det_updateRow_add_self ** m : Type u_1 n : Type u_2 inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m R : Type v inst\u271d : CommRing R A : Matrix n n R i j : n hij : i \u2260 j \u22a2 det (updateRow A i (A i + A j)) = det A ** simp [det_updateRow_add,\n det_zero_of_row_eq hij (updateRow_self.trans (updateRow_ne hij.symm).symm)] ** Qed", + "informal": "" + }, + { + "formal": "IsLocalization.isLocalization_of_base_ringEquiv ** R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P \u22a2 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S ** letI : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra ** R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) \u22a2 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S ** constructor ** case map_units' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) \u22a2 \u2200 (y : { x // x \u2208 Submonoid.map (RingEquiv.toMonoidHom h) M }), IsUnit (\u2191(algebraMap P S) \u2191y) ** rintro \u27e8_, \u27e8y, hy, rfl\u27e9\u27e9 ** case map_units'.mk.intro.intro R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) y : R hy : y \u2208 \u2191M \u22a2 IsUnit (\u2191(algebraMap P S) \u2191{ val := \u2191(RingEquiv.toMonoidHom h) y, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) y) }) ** convert IsLocalization.map_units S \u27e8y, hy\u27e9 ** case h.e'_3 R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) y : R hy : y \u2208 \u2191M \u22a2 \u2191(algebraMap P S) \u2191{ val := \u2191(RingEquiv.toMonoidHom h) y, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) y) } = \u2191(algebraMap R S) \u2191{ val := y, property := hy } ** dsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply] ** case h.e'_3 R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) y : R hy : y \u2208 \u2191M \u22a2 \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) y)) = \u2191(algebraMap R S) y ** exact congr_arg _ (h.symm_apply_apply _) ** case surj' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) \u22a2 \u2200 (z : S), \u2203 x, z * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1 ** intro y ** case surj' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) y : S \u22a2 \u2203 x, y * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1 ** obtain \u27e8\u27e8x, s\u27e9, e\u27e9 := IsLocalization.surj M y ** case surj'.intro.mk R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) y : S x : R s : { x // x \u2208 M } e : y * \u2191(algebraMap R S) \u2191(x, s).2 = \u2191(algebraMap R S) (x, s).1 \u22a2 \u2203 x, y * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1 ** refine' \u27e8\u27e8h x, _, _, s.prop, rfl\u27e9, _\u27e9 ** case surj'.intro.mk R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) y : S x : R s : { x // x \u2208 M } e : y * \u2191(algebraMap R S) \u2191(x, s).2 = \u2191(algebraMap R S) (x, s).1 \u22a2 y * \u2191(algebraMap P S) \u2191(\u2191h x, { val := \u2191(RingEquiv.toMonoidHom h) \u2191s, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) \u2191s) }).2 = \u2191(algebraMap P S) (\u2191h x, { val := \u2191(RingEquiv.toMonoidHom h) \u2191s, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) \u2191s) }).1 ** dsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply] at e \u22a2 ** case surj'.intro.mk R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) y : S x : R s : { x // x \u2208 M } e : y * \u2191(algebraMap R S) \u2191s = \u2191(algebraMap R S) x \u22a2 y * \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) \u2191s)) = \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191h x)) ** convert e <;> exact h.symm_apply_apply _ ** case eq_iff_exists' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) \u22a2 \u2200 {x y : P}, \u2191(algebraMap P S) x = \u2191(algebraMap P S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y ** intro x y ** case eq_iff_exists' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) x y : P \u22a2 \u2191(algebraMap P S) x = \u2191(algebraMap P S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y ** rw [RingHom.algebraMap_toAlgebra, RingHom.comp_apply, RingHom.comp_apply,\n IsLocalization.eq_iff_exists M S] ** case eq_iff_exists' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) x y : P \u22a2 (\u2203 c, \u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) x = \u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) y) \u2194 \u2203 c, \u2191c * x = \u2191c * y ** simp_rw [\u2190 h.toEquiv.apply_eq_iff_eq] ** case eq_iff_exists' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) x y : P \u22a2 (\u2203 c, \u2191h.toEquiv (\u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) x) = \u2191h.toEquiv (\u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) y)) \u2194 \u2203 c, \u2191c * x = \u2191c * y ** change (\u2203 c : M, h (c * h.symm x) = h (c * h.symm y)) \u2194 _ ** case eq_iff_exists' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) x y : P \u22a2 (\u2203 c, \u2191h (\u2191c * \u2191(RingEquiv.symm h) x) = \u2191h (\u2191c * \u2191(RingEquiv.symm h) y)) \u2194 \u2203 c, \u2191c * x = \u2191c * y ** simp only [RingEquiv.apply_symm_apply, RingEquiv.map_mul] ** case eq_iff_exists' R : Type u_1 inst\u271d\u2074 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R S P : Type u_3 inst\u271d\u00b9 : CommSemiring P inst\u271d : IsLocalization M S h : R \u2243+* P this : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h))) x y : P \u22a2 (\u2203 c, \u2191h \u2191c * x = \u2191h \u2191c * y) \u2194 \u2203 c, \u2191c * x = \u2191c * y ** 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 ** Qed", + "informal": "" + }, + { + "formal": "Nat.self_add_sub_one ** m n\u271d k n : \u2115 \u22a2 n + (n - 1) = 2 * n - 1 ** cases n ** case zero m n k : \u2115 \u22a2 zero + (zero - 1) = 2 * zero - 1 ** rfl ** case succ m n k n\u271d : \u2115 \u22a2 succ n\u271d + (succ n\u271d - 1) = 2 * succ n\u271d - 1 ** rw [two_mul] ** case succ m n k n\u271d : \u2115 \u22a2 succ n\u271d + (succ n\u271d - 1) = succ n\u271d + succ n\u271d - 1 ** convert (add_succ_sub_one (Nat.succ _) _).symm ** Qed", + "informal": "" + }, + { + "formal": "GaussianInt.norm_le_norm_mul_left ** x y : \u2124[i] hy : y \u2260 0 \u22a2 Int.natAbs (norm x) \u2264 Int.natAbs (norm (x * y)) ** rw [Zsqrtd.norm_mul, Int.natAbs_mul] ** x y : \u2124[i] hy : y \u2260 0 \u22a2 Int.natAbs (norm x) \u2264 Int.natAbs (norm x) * Int.natAbs (norm y) ** 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))) ** x y : \u2124[i] hy : y \u2260 0 \u22a2 \u21911 \u2264 \u2191(Int.natAbs (norm y)) ** rw [abs_coe_nat_norm] ** x y : \u2124[i] hy : y \u2260 0 \u22a2 \u21911 \u2264 norm y ** exact Int.add_one_le_of_lt (norm_pos.2 hy) ** Qed", + "informal": "" + }, + { + "formal": "Multiset.count_eq_of_nodup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d t : Multiset \u03b1 a\u271d : \u03b1 inst\u271d : DecidableEq \u03b1 a : \u03b1 s : Multiset \u03b1 d : Nodup s \u22a2 count a s = if a \u2208 s then 1 else 0 ** split_ifs with h ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d t : Multiset \u03b1 a\u271d : \u03b1 inst\u271d : DecidableEq \u03b1 a : \u03b1 s : Multiset \u03b1 d : Nodup s h : a \u2208 s \u22a2 count a s = 1 ** exact count_eq_one_of_mem d h ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d t : Multiset \u03b1 a\u271d : \u03b1 inst\u271d : DecidableEq \u03b1 a : \u03b1 s : Multiset \u03b1 d : Nodup s h : \u00aca \u2208 s \u22a2 count a s = 0 ** exact count_eq_zero_of_not_mem h ** Qed", + "informal": "" + }, + { + "formal": "Int.even_add ** m n : \u2124 \u22a2 Even (m + n) \u2194 (Even m \u2194 Even n) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Int.sub_neg ** a b : Int \u22a2 a - -b = a + b ** simp [Int.sub_eq_add_neg] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Walk.cons_copy ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' u v w v' w' : V h : Adj G u v p : Walk G v' w' hv : v' = v hw : w' = w \u22a2 cons h (Walk.copy p hv hw) = Walk.copy (cons (_ : Adj G u v') p) (_ : u = u) hw ** subst_vars ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' u v' w' : V p : Walk G v' w' h : Adj G u v' \u22a2 cons h (Walk.copy p (_ : v' = v') (_ : w' = w')) = Walk.copy (cons (_ : Adj G u v') p) (_ : u = u) (_ : w' = w') ** rfl ** Qed", + "informal": "" + }, + { + "formal": "sSup_inv ** \u03b1 : Type u_1 inst\u271d\u00b3 : CompleteLattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 s\u271d t s : Set \u03b1 \u22a2 sSup s\u207b\u00b9 = (sInf s)\u207b\u00b9 ** rw [\u2190 image_inv, sSup_image] ** \u03b1 : Type u_1 inst\u271d\u00b3 : CompleteLattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 s\u271d t s : Set \u03b1 \u22a2 \u2a06 a \u2208 s, a\u207b\u00b9 = (sInf s)\u207b\u00b9 ** exact ((OrderIso.inv \u03b1).map_sInf _).symm ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.range_comp_of_range_eq_top ** R : Type u_1 R\u2081 : Type u_2 R\u2082 : Type u_3 R\u2083 : Type u_4 R\u2084 : Type u_5 S : Type u_6 K : Type u_7 K\u2082 : Type u_8 M : Type u_9 M' : Type u_10 M\u2081 : Type u_11 M\u2082 : Type u_12 M\u2083 : Type u_13 M\u2084 : Type u_14 N : Type u_15 N\u2082 : Type u_16 \u03b9 : Type u_17 V : Type u_18 V\u2082 : Type u_19 inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : Semiring R\u2082 inst\u271d\u00b9\u2070 : Semiring R\u2083 inst\u271d\u2079 : AddCommMonoid M inst\u271d\u2078 : AddCommMonoid M\u2082 inst\u271d\u2077 : AddCommMonoid M\u2083 inst\u271d\u2076 : Module R M inst\u271d\u2075 : Module R\u2082 M\u2082 inst\u271d\u2074 : Module R\u2083 M\u2083 \u03c4\u2081\u2082 : R \u2192+* R\u2082 \u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083 \u03c4\u2081\u2083 : R \u2192+* R\u2083 inst\u271d\u00b3 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083 inst\u271d\u00b2 : RingHomSurjective \u03c4\u2081\u2082 inst\u271d\u00b9 : RingHomSurjective \u03c4\u2082\u2083 inst\u271d : RingHomSurjective \u03c4\u2081\u2083 f : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082 g : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083 hf : range f = \u22a4 \u22a2 range (comp g f) = range g ** rw [range_comp, hf, Submodule.map_top] ** Qed", + "informal": "" + }, + { + "formal": "Real.arccos_one ** \u22a2 arccos 1 = 0 ** simp [arccos] ** Qed", + "informal": "" + }, + { + "formal": "UniformCauchySeqOn.mul ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : UniformSpace \u03b1 inst\u271d\u00b9 : Group \u03b1 inst\u271d : UniformGroup \u03b1 \u03b9 : Type u_3 l : Filter \u03b9 l' : Filter \u03b2 f f' : \u03b9 \u2192 \u03b2 \u2192 \u03b1 g g' : \u03b2 \u2192 \u03b1 s : Set \u03b2 hf : UniformCauchySeqOn f l s hf' : UniformCauchySeqOn f' l s u : Set (\u03b1 \u00d7 \u03b1) hu : u \u2208 \ud835\udce4 \u03b1 \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 ** simpa using (uniformContinuous_mul.comp_uniformCauchySeqOn (hf.prod' hf')) u hu ** Qed", + "informal": "" + }, + { + "formal": "PadicInt.isCauSeq_nthHom ** p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 r : R \u22a2 IsCauSeq (padicNorm p) fun n => \u2191(nthHom f r n) ** intro \u03b5 h\u03b5 ** p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 r : R \u03b5 : \u211a h\u03b5 : \u03b5 > 0 \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 ** obtain \u27e8k, hk\u27e9 : \u2203 k : \u2115, (p : \u211a) ^ (-((k : \u2115) : \u2124)) < \u03b5 := exists_pow_neg_lt_rat p h\u03b5 ** case intro p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 r : R \u03b5 : \u211a h\u03b5 : \u03b5 > 0 k : \u2115 hk : \u2191p ^ (-\u2191k) < \u03b5 \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 ** use k ** case h p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 r : R \u03b5 : \u211a h\u03b5 : \u03b5 > 0 k : \u2115 hk : \u2191p ^ (-\u2191k) < \u03b5 \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 ** intro j hj ** case h p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 r : R \u03b5 : \u211a h\u03b5 : \u03b5 > 0 k : \u2115 hk : \u2191p ^ (-\u2191k) < \u03b5 j : \u2115 hj : j \u2265 k \u22a2 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) k) < \u03b5 ** refine' lt_of_le_of_lt _ hk ** case h p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 r : R \u03b5 : \u211a h\u03b5 : \u03b5 > 0 k : \u2115 hk : \u2191p ^ (-\u2191k) < \u03b5 j : \u2115 hj : j \u2265 k \u22a2 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) k) \u2264 \u2191p ^ (-\u2191k) ** norm_cast ** case h p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 r : R \u03b5 : \u211a h\u03b5 : \u03b5 > 0 k : \u2115 hk : \u2191p ^ (-\u2191k) < \u03b5 j : \u2115 hj : j \u2265 k \u22a2 padicNorm p \u2191(nthHom f r j - nthHom f r k) \u2264 \u2191p ^ (-\u2191k) ** rw [\u2190 padicNorm.dvd_iff_norm_le] ** case h p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 r : R \u03b5 : \u211a h\u03b5 : \u03b5 > 0 k : \u2115 hk : \u2191p ^ (-\u2191k) < \u03b5 j : \u2115 hj : j \u2265 k \u22a2 \u2191(p ^ k) \u2223 nthHom f r j - nthHom f r k ** exact_mod_cast pow_dvd_nthHom_sub f_compat r k j hj ** Qed", + "informal": "" + }, + { + "formal": "List.map_fst_zip ** \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 head\u271d\u00b9 : \u03b1 as : List \u03b1 head\u271d : \u03b2 bs : List \u03b2 h : length (head\u271d\u00b9 :: as) \u2264 length (head\u271d :: bs) \u22a2 map Prod.fst (zip (head\u271d\u00b9 :: as) (head\u271d :: bs)) = head\u271d\u00b9 :: as ** simp [succ_le_succ_iff] at h ** \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 head\u271d\u00b9 : \u03b1 as : List \u03b1 head\u271d : \u03b2 bs : List \u03b2 h : length as \u2264 length bs \u22a2 map Prod.fst (zip (head\u271d\u00b9 :: as) (head\u271d :: bs)) = head\u271d\u00b9 :: as ** change _ :: map Prod.fst (zip as bs) = _ :: as ** \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 head\u271d\u00b9 : \u03b1 as : List \u03b1 head\u271d : \u03b2 bs : List \u03b2 h : length as \u2264 length bs \u22a2 (head\u271d\u00b9, head\u271d).1 :: map Prod.fst (zip as bs) = head\u271d\u00b9 :: as ** rw [map_fst_zip as bs h] ** \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 a : \u03b1 as : List \u03b1 h : length (a :: as) \u2264 length [] \u22a2 map Prod.fst (zip (a :: as) []) = a :: as ** simp at h ** Qed", + "informal": "" + }, + { + "formal": "NonUnitalSubalgebra.prod_top ** R : Type u A : Type v B : Type w inst\u271d\u2078 : CommSemiring R inst\u271d\u2077 : NonUnitalNonAssocSemiring A inst\u271d\u2076 : Module R A inst\u271d\u2075 : IsScalarTower R A A inst\u271d\u2074 : SMulCommClass R A A inst\u271d\u00b3 : NonUnitalNonAssocSemiring B inst\u271d\u00b2 : Module R B inst\u271d\u00b9 : IsScalarTower R B B inst\u271d : SMulCommClass R B B S : NonUnitalSubalgebra R A S\u2081 : NonUnitalSubalgebra R B \u22a2 prod \u22a4 \u22a4 = \u22a4 ** ext ** case h R : Type u A : Type v B : Type w inst\u271d\u2078 : CommSemiring R inst\u271d\u2077 : NonUnitalNonAssocSemiring A inst\u271d\u2076 : Module R A inst\u271d\u2075 : IsScalarTower R A A inst\u271d\u2074 : SMulCommClass R A A inst\u271d\u00b3 : NonUnitalNonAssocSemiring B inst\u271d\u00b2 : Module R B inst\u271d\u00b9 : IsScalarTower R B B inst\u271d : SMulCommClass R B B S : NonUnitalSubalgebra R A S\u2081 : NonUnitalSubalgebra R B x\u271d : A \u00d7 B \u22a2 x\u271d \u2208 prod \u22a4 \u22a4 \u2194 x\u271d \u2208 \u22a4 ** simp ** Qed", + "informal": "" + }, + { + "formal": "omegaLimit_subset_of_tendsto ** \u03c4 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 f : Filter \u03c4 \u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2 s s\u2081 s\u2082 : Set \u03b1 m : \u03c4 \u2192 \u03c4 f\u2081 f\u2082 : Filter \u03c4 hf : Tendsto m f\u2081 f\u2082 \u22a2 \u03c9 f\u2081 (fun t x => \u03d5 (m t) x) s \u2286 \u03c9 f\u2082 \u03d5 s ** refine' iInter\u2082_mono' fun u hu \u21a6 \u27e8m \u207b\u00b9' u, tendsto_def.mp hf _ hu, _\u27e9 ** \u03c4 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 f : Filter \u03c4 \u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2 s s\u2081 s\u2082 : Set \u03b1 m : \u03c4 \u2192 \u03c4 f\u2081 f\u2082 : Filter \u03c4 hf : Tendsto m f\u2081 f\u2082 u : Set \u03c4 hu : u \u2208 f\u2082 \u22a2 closure (image2 (fun t x => \u03d5 (m t) x) (m \u207b\u00b9' u) s) \u2286 closure (image2 \u03d5 u s) ** rw [\u2190 image2_image_left] ** \u03c4 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b9 : Type u_4 inst\u271d : TopologicalSpace \u03b2 f : Filter \u03c4 \u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2 s s\u2081 s\u2082 : Set \u03b1 m : \u03c4 \u2192 \u03c4 f\u2081 f\u2082 : Filter \u03c4 hf : Tendsto m f\u2081 f\u2082 u : Set \u03c4 hu : u \u2208 f\u2082 \u22a2 closure (image2 \u03d5 ((fun t => m t) '' (m \u207b\u00b9' u)) s) \u2286 closure (image2 \u03d5 u s) ** exact closure_mono (image2_subset (image_preimage_subset _ _) Subset.rfl) ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.bind\u2082_C_left ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S inst\u271d : CommSemiring T f : \u03c3 \u2192 MvPolynomial \u03c4 R \u22a2 bind\u2082 C = RingHom.id (MvPolynomial \u03c3 R) ** ext : 2 <;> simp ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.flat_iff_lan_flat ** C D : Type u\u2081 inst\u271d\u2079 : SmallCategory C inst\u271d\u2078 : SmallCategory D E : Type u\u2082 inst\u271d\u2077 : Category.{u\u2081, u\u2082} E inst\u271d\u2076 : ConcreteCategory E inst\u271d\u2075 : HasLimits E inst\u271d\u2074 : HasColimits E inst\u271d\u00b3 : ReflectsLimits (forget E) inst\u271d\u00b2 : PreservesFilteredColimits (forget E) inst\u271d\u00b9 : PreservesLimits (forget E) inst\u271d : HasFiniteLimits C F : C \u2964 D H : RepresentablyFlat (lan F.op) \u22a2 RepresentablyFlat F ** haveI := preservesFiniteLimitsOfFlat (lan F.op : _ \u2964 D\u1d52\u1d56 \u2964 Type u\u2081) ** C D : Type u\u2081 inst\u271d\u2079 : SmallCategory C inst\u271d\u2078 : SmallCategory D E : Type u\u2082 inst\u271d\u2077 : Category.{u\u2081, u\u2082} E inst\u271d\u2076 : ConcreteCategory E inst\u271d\u2075 : HasLimits E inst\u271d\u2074 : HasColimits E inst\u271d\u00b3 : ReflectsLimits (forget E) inst\u271d\u00b2 : PreservesFilteredColimits (forget E) inst\u271d\u00b9 : PreservesLimits (forget E) inst\u271d : HasFiniteLimits C F : C \u2964 D H : RepresentablyFlat (lan F.op) this : PreservesFiniteLimits (lan F.op) \u22a2 RepresentablyFlat F ** haveI : PreservesFiniteLimits F := by\n apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{u\u2081}\n intros; skip; apply preservesLimitOfLanPreservesLimit ** C D : Type u\u2081 inst\u271d\u2079 : SmallCategory C inst\u271d\u2078 : SmallCategory D E : Type u\u2082 inst\u271d\u2077 : Category.{u\u2081, u\u2082} E inst\u271d\u2076 : ConcreteCategory E inst\u271d\u2075 : HasLimits E inst\u271d\u2074 : HasColimits E inst\u271d\u00b3 : ReflectsLimits (forget E) inst\u271d\u00b2 : PreservesFilteredColimits (forget E) inst\u271d\u00b9 : PreservesLimits (forget E) inst\u271d : HasFiniteLimits C F : C \u2964 D H : RepresentablyFlat (lan F.op) this\u271d : PreservesFiniteLimits (lan F.op) this : PreservesFiniteLimits F \u22a2 RepresentablyFlat F ** apply flat_of_preservesFiniteLimits ** C D : Type u\u2081 inst\u271d\u2079 : SmallCategory C inst\u271d\u2078 : SmallCategory D E : Type u\u2082 inst\u271d\u2077 : Category.{u\u2081, u\u2082} E inst\u271d\u2076 : ConcreteCategory E inst\u271d\u2075 : HasLimits E inst\u271d\u2074 : HasColimits E inst\u271d\u00b3 : ReflectsLimits (forget E) inst\u271d\u00b2 : PreservesFilteredColimits (forget E) inst\u271d\u00b9 : PreservesLimits (forget E) inst\u271d : HasFiniteLimits C F : C \u2964 D H : RepresentablyFlat (lan F.op) this : PreservesFiniteLimits (lan F.op) \u22a2 PreservesFiniteLimits F ** apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{u\u2081} ** case h C D : Type u\u2081 inst\u271d\u2079 : SmallCategory C inst\u271d\u2078 : SmallCategory D E : Type u\u2082 inst\u271d\u2077 : Category.{u\u2081, u\u2082} E inst\u271d\u2076 : ConcreteCategory E inst\u271d\u2075 : HasLimits E inst\u271d\u2074 : HasColimits E inst\u271d\u00b3 : ReflectsLimits (forget E) inst\u271d\u00b2 : PreservesFilteredColimits (forget E) inst\u271d\u00b9 : PreservesLimits (forget E) inst\u271d : HasFiniteLimits C F : C \u2964 D H : RepresentablyFlat (lan F.op) this : PreservesFiniteLimits (lan F.op) \u22a2 (J : Type u\u2081) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesLimitsOfShape J F ** intros ** case h C D : Type u\u2081 inst\u271d\u2079 : SmallCategory C inst\u271d\u2078 : SmallCategory D E : Type u\u2082 inst\u271d\u2077 : Category.{u\u2081, u\u2082} E inst\u271d\u2076 : ConcreteCategory E inst\u271d\u2075 : HasLimits E inst\u271d\u2074 : HasColimits E inst\u271d\u00b3 : ReflectsLimits (forget E) inst\u271d\u00b2 : PreservesFilteredColimits (forget E) inst\u271d\u00b9 : PreservesLimits (forget E) inst\u271d : HasFiniteLimits C F : C \u2964 D H : RepresentablyFlat (lan F.op) this : PreservesFiniteLimits (lan F.op) J\u271d : Type u\u2081 \ud835\udca5\u271d : SmallCategory J\u271d x\u271d : FinCategory J\u271d \u22a2 PreservesLimitsOfShape J\u271d F ** apply preservesLimitOfLanPreservesLimit ** Qed", + "informal": "" + }, + { + "formal": "QuotientRing.isOpenMap_coe ** R : Type u_1 inst\u271d\u00b2 : TopologicalSpace R inst\u271d\u00b9 : CommRing R N : Ideal R inst\u271d : TopologicalRing R \u22a2 IsOpenMap \u2191(mk N) ** intro s s_op ** R : Type u_1 inst\u271d\u00b2 : TopologicalSpace R inst\u271d\u00b9 : CommRing R N : Ideal R inst\u271d : TopologicalRing R s : Set R s_op : IsOpen s \u22a2 IsOpen (\u2191(mk N) '' s) ** change IsOpen (mk N \u207b\u00b9' (mk N '' s)) ** R : Type u_1 inst\u271d\u00b2 : TopologicalSpace R inst\u271d\u00b9 : CommRing R N : Ideal R inst\u271d : TopologicalRing R s : Set R s_op : IsOpen s \u22a2 IsOpen (\u2191(mk N) \u207b\u00b9' (\u2191(mk N) '' s)) ** rw [quotient_ring_saturate] ** R : Type u_1 inst\u271d\u00b2 : TopologicalSpace R inst\u271d\u00b9 : CommRing R N : Ideal R inst\u271d : TopologicalRing R s : Set R s_op : IsOpen s \u22a2 IsOpen (\u22c3 x, (fun y => \u2191x + y) '' s) ** exact isOpen_iUnion fun \u27e8n, _\u27e9 => isOpenMap_add_left n s s_op ** Qed", + "informal": "" + }, + { + "formal": "Nat.div2_succ ** n : \u2115 \u22a2 div2 (succ n) = bif bodd n then succ (div2 n) else div2 n ** simp only [bodd, boddDiv2, div2] ** n : \u2115 \u22a2 (match boddDiv2 n with | (false, m) => (true, m) | (true, m) => (false, succ m)).snd = bif (boddDiv2 n).fst then succ (boddDiv2 n).snd else (boddDiv2 n).snd ** cases' boddDiv2 n with fst snd ** case mk n : \u2115 fst : Bool snd : \u2115 \u22a2 (match (fst, snd) with | (false, m) => (true, m) | (true, m) => (false, succ m)).snd = bif (fst, snd).fst then succ (fst, snd).snd else (fst, snd).snd ** cases fst ** case mk.false n snd : \u2115 \u22a2 (match (false, snd) with | (false, m) => (true, m) | (true, m) => (false, succ m)).snd = bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd case mk.true n snd : \u2115 \u22a2 (match (true, snd) with | (false, m) => (true, m) | (true, m) => (false, succ m)).snd = bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd ** case mk.false =>\n simp ** case mk.true n snd : \u2115 \u22a2 (match (true, snd) with | (false, m) => (true, m) | (true, m) => (false, succ m)).snd = bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd ** case mk.true =>\n simp ** n snd : \u2115 \u22a2 (match (false, snd) with | (false, m) => (true, m) | (true, m) => (false, succ m)).snd = bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd ** simp ** n snd : \u2115 \u22a2 (match (true, snd) with | (false, m) => (true, m) | (true, m) => (false, succ m)).snd = bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd ** simp ** Qed", + "informal": "" + }, + { + "formal": "TopCat.openEmbedding_iff_comp_isIso' ** X Y Z : TopCat f : X \u27f6 Y g : Y \u27f6 Z inst\u271d : IsIso g \u22a2 OpenEmbedding ((forget TopCat).map f \u226b (forget TopCat).map g) \u2194 OpenEmbedding \u2191f ** simp only [\u2190Functor.map_comp] ** X Y Z : TopCat f : X \u27f6 Y g : Y \u27f6 Z inst\u271d : IsIso g \u22a2 OpenEmbedding ((forget TopCat).map (f \u226b g)) \u2194 OpenEmbedding \u2191f ** exact openEmbedding_iff_comp_isIso f g ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.eq_zero_of_eq_zero ** R : Type u a b : R m n : \u2115 inst\u271d : Semiring R p\u271d q : R[X] h : 0 = 1 p : R[X] \u22a2 p = 0 ** rw [\u2190 one_smul R p, \u2190 h, zero_smul] ** Qed", + "informal": "" + }, + { + "formal": "Affine.Triangle.orthocenter_mem_altitude ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P t : Triangle \u211d P i\u2081 : Fin 3 \u22a2 orthocenter t \u2208 altitude t i\u2081 ** 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 ** case intro.intro.intro.intro V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P t : Triangle \u211d P i\u2081 i\u2082 i\u2083 : Fin 3 h\u2081\u2082 : i\u2081 \u2260 i\u2082 h\u2082\u2083 : i\u2082 \u2260 i\u2083 h\u2081\u2083 : i\u2081 \u2260 i\u2083 \u22a2 orthocenter t \u2208 altitude t i\u2081 ** rw [orthocenter_eq_mongePoint, t.altitude_eq_mongePlane h\u2081\u2082 h\u2081\u2083 h\u2082\u2083] ** case intro.intro.intro.intro V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P t : Triangle \u211d P i\u2081 i\u2082 i\u2083 : Fin 3 h\u2081\u2082 : i\u2081 \u2260 i\u2082 h\u2082\u2083 : i\u2082 \u2260 i\u2083 h\u2081\u2083 : i\u2081 \u2260 i\u2083 \u22a2 mongePoint t \u2208 mongePlane t i\u2082 i\u2083 ** exact t.mongePoint_mem_mongePlane ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P t : Triangle \u211d P i\u2081 : Fin 3 \u22a2 \u2203 i\u2082 i\u2083, i\u2081 \u2260 i\u2082 \u2227 i\u2082 \u2260 i\u2083 \u2227 i\u2081 \u2260 i\u2083 ** fin_cases i\u2081 <;> decide ** Qed", + "informal": "" + }, + { + "formal": "Finset.Ico_union_Ico_eq_Ico ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d b\u271d a b c : \u03b1 hab : a \u2264 b hbc : b \u2264 c \u22a2 Ico a b \u222a Ico b c = Ico a c ** rw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico_eq_Ico hab hbc] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.nondegenerate_toBilin_iff ** R : Type u_1 M\u271d : Type u_2 inst\u271d\u00b9\u2079 : Semiring R inst\u271d\u00b9\u2078 : AddCommMonoid M\u271d inst\u271d\u00b9\u2077 : Module R M\u271d R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u2076 : Ring R\u2081 inst\u271d\u00b9\u2075 : AddCommGroup M\u2081 inst\u271d\u00b9\u2074 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u00b3 : CommSemiring R\u2082 inst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082 inst\u271d\u00b9\u00b9 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u00b9\u2070 : CommRing R\u2083 inst\u271d\u2079 : AddCommGroup M\u2083 inst\u271d\u2078 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2077 : Field K inst\u271d\u2076 : AddCommGroup V inst\u271d\u2075 : Module K V B : BilinForm R M\u271d B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 A : Type u_11 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : IsDomain A inst\u271d\u00b2 : Module A M\u2083 B\u2083 : BilinForm A M\u2083 \u03b9 : Type u_12 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 M : Matrix \u03b9 \u03b9 R\u2083 b : Basis \u03b9 R\u2083 M\u2083 \u22a2 Nondegenerate (\u2191(toBilin b) M) \u2194 Matrix.Nondegenerate M ** rw [\u2190 Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin, Matrix.nondegenerate_toBilin'_iff] ** Qed", + "informal": "" + }, + { + "formal": "Orientation.oangle_sign_add_left ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) x y : V \u22a2 Real.Angle.sign (oangle o (x + y) y) = Real.Angle.sign (oangle o x y) ** rw [\u2190 o.oangle_sign_add_smul_left x y 1, one_smul] ** Qed", + "informal": "" + }, + { + "formal": "SetTheory.PGame.birthday_add_zero ** a b x : PGame \u22a2 birthday (a + 0) = birthday a ** simp ** Qed", + "informal": "" + }, + { + "formal": "Affine.Simplex.point_eq_affineCombination_of_pointsWithCircumcenter ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i : Fin (n + 1) \u22a2 points s i = \u2191(affineCombination \u211d univ (pointsWithCircumcenter s)) (pointWeightsWithCircumcenter i) ** rw [\u2190 pointsWithCircumcenter_point] ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i : Fin (n + 1) \u22a2 pointsWithCircumcenter s (point_index i) = \u2191(affineCombination \u211d univ (pointsWithCircumcenter s)) (pointWeightsWithCircumcenter i) ** symm ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i : Fin (n + 1) \u22a2 \u2191(affineCombination \u211d univ (pointsWithCircumcenter s)) (pointWeightsWithCircumcenter i) = pointsWithCircumcenter s (point_index i) ** refine'\n affineCombination_of_eq_one_of_eq_zero _ _ _ (mem_univ _)\n (by simp [pointWeightsWithCircumcenter]) _ ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i : Fin (n + 1) \u22a2 \u2200 (i2 : PointsWithCircumcenterIndex n), i2 \u2208 univ \u2192 i2 \u2260 point_index i \u2192 pointWeightsWithCircumcenter i i2 = 0 ** intro i hi hn ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i\u271d : Fin (n + 1) i : PointsWithCircumcenterIndex n hi : i \u2208 univ hn : i \u2260 point_index i\u271d \u22a2 pointWeightsWithCircumcenter i\u271d i = 0 ** cases i ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i : Fin (n + 1) \u22a2 pointWeightsWithCircumcenter i (point_index i) = 1 ** simp [pointWeightsWithCircumcenter] ** case point_index V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i a\u271d : Fin (n + 1) hi : point_index a\u271d \u2208 univ hn : point_index a\u271d \u2260 point_index i \u22a2 pointWeightsWithCircumcenter i (point_index a\u271d) = 0 ** have h : _ \u2260 i := fun h => hn (h \u25b8 rfl) ** case point_index V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i a\u271d : Fin (n + 1) hi : point_index a\u271d \u2208 univ hn : point_index a\u271d \u2260 point_index i h : a\u271d \u2260 i \u22a2 pointWeightsWithCircumcenter i (point_index a\u271d) = 0 ** simp [pointWeightsWithCircumcenter, h] ** case circumcenter_index V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s : Simplex \u211d P n i : Fin (n + 1) hi : circumcenter_index \u2208 univ hn : circumcenter_index \u2260 point_index i \u22a2 pointWeightsWithCircumcenter i circumcenter_index = 0 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Complex.GammaSeq_eq_approx_Gamma_integral ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 \u22a2 GammaSeq s n = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191x ^ (s - 1) ** have : \u2200 x : \u211d, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this : \u2200 (x : \u211d), x = x / \u2191n * \u2191n \u22a2 GammaSeq s n = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191x ^ (s - 1) ** conv_rhs => enter [1, x, 2, 1]; rw [this x] ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this : \u2200 (x : \u211d), x = x / \u2191n * \u2191n \u22a2 GammaSeq s n = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) ** rw [GammaSeq_eq_betaIntegral_of_re_pos hs] ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this : \u2200 (x : \u211d), x = x / \u2191n * \u2191n \u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) ** 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) ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, (fun x => \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) (x / \u2191n) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, (fun x => \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) x \u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) ** dsimp only at this ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) \u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) ** rw [betaIntegral, this, real_smul, zero_div, div_self, add_sub_cancel,\n \u2190 intervalIntegral.integral_const_mul, \u2190 intervalIntegral.integral_const_mul] ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) \u22a2 \u222b (x : \u211d) in 0 ..1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u222b (x : \u211d) in 0 ..1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) \u22a2 \u2191n \u2260 0 ** swap ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) \u22a2 \u222b (x : \u211d) in 0 ..1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u222b (x : \u211d) in 0 ..1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) ** simp_rw [intervalIntegral.integral_of_le zero_le_one] ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) \u22a2 \u222b (x : \u211d) in Ioc 0 1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u222b (x : \u211d) in Ioc 0 1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) ** refine' set_integral_congr measurableSet_Ioc fun x hx => _ ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 \u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) ** push_cast ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 \u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1)) ** have hn' : (n : \u2102) \u2260 0 := Nat.cast_ne_zero.mpr hn ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 hn' : \u2191n \u2260 0 \u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1)) ** 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 ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 hn' : \u2191n \u2260 0 A : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n \u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1)) ** 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] ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 hn' : \u2191n \u2260 0 A : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n B : (\u2191x * \u2191n) ^ (s - 1) = \u2191x ^ (s - 1) * \u2191n ^ (s - 1) \u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1)) ** rw [A, B, cpow_nat_cast] ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 hn' : \u2191n \u2260 0 A : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n B : (\u2191x * \u2191n) ^ (s - 1) = \u2191x ^ (s - 1) * \u2191n ^ (s - 1) \u22a2 \u2191n ^ (s - 1) * \u2191n * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x ^ (s - 1) * \u2191n ^ (s - 1))) ** ring ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 \u22a2 \u2200 (x : \u211d), x = x / \u2191n * \u2191n ** intro x ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 x : \u211d \u22a2 x = x / \u2191n * \u2191n ** rw [div_mul_cancel] ** case h s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 x : \u211d \u22a2 \u2191n \u2260 0 ** exact Nat.cast_ne_zero.mpr hn ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) \u22a2 \u2191n \u2260 0 ** exact Nat.cast_ne_zero.mpr hn ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 hn' : \u2191n \u2260 0 \u22a2 \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n ** conv_lhs => rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn'] ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 hn' : \u2191n \u2260 0 \u22a2 \u2191n ^ (s - 1) * \u2191n ^ 1 = \u2191n ^ (s - 1) * \u2191n ** simp ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 hn' : \u2191n \u2260 0 \u22a2 s = s - 1 + 1 ** ring ** s : \u2102 hs : 0 < s.re n : \u2115 hn : n \u2260 0 this\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n this : \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) = \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) x : \u211d hx : x \u2208 Ioc 0 1 hn' : \u2191n \u2260 0 A : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n \u22a2 (\u2191x * \u2191n) ^ (s - 1) = \u2191x ^ (s - 1) * \u2191n ^ (s - 1) ** rw [\u2190 ofReal_nat_cast,\n mul_cpow_ofReal_nonneg hx.1.le (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)).le] ** Qed", + "informal": "" + }, + { + "formal": "add_tsub_add_eq_tsub_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2075 : PartialOrder \u03b1 inst\u271d\u2074 : AddCommSemigroup \u03b1 inst\u271d\u00b3 : Sub \u03b1 inst\u271d\u00b2 : OrderedSub \u03b1 a\u271d b\u271d c\u271d d : \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 inst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 a b c : \u03b1 \u22a2 a + b - (a + c) = b - c ** rw [add_comm a b, add_comm a c, add_tsub_add_eq_tsub_right] ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Language.BoundedFormula.realize_sup ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b2 : Structure L M inst\u271d\u00b9 : Structure L N inst\u271d : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n l : \u2115 \u03c6 \u03c8 : BoundedFormula L \u03b1 l \u03b8 : BoundedFormula L \u03b1 (Nat.succ l) v : \u03b1 \u2192 M xs : Fin l \u2192 M \u22a2 Realize (\u03c6 \u2294 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2228 Realize \u03c8 v xs ** simp only [realize, Sup.sup, realize_not, eq_iff_iff] ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b2 : Structure L M inst\u271d\u00b9 : Structure L N inst\u271d : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n l : \u2115 \u03c6 \u03c8 : BoundedFormula L \u03b1 l \u03b8 : BoundedFormula L \u03b1 (Nat.succ l) v : \u03b1 \u2192 M xs : Fin l \u2192 M \u22a2 Realize (\u223c\u03c6 \u27f9 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2228 Realize \u03c8 v xs ** tauto ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Abelian.Functor.exact_of_map_injectiveResolution ** C : Type u inst\u271d\u2075 : Category.{w, u} C D : Type u inst\u271d\u2074 : Category.{w, u} D F : C \u2964 D X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z inst\u271d\u00b3 : Abelian C inst\u271d\u00b2 : Abelian D inst\u271d\u00b9 : Functor.Additive F P : InjectiveResolution X inst\u271d : PreservesFiniteLimits F \u22a2 (Iso.refl (F.obj (HomologicalComplex.X ((CochainComplex.single\u2080 C).obj X) 0))).hom \u226b F.map (HomologicalComplex.Hom.f P.\u03b9 0) = F.map (HomologicalComplex.Hom.f P.\u03b9 0) \u226b (Iso.refl (F.obj (HomologicalComplex.X P.cocomplex 0))).hom ** simp ** C : Type u inst\u271d\u2075 : Category.{w, u} C D : Type u inst\u271d\u2074 : Category.{w, u} D F : C \u2964 D X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z inst\u271d\u00b3 : Abelian C inst\u271d\u00b2 : Abelian D inst\u271d\u00b9 : Functor.Additive F P : InjectiveResolution X inst\u271d : PreservesFiniteLimits F \u22a2 (Iso.refl (F.obj (HomologicalComplex.X P.cocomplex 0))).hom \u226b HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up \u2115)).obj P.cocomplex) 0 = F.map (HomologicalComplex.d P.cocomplex 0 1) \u226b (HomologicalComplex.xNextIso ((mapHomologicalComplex F (ComplexShape.up \u2115)).obj P.cocomplex) (_ : 0 + 1 = 0 + 1)).symm.hom ** rw [Iso.refl_hom, Category.id_comp, Iso.symm_hom, HomologicalComplex.dFrom_eq] <;> congr ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.hasseDeriv_C ** R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] r : R hk : 0 < k \u22a2 \u2191(hasseDeriv k) (\u2191C r) = 0 ** rw [\u2190 monomial_zero_left, hasseDeriv_monomial, Nat.choose_eq_zero_of_lt hk, Nat.cast_zero,\n zero_mul, monomial_zero_right] ** Qed", + "informal": "" + }, + { + "formal": "MvPowerSeries.coeff_monomial_mul ** \u03c3 : Type u_1 R : Type u_2 inst\u271d : Semiring R m n : \u03c3 \u2192\u2080 \u2115 \u03c6 \u03c8 : MvPowerSeries \u03c3 R a : R \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 ** 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] ** \u03c3 : Type u_1 R : Type u_2 inst\u271d : Semiring R m n : \u03c3 \u2192\u2080 \u2115 \u03c6 \u03c8 : MvPowerSeries \u03c3 R a : R \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 ** 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) ** \u03c3 : Type u_1 R : Type u_2 inst\u271d : Semiring R m n : \u03c3 \u2192\u2080 \u2115 \u03c6 \u03c8 : MvPowerSeries \u03c3 R a : R this : \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 \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 ** rw [coeff_mul, \u2190 Finset.sum_filter_of_ne this, antidiagonal_filter_fst_eq, Finset.sum_ite_index] ** \u03c3 : Type u_1 R : Type u_2 inst\u271d : Semiring R m n : \u03c3 \u2192\u2080 \u2115 \u03c6 \u03c8 : MvPowerSeries \u03c3 R a : R this : \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 \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 else \u2211 x in \u2205, \u2191(coeff R x.1) (\u2191(monomial R n) a) * \u2191(coeff R x.2) \u03c6) = if n \u2264 m then a * \u2191(coeff R (m - n)) \u03c6 else 0 ** simp only [Finset.sum_singleton, coeff_monomial_same, Finset.sum_empty] ** Qed", + "informal": "" + }, + { + "formal": "Array.get?_push_eq ** \u03b1 : Type u_1 a : Array \u03b1 x : \u03b1 \u22a2 (push a x)[size a]? = some x ** rw [getElem?_pos, get_push_eq] ** Qed", + "informal": "" + }, + { + "formal": "SetTheory.PGame.Numeric.moveLeft ** x : PGame o : Numeric x i : LeftMoves x \u22a2 Numeric (PGame.moveLeft x i) ** cases x ** case mk \u03b1\u271d \u03b2\u271d : Type u_1 a\u271d\u00b9 : \u03b1\u271d \u2192 PGame a\u271d : \u03b2\u271d \u2192 PGame o : Numeric (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i : LeftMoves (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) \u22a2 Numeric (PGame.moveLeft (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i) ** exact o.2.1 i ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.snorm'_measure_zero_of_pos ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 F hq_pos : 0 < q \u22a2 snorm' f q 0 = 0 ** simp [snorm', hq_pos] ** Qed", + "informal": "" + }, + { + "formal": "CompositionSeries.ext_fun ** X : Type u inst\u271d\u00b9 : Lattice X inst\u271d : JordanHolderLattice X s\u2081 s\u2082 : CompositionSeries X hl : s\u2081.length = s\u2082.length h : \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) \u22a2 s\u2081 = s\u2082 ** cases s\u2081 ** case mk X : Type u inst\u271d\u00b9 : Lattice X inst\u271d : JordanHolderLattice X s\u2082 : CompositionSeries X length\u271d : \u2115 series\u271d : Fin (length\u271d + 1) \u2192 X step'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i)) hl : { length := length\u271d, series := series\u271d, step' := step'\u271d }.length = s\u2082.length h : \u2200 (i : Fin ({ length := length\u271d, series := series\u271d, step' := step'\u271d }.length + 1)), series { length := length\u271d, series := series\u271d, step' := step'\u271d } i = series s\u2082 (Fin.cast (_ : Nat.succ { length := length\u271d, series := series\u271d, step' := step'\u271d }.length = Nat.succ s\u2082.length) i) \u22a2 { length := length\u271d, series := series\u271d, step' := step'\u271d } = s\u2082 ** cases s\u2082 ** case mk.mk X : Type u inst\u271d\u00b9 : Lattice X inst\u271d : JordanHolderLattice X length\u271d\u00b9 : \u2115 series\u271d\u00b9 : Fin (length\u271d\u00b9 + 1) \u2192 X step'\u271d\u00b9 : \u2200 (i : Fin length\u271d\u00b9), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i)) length\u271d : \u2115 series\u271d : Fin (length\u271d + 1) \u2192 X step'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i)) hl : { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length = { length := length\u271d, series := series\u271d, step' := step'\u271d }.length h : \u2200 (i : Fin ({ length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length + 1)), series { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } i = series { length := length\u271d, series := series\u271d, step' := step'\u271d } (Fin.cast (_ : Nat.succ { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length = Nat.succ { length := length\u271d, series := series\u271d, step' := step'\u271d }.length) i) \u22a2 { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } = { length := length\u271d, series := series\u271d, step' := step'\u271d } ** dsimp at hl h ** case mk.mk X : Type u inst\u271d\u00b9 : Lattice X inst\u271d : JordanHolderLattice X length\u271d\u00b9 : \u2115 series\u271d\u00b9 : Fin (length\u271d\u00b9 + 1) \u2192 X step'\u271d\u00b9 : \u2200 (i : Fin length\u271d\u00b9), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i)) length\u271d : \u2115 series\u271d : Fin (length\u271d + 1) \u2192 X step'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i)) hl : length\u271d\u00b9 = length\u271d h : \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) \u22a2 { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } = { length := length\u271d, series := series\u271d, step' := step'\u271d } ** subst hl ** case mk.mk X : Type u inst\u271d\u00b9 : Lattice X inst\u271d : JordanHolderLattice X length\u271d : \u2115 series\u271d\u00b9 : Fin (length\u271d + 1) \u2192 X step'\u271d\u00b9 : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i)) series\u271d : Fin (length\u271d + 1) \u2192 X step'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i)) h : \u2200 (i : Fin (length\u271d + 1)), series\u271d\u00b9 i = series\u271d (Fin.cast (_ : Nat.succ length\u271d = Nat.succ length\u271d) i) \u22a2 { length := length\u271d, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } = { length := length\u271d, series := series\u271d, step' := step'\u271d } ** simpa [Function.funext_iff] using h ** Qed", + "informal": "" + }, + { + "formal": "Nat.pos_of_lt_add_left ** n k : Nat h : n < k + n \u22a2 0 + ?m.18605 h < k + ?m.18605 h ** rw [Nat.zero_add] ** n k : Nat h : n < k + n \u22a2 ?m.18605 h < k + ?m.18605 h \u22a2 {n k : Nat} \u2192 n < k + n \u2192 Nat ** exact h ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.map_prod_map ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SigmaFinite \u03bd inst\u271d\u00b9 : SigmaFinite \u03bc \u03b4 : Type u_7 inst\u271d : MeasurableSpace \u03b4 f : \u03b1 \u2192 \u03b2 g : \u03b3 \u2192 \u03b4 \u03bca : Measure \u03b1 \u03bcc : Measure \u03b3 hfa : SigmaFinite (map f \u03bca) hgc : SigmaFinite (map g \u03bcc) hf : Measurable f hg : Measurable g \u22a2 Measure.prod (map f \u03bca) (map g \u03bcc) = map (Prod.map f g) (Measure.prod \u03bca \u03bcc) ** haveI := hgc.of_map \u03bcc hg.aemeasurable ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SigmaFinite \u03bd inst\u271d\u00b9 : SigmaFinite \u03bc \u03b4 : Type u_7 inst\u271d : MeasurableSpace \u03b4 f : \u03b1 \u2192 \u03b2 g : \u03b3 \u2192 \u03b4 \u03bca : Measure \u03b1 \u03bcc : Measure \u03b3 hfa : SigmaFinite (map f \u03bca) hgc : SigmaFinite (map g \u03bcc) hf : Measurable f hg : Measurable g this : SigmaFinite \u03bcc \u22a2 Measure.prod (map f \u03bca) (map g \u03bcc) = map (Prod.map f g) (Measure.prod \u03bca \u03bcc) ** refine' prod_eq fun s t hs ht => _ ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SigmaFinite \u03bd inst\u271d\u00b9 : SigmaFinite \u03bc \u03b4 : Type u_7 inst\u271d : MeasurableSpace \u03b4 f : \u03b1 \u2192 \u03b2 g : \u03b3 \u2192 \u03b4 \u03bca : Measure \u03b1 \u03bcc : Measure \u03b3 hfa : SigmaFinite (map f \u03bca) hgc : SigmaFinite (map g \u03bcc) hf : Measurable f hg : Measurable g this : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t \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 ** rw [map_apply (hf.prod_map hg) (hs.prod ht), map_apply hf hs, map_apply hg ht] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SigmaFinite \u03bd inst\u271d\u00b9 : SigmaFinite \u03bc \u03b4 : Type u_7 inst\u271d : MeasurableSpace \u03b4 f : \u03b1 \u2192 \u03b2 g : \u03b3 \u2192 \u03b4 \u03bca : Measure \u03b1 \u03bcc : Measure \u03b3 hfa : SigmaFinite (map f \u03bca) hgc : SigmaFinite (map g \u03bcc) hf : Measurable f hg : Measurable g this : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t \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) ** exact prod_prod (f \u207b\u00b9' s) (g \u207b\u00b9' t) ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.kernel.compProd_apply_eq_compProdFun ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = compProdFun \u03ba \u03b7 a s ** rw [compProd, dif_pos] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191{ val := fun a => Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0) (_ : \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)), (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)), property := (_ : Measurable fun a => Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0) (_ : \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)), (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) } a) s = compProdFun \u03ba \u03b7 a s case hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7 ** swap ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191{ val := fun a => Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0) (_ : \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)), (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)), property := (_ : Measurable fun a => Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0) (_ : \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)), (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) } a) s = compProdFun \u03ba \u03b7 a s ** 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 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0) (_ : \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)), (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) s = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a ** rw [Measure.ofMeasurable_apply _ hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \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 ** rfl ** case hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7 ** constructor <;> infer_instance ** Qed", + "informal": "" + }, + { + "formal": "Multiset.coe_bind ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 \u03b4 : Type u_3 a : \u03b1 s t : Multiset \u03b1 f\u271d g : \u03b1 \u2192 Multiset \u03b2 l : List \u03b1 f : \u03b1 \u2192 List \u03b2 \u22a2 (bind \u2191l fun a => \u2191(f a)) = \u2191(List.bind l f) ** rw [List.bind, \u2190 coe_join, List.map_map] ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 \u03b4 : Type u_3 a : \u03b1 s t : Multiset \u03b1 f\u271d g : \u03b1 \u2192 Multiset \u03b2 l : List \u03b1 f : \u03b1 \u2192 List \u03b2 \u22a2 (bind \u2191l fun a => \u2191(f a)) = join \u2191(List.map (ofList \u2218 f) l) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "TopCat.Presheaf.stalkPushforward.id ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasColimits C X Y Z : TopCat \u2131 : Presheaf C X x : \u2191X \u22a2 stalkPushforward C (\ud835\udfd9 X) \u2131 x = (stalkFunctor C x).map (Pushforward.id \u2131).hom ** ext1 j ** case w C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasColimits C X Y Z : TopCat \u2131 : Presheaf C X x : \u2191X j : (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 \u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj (\ud835\udfd9 X _* \u2131)) j \u226b stalkPushforward C (\ud835\udfd9 X) \u2131 x = colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj (\ud835\udfd9 X _* \u2131)) j \u226b (stalkFunctor C x).map (Pushforward.id \u2131).hom ** induction' j with j ** case w.h C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasColimits C X Y Z : TopCat \u2131 : Presheaf C X x : \u2191X j : OpenNhds (\u2191(\ud835\udfd9 X) x) \u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj (\ud835\udfd9 X _* \u2131)) (op j) \u226b stalkPushforward C (\ud835\udfd9 X) \u2131 x = colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj (\ud835\udfd9 X _* \u2131)) (op j) \u226b (stalkFunctor C x).map (Pushforward.id \u2131).hom ** rcases j with \u27e8\u27e8_, _\u27e9, _\u27e9 ** case w.h.mk.mk C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasColimits C X Y Z : TopCat \u2131 : Presheaf C X x : \u2191X carrier\u271d : Set \u2191X is_open'\u271d : IsOpen carrier\u271d property\u271d : \u2191(\ud835\udfd9 X) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d } \u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj (\ud835\udfd9 X _* \u2131)) (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b stalkPushforward C (\ud835\udfd9 X) \u2131 x = colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj (\ud835\udfd9 X _* \u2131)) (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b (stalkFunctor C x).map (Pushforward.id \u2131).hom ** erw [colimit.\u03b9_map_assoc] ** case w.h.mk.mk C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasColimits C X Y Z : TopCat \u2131 : Presheaf C X x : \u2191X carrier\u271d : Set \u2191X is_open'\u271d : IsOpen carrier\u271d property\u271d : \u2191(\ud835\udfd9 X) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d } \u22a2 (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (\ud835\udfd9 X) x).inv) \u2131).app (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b colimit.\u03b9 ((OpenNhds.map (\ud835\udfd9 X) x).op \u22d9 ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj \u2131) (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b colimit.pre (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj \u2131) (OpenNhds.map (\ud835\udfd9 X) x).op = colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj (\ud835\udfd9 X _* \u2131)) (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b (stalkFunctor C x).map (Pushforward.id \u2131).hom ** simp [stalkFunctor, stalkPushforward] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** have fmeas : AEMeasurable f \u03bc := by\n convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable\n simp only [Real.toNNReal_coe] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 fmeas : AEMeasurable f \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** lift \u03b5 to \u211d\u22650 using \u03b5pos.le ** case intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u03b5\u27e9 : \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227 \u03b4 < \u03b5 ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 < \u03b5 case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** exact exists_between \u03b5pos ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** have int_f_ne_top : (\u222b\u207b a : \u03b1, f a \u2202\u03bc) \u2260 \u221e :=\n (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** 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 ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** have gint_ne : (\u222b\u207b x : \u03b1, g x \u2202\u03bc) \u2260 \u221e := ne_top_of_le_ne_top (by simpa) gint ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** have g_lt_top : \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u221e := ae_lt_top gcont.measurable gint_ne ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** 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] ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 (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 ** refine' \u27e8g, f_lt_g, gcont, g_lt_top, _, _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 AEMeasurable f ** convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable ** case h.e'_5.h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 x\u271d : \u03b1 \u22a2 f x\u271d = Real.toNNReal \u2191(f x\u271d) ** simp only [Real.toNNReal_coe] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4 ** simpa ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc ** apply lintegral_congr_ae ** case h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 \u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (g a))) =\u1da0[ae \u03bc] fun a => g a ** filter_upwards [g_lt_top] with _ hx ** case h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 a\u271d : \u03b1 hx : g a\u271d < \u22a4 \u22a2 ENNReal.ofReal (ENNReal.toReal (g a\u271d)) = g a\u271d ** simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff] ** case intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 Integrable fun x => ENNReal.toReal (g x) ** refine' \u27e8gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9 ** case intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 HasFiniteIntegral fun x => ENNReal.toReal (g x) ** simp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg] ** case intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc < \u22a4 ** convert gint_ne.lt_top using 1 ** case intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae] ** case intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) < ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5 ** 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 ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \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) ** congr 1 ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \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) ** apply ENNReal.toReal_mono _ gint ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4 ** simpa using int_f_ne_top ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \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 ** rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \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 ** simp ** case intro.intro.intro.intro.intro.intro.refine'_2.hf \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x) ** apply Filter.eventually_of_forall fun x => _ ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x ** simp ** case intro.intro.intro.intro.intro.intro.refine'_2.hfm \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc ** exact fmeas.coe_nnreal_real.aestronglyMeasurable ** case intro.intro.intro.intro.intro.intro.refine'_2.hf \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 0 \u2264\u1da0[ae \u03bc] fun x => ENNReal.toReal (g x) ** apply Filter.eventually_of_forall fun x => _ ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => ENNReal.toReal (g x)) x ** simp ** case intro.intro.intro.intro.intro.intro.refine'_2.hfm \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) \u03bc ** apply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable ** Qed", + "informal": "" + }, + { + "formal": "List.Nat.antidiagonalTuple_pairwise_pi_lex ** k n : \u2115 \u22a2 Pairwise (Pi.Lex (fun x x_1 => x < x_1) fun x x x_1 => x < x_1) (antidiagonalTuple (k + 1) n) ** simp_rw [antidiagonalTuple, List.pairwise_bind, List.pairwise_map, List.mem_map,\n forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082] ** k n : \u2115 \u22a2 (\u2200 (a : \u2115 \u00d7 \u2115), a \u2208 antidiagonal n \u2192 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.1 a_2) (Fin.cons a.1 b)) (antidiagonalTuple (Nat.add k 0) a.2)) \u2227 Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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)) (antidiagonal n) ** simp only [mem_antidiagonal, Prod.forall, and_imp, forall_apply_eq_imp_iff\u2082] ** k n : \u2115 \u22a2 (\u2200 (a b : \u2115), a + b = n \u2192 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)) (antidiagonalTuple (Nat.add k 0) b)) \u2227 Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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)) (antidiagonal n) ** simp only [Fin.pi_lex_lt_cons_cons, eq_self_iff_true, true_and_iff, lt_self_iff_false,\n false_or_iff] ** k n : \u2115 \u22a2 (\u2200 (a b : \u2115), a + b = n \u2192 Pairwise (fun a b => Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a b) (antidiagonalTuple (Nat.add k 0) b)) \u2227 Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) ** refine' \u27e8fun _ _ _ => antidiagonalTuple_pairwise_pi_lex k _, _\u27e9 ** k n : \u2115 \u22a2 Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) ** induction' n with n n_ih ** case zero k : \u2115 \u22a2 Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal Nat.zero) ** rw [antidiagonal_zero] ** case zero k : \u2115 \u22a2 Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) [(0, 0)] ** exact List.pairwise_singleton _ _ ** case succ k n : \u2115 n_ih : Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) \u22a2 Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal (Nat.succ n)) ** rw [antidiagonal_succ, List.pairwise_cons, List.pairwise_map] ** case succ k n : \u2115 n_ih : Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) \u22a2 (\u2200 (a' : \u2115 \u00d7 \u2115), a' \u2208 map (Prod.map Nat.succ id) (antidiagonal n) \u2192 \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a'.2 \u2192 (0, n + 1).1 < a'.1 \u2228 (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 Pairwise (fun a b => \u2200 (a_1 : Fin (Nat.add k 0) \u2192 \u2115), a_1 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id a).2 \u2192 \u2200 (a_3 : Fin (Nat.add k 0) \u2192 \u2115), a_3 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id b).2 \u2192 (Prod.map Nat.succ id a).1 < (Prod.map Nat.succ id b).1 \u2228 (Prod.map Nat.succ id a).1 = (Prod.map Nat.succ id b).1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3) (antidiagonal n) ** refine' \u27e8fun p hp x hx y hy => _, _\u27e9 ** case succ.refine'_2 k n : \u2115 n_ih : Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) \u22a2 Pairwise (fun a b => \u2200 (a_1 : Fin (Nat.add k 0) \u2192 \u2115), a_1 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id a).2 \u2192 \u2200 (a_3 : Fin (Nat.add k 0) \u2192 \u2115), a_3 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id b).2 \u2192 (Prod.map Nat.succ id a).1 < (Prod.map Nat.succ id b).1 \u2228 (Prod.map Nat.succ id a).1 = (Prod.map Nat.succ id b).1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3) (antidiagonal n) ** dsimp ** case succ.refine'_2 k n : \u2115 n_ih : Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) \u22a2 Pairwise (fun a b => \u2200 (a_1 : Fin k \u2192 \u2115), a_1 \u2208 antidiagonalTuple k a.2 \u2192 \u2200 (a_3 : Fin k \u2192 \u2115), a_3 \u2208 antidiagonalTuple k b.2 \u2192 Nat.succ a.1 < Nat.succ b.1 \u2228 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) (antidiagonal n) ** simp_rw [Nat.succ_inj', Nat.succ_lt_succ_iff] ** case succ.refine'_2 k n : \u2115 n_ih : Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) \u22a2 Pairwise (fun a b => \u2200 (a_1 : Fin k \u2192 \u2115), a_1 \u2208 antidiagonalTuple k a.2 \u2192 \u2200 (a_3 : Fin k \u2192 \u2115), a_3 \u2208 antidiagonalTuple k b.2 \u2192 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) (antidiagonal n) ** exact n_ih ** case succ.refine'_1 k n : \u2115 n_ih : Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) p : \u2115 \u00d7 \u2115 hp : p \u2208 map (Prod.map Nat.succ id) (antidiagonal n) x : Fin (Nat.add k 0) \u2192 \u2115 hx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2 y : Fin (Nat.add k 0) \u2192 \u2115 hy : y \u2208 antidiagonalTuple (Nat.add k 0) p.2 \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 ** rw [List.mem_map, Prod.exists] at hp ** case succ.refine'_1 k n : \u2115 n_ih : Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) p : \u2115 \u00d7 \u2115 hp : \u2203 a b, (a, b) \u2208 antidiagonal n \u2227 Prod.map Nat.succ id (a, b) = p x : Fin (Nat.add k 0) \u2192 \u2115 hx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2 y : Fin (Nat.add k 0) \u2192 \u2115 hy : y \u2208 antidiagonalTuple (Nat.add k 0) p.2 \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 ** obtain \u27e8a, b, _, rfl : (Nat.succ a, b) = p\u27e9 := hp ** case succ.refine'_1.intro.intro.intro k n : \u2115 n_ih : Pairwise (fun a\u2081 a\u2082 => \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115), a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192 \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115), a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192 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) (antidiagonal n) x : Fin (Nat.add k 0) \u2192 \u2115 hx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2 y : Fin (Nat.add k 0) \u2192 \u2115 a b : \u2115 left\u271d : (a, b) \u2208 antidiagonal n hy : y \u2208 antidiagonalTuple (Nat.add k 0) (Nat.succ a, b).2 \u22a2 (0, n + 1).1 < (Nat.succ a, b).1 \u2228 (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 ** exact Or.inl (Nat.zero_lt_succ _) ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.measure_diff' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t s : Set \u03b1 hm : MeasurableSet t h_fin : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc (s \\ t) + \u2191\u2191\u03bc t = \u2191\u2191\u03bc (s \u222a t) ** rw [add_comm, measure_add_diff hm, union_comm] ** Qed", + "informal": "" + }, + { + "formal": "ContractingWith.isFixedPt_fixedPoint_iterate ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] \u22a2 IsFixedPt f (fixedPoint f^[n] hf) ** set x := hf.fixedPoint f^[n] ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] x : \u03b1 := fixedPoint f^[n] hf \u22a2 IsFixedPt f x ** have hx : f^[n] x = x := hf.fixedPoint_isFixedPt ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] x : \u03b1 := fixedPoint f^[n] hf hx : f^[n] x = x \u22a2 IsFixedPt f x ** have := hf.toLipschitzWith.dist_le_mul x (f x) ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] x : \u03b1 := fixedPoint f^[n] hf hx : f^[n] x = x this : dist (f^[n] x) (f^[n] (f x)) \u2264 \u2191K * dist x (f x) \u22a2 IsFixedPt f x ** rw [\u2190 iterate_succ_apply, iterate_succ_apply', hx] at this ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] x : \u03b1 := fixedPoint f^[n] hf hx : f^[n] x = x this : dist x (f x) \u2264 \u2191K * dist x (f x) \u22a2 IsFixedPt f x ** revert this ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] x : \u03b1 := fixedPoint f^[n] hf hx : f^[n] x = x \u22a2 dist x (f x) \u2264 \u2191K * dist x (f x) \u2192 IsFixedPt f x ** contrapose! ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] x : \u03b1 := fixedPoint f^[n] hf hx : f^[n] x = x \u22a2 \u00acIsFixedPt f x \u2192 \u2191K * dist x (f x) < dist x (f x) ** intro this ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] x : \u03b1 := fixedPoint f^[n] hf hx : f^[n] x = x this : \u00acIsFixedPt f x \u22a2 \u2191K * dist x (f x) < dist x (f x) ** have := dist_pos.2 (Ne.symm this) ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 K : \u211d\u22650 f : \u03b1 \u2192 \u03b1 hf\u271d : ContractingWith K f inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : CompleteSpace \u03b1 n : \u2115 hf : ContractingWith K f^[n] x : \u03b1 := fixedPoint f^[n] hf hx : f^[n] x = x this\u271d : \u00acIsFixedPt f x this : 0 < dist x (f x) \u22a2 \u2191K * dist x (f x) < dist x (f x) ** simpa only [NNReal.coe_one, one_mul, NNReal.val_eq_coe] using (mul_lt_mul_right this).mpr hf.left ** Qed", + "informal": "" + }, + { + "formal": "multiplicity.finite_pow ** \u03b1 : Type u_1 inst\u271d : CancelCommMonoidWithZero \u03b1 p a : \u03b1 hp : Prime p x\u271d : Finite p a \u22a2 \u00acp ^ (0 + 1) \u2223 a ^ 0 ** simp [mt isUnit_iff_dvd_one.2 hp.2.1] ** \u03b1 : Type u_1 inst\u271d : CancelCommMonoidWithZero \u03b1 p a : \u03b1 hp : Prime p k : \u2115 ha : Finite p a \u22a2 Finite p (a ^ (k + 1)) ** rw [_root_.pow_succ] ** \u03b1 : Type u_1 inst\u271d : CancelCommMonoidWithZero \u03b1 p a : \u03b1 hp : Prime p k : \u2115 ha : Finite p a \u22a2 Finite p (a * a ^ k) ** exact finite_mul hp ha (finite_pow hp ha) ** Qed", + "informal": "" + }, + { + "formal": "NonUnitalSubring.toSubring_toNonUnitalSubring ** R : Type u_1 inst\u271d : Ring R S : NonUnitalSubring R h1 : 1 \u2208 S \u22a2 Subring.toNonUnitalSubring (toSubring S h1) = S ** cases S ** case mk R : Type u_1 inst\u271d : Ring R toNonUnitalSubsemiring\u271d : NonUnitalSubsemiring R neg_mem'\u271d : \u2200 {x : R}, x \u2208 toNonUnitalSubsemiring\u271d.carrier \u2192 -x \u2208 toNonUnitalSubsemiring\u271d.carrier h1 : 1 \u2208 { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d } \u22a2 Subring.toNonUnitalSubring (toSubring { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d } h1) = { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d } ** rfl ** Qed", + "informal": "" + }, + { + "formal": "nat_abs_sum_le ** \u03b9\u271d : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 \u03b9 : Type u_2 s : Finset \u03b9 f : \u03b9 \u2192 \u2124 \u22a2 Int.natAbs (\u2211 i in s, f i) \u2264 \u2211 i in s, Int.natAbs (f i) ** induction' s using Finset.induction_on with i s his IH ** case empty \u03b9\u271d : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 \u03b9 : Type u_2 f : \u03b9 \u2192 \u2124 \u22a2 Int.natAbs (\u2211 i in \u2205, f i) \u2264 \u2211 i in \u2205, Int.natAbs (f i) ** simp only [Finset.sum_empty, Int.natAbs_zero] ** case insert \u03b9\u271d : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 \u03b9 : Type u_2 f : \u03b9 \u2192 \u2124 i : \u03b9 s : Finset \u03b9 his : \u00aci \u2208 s IH : Int.natAbs (\u2211 i in s, f i) \u2264 \u2211 i in s, Int.natAbs (f i) \u22a2 Int.natAbs (\u2211 i in insert i s, f i) \u2264 \u2211 i in insert i s, Int.natAbs (f i) ** simp only [his, Finset.sum_insert, not_false_iff] ** case insert \u03b9\u271d : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 \u03b9 : Type u_2 f : \u03b9 \u2192 \u2124 i : \u03b9 s : Finset \u03b9 his : \u00aci \u2208 s IH : Int.natAbs (\u2211 i in s, f i) \u2264 \u2211 i in s, Int.natAbs (f i) \u22a2 Int.natAbs (f i + \u2211 i in s, f i) \u2264 Int.natAbs (f i) + \u2211 i in s, Int.natAbs (f i) ** exact (Int.natAbs_add_le _ _).trans (add_le_add le_rfl IH) ** Qed", + "informal": "" + }, + { + "formal": "RingHom.StableUnderBaseChange.pushout_inl ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : StableUnderBaseChange P hP' : RespectsIso P R S T : CommRingCat f : R \u27f6 S g : R \u27f6 T H : P g \u22a2 P pushout.inl ** 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] ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : StableUnderBaseChange P hP' : RespectsIso P R S T : CommRingCat f : R \u27f6 S g : R \u27f6 T H : P g \u22a2 P ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app WalkingSpan.left) ** letI := f.toAlgebra ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : StableUnderBaseChange P hP' : RespectsIso P R S T : CommRingCat f : R \u27f6 S g : R \u27f6 T H : P g this : Algebra \u2191R \u2191S := toAlgebra f \u22a2 P ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app WalkingSpan.left) ** letI := g.toAlgebra ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : StableUnderBaseChange P hP' : RespectsIso P R S T : CommRingCat f : R \u27f6 S g : R \u27f6 T H : P g this\u271d : Algebra \u2191R \u2191S := toAlgebra f this : Algebra \u2191R \u2191T := toAlgebra g \u22a2 P ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app WalkingSpan.left) ** dsimp only [CommRingCat.pushoutCocone_inl, PushoutCocone.\u03b9_app_left] ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : StableUnderBaseChange P hP' : RespectsIso P R S T : CommRingCat f : R \u27f6 S g : R \u27f6 T H : P g this\u271d : Algebra \u2191R \u2191S := toAlgebra f this : Algebra \u2191R \u2191T := toAlgebra g \u22a2 P Algebra.TensorProduct.includeLeftRingHom ** apply hP R T S (TensorProduct R S T) ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : StableUnderBaseChange P hP' : RespectsIso P R S T : CommRingCat f : R \u27f6 S g : R \u27f6 T H : P g this\u271d : Algebra \u2191R \u2191S := toAlgebra f this : Algebra \u2191R \u2191T := toAlgebra g \u22a2 P (algebraMap \u2191R \u2191T) ** exact H ** Qed", + "informal": "" + }, + { + "formal": "Set.unbounded_lt_inter_le ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop s t : Set \u03b1 inst\u271d : LinearOrder \u03b1 a : \u03b1 \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 ** convert @unbounded_lt_inter_not_lt _ s _ a ** case h.e'_1.h.e'_3.h.e'_4.h.e'_2.h.a \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop s t : Set \u03b1 inst\u271d : LinearOrder \u03b1 a x\u271d : \u03b1 \u22a2 a \u2264 x\u271d \u2194 \u00acx\u271d < a ** exact not_lt.symm ** Qed", + "informal": "" + }, + { + "formal": "Orientation.map_eq_neg_iff_det_neg ** R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M \u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0 ** cases isEmpty_or_nonempty \u03b9 ** case inr R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : Nonempty \u03b9 \u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0 ** have H : 0 < finrank R M := by\n rw [\u2190 h]\n exact Fintype.card_pos ** case inr R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : Nonempty \u03b9 H : 0 < finrank R M \u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0 ** haveI : FiniteDimensional R M := finiteDimensional_of_finrank H ** case inr R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : Nonempty \u03b9 H : 0 < finrank R M this : FiniteDimensional R M \u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0 ** rw [map_eq_det_inv_smul _ _ h, units_inv_smul, units_smul_eq_neg_iff, LinearEquiv.coe_det] ** case inl R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : IsEmpty \u03b9 \u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0 ** have H : finrank R M = 0 := by\n refine' h.symm.trans _\n convert @Fintype.card_of_isEmpty \u03b9 _ ** case inl R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : IsEmpty \u03b9 H : finrank R M = 0 \u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0 ** simp [LinearMap.det_eq_one_of_finrank_eq_zero H, Module.Ray.ne_neg_self x] ** R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : IsEmpty \u03b9 \u22a2 finrank R M = 0 ** refine' h.symm.trans _ ** R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : IsEmpty \u03b9 \u22a2 Fintype.card \u03b9 = 0 ** convert @Fintype.card_of_isEmpty \u03b9 _ ** R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : Nonempty \u03b9 \u22a2 0 < finrank R M ** rw [\u2190 h] ** R : Type u_1 inst\u271d\u00b3 : LinearOrderedField R M : Type u_2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 _i : FiniteDimensional R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M h : Fintype.card \u03b9 = finrank R M h\u271d : Nonempty \u03b9 \u22a2 0 < Fintype.card \u03b9 ** exact Fintype.card_pos ** Qed", + "informal": "" + }, + { + "formal": "WittVector.mul_polyOfInterest_aux3 ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 \u22a2 wittPolyProd p (n + 1) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * \u2191(rename (Prod.mk 1)) (wittPolynomial p \u2124 (n + 1)) + \u2191p ^ (n + 1) * X (1, n + 1) * \u2191(rename (Prod.mk 0)) (wittPolynomial p \u2124 (n + 1)) + remainder p n ** have mvpz : (p : \ud835\udd44) ^ (n + 1) = MvPolynomial.C ((p : \u2124) ^ (n + 1)) := by simp only; norm_cast ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 wittPolyProd p (n + 1) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * \u2191(rename (Prod.mk 1)) (wittPolynomial p \u2124 (n + 1)) + \u2191p ^ (n + 1) * X (1, n + 1) * \u2191(rename (Prod.mk 0)) (wittPolynomial p \u2124 (n + 1)) + remainder p n ** rw [wittPolyProd, wittPolynomial, AlgHom.map_sum, AlgHom.map_sum] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 (\u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + remainder p n ** 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] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1)) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + remainder p n ** 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] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1)) * (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1)) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + remainder p n ** 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] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1)) * (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1)) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (1, n + 1) * \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + remainder p n ** 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] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1)) * (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1)) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (1, n + 1) * (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1)) + remainder p n ** simp only [add_mul, mul_add] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + (\u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) + (\u2191p ^ (n + 1) * X (1, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1))) + remainder p n ** rw [add_comm _ (remainder p n)] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) = remainder p n + (-(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + (\u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) + (\u2191p ^ (n + 1) * X (1, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1)))) ** simp only [add_assoc] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + (\u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)))) = remainder p n + (-(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + (\u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + (\u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)) + (\u2191p ^ (n + 1) * X (1, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1)))))) ** apply congrArg (Add.add _) ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 mvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) \u22a2 \u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * (\u2191p ^ (n + 1) * X (1, n + 1)) + \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) = -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) + (\u2191p ^ (n + 1) * X (0, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + (\u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)) + (\u2191p ^ (n + 1) * X (1, n + 1) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) + \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1))))) ** ring ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 \u22a2 \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1)) ** norm_cast ** Qed", + "informal": "" + }, + { + "formal": "GeneralizedContinuedFraction.continuantsAux_recurrence ** K : Type u_1 g : GeneralizedContinuedFraction K n : \u2115 inst\u271d : DivisionRing K gp ppred pred : Pair K nth_s_eq : Stream'.Seq.get? g.s n = some gp nth_conts_aux_eq : continuantsAux g n = ppred succ_nth_conts_aux_eq : continuantsAux g (n + 1) = pred \u22a2 continuantsAux g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b } ** simp [*, continuantsAux, nextContinuants, nextDenominator, nextNumerator] ** Qed", + "informal": "" + }, + { + "formal": "Bool.xor_iff_ne ** \u22a2 \u2200 {x y : Bool}, xor x y = true \u2194 x \u2260 y ** decide ** Qed", + "informal": "" + }, + { + "formal": "Nat.doubleFactorial_add_one ** n : \u2115 \u22a2 (n + 1)\u203c = (n + 1) * (n - 1)\u203c ** cases n <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "CommRingCat.pushoutCocone_inl ** R A B : CommRingCat f : R \u27f6 A g : R \u27f6 B \u22a2 A \u27f6 (pushoutCocone f g).pt ** letI := f.toAlgebra ** R A B : CommRingCat f : R \u27f6 A g : R \u27f6 B this : Algebra \u2191R \u2191A := RingHom.toAlgebra f \u22a2 A \u27f6 (pushoutCocone f g).pt ** letI := g.toAlgebra ** R A B : CommRingCat f : R \u27f6 A g : R \u27f6 B this\u271d : Algebra \u2191R \u2191A := RingHom.toAlgebra f this : Algebra \u2191R \u2191B := RingHom.toAlgebra g \u22a2 A \u27f6 (pushoutCocone f g).pt ** exact Algebra.TensorProduct.includeLeftRingHom ** Qed", + "informal": "" + }, + { + "formal": "List.get?_set_ne ** \u03b1 : Type u_1 a : \u03b1 m n : Nat l : List \u03b1 h : m \u2260 n \u22a2 get? (set l m a) n = get? l n ** simp only [set_eq_modifyNth, get?_modifyNth_ne _ _ h] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.toBilin_basisFun ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2077 : Semiring R inst\u271d\u00b9\u2076 : AddCommMonoid M inst\u271d\u00b9\u2075 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u2074 : Ring R\u2081 inst\u271d\u00b9\u00b3 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u00b9 : CommSemiring R\u2082 inst\u271d\u00b9\u2070 : AddCommMonoid M\u2082 inst\u271d\u2079 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2078 : CommRing R\u2083 inst\u271d\u2077 : AddCommGroup M\u2083 inst\u271d\u2076 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2075 : Field K inst\u271d\u2074 : AddCommGroup V inst\u271d\u00b3 : Module K V B : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 o : Type u_12 inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : Fintype o inst\u271d : DecidableEq n b : Basis n R\u2082 M\u2082 \u22a2 toBilin (Pi.basisFun R\u2082 n) = toBilin' ** ext M ** case h.H R : Type u_1 M\u271d : Type u_2 inst\u271d\u00b9\u2077 : Semiring R inst\u271d\u00b9\u2076 : AddCommMonoid M\u271d inst\u271d\u00b9\u2075 : Module R M\u271d R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u2074 : Ring R\u2081 inst\u271d\u00b9\u00b3 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u00b9 : CommSemiring R\u2082 inst\u271d\u00b9\u2070 : AddCommMonoid M\u2082 inst\u271d\u2079 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2078 : CommRing R\u2083 inst\u271d\u2077 : AddCommGroup M\u2083 inst\u271d\u2076 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2075 : Field K inst\u271d\u2074 : AddCommGroup V inst\u271d\u00b3 : Module K V B : BilinForm R M\u271d B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 o : Type u_12 inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : Fintype o inst\u271d : DecidableEq n b : Basis n R\u2082 M\u2082 M : Matrix n n R\u2082 x\u271d y\u271d : n \u2192 R\u2082 \u22a2 bilin (\u2191(toBilin (Pi.basisFun R\u2082 n)) M) x\u271d y\u271d = bilin (\u2191toBilin' M) x\u271d y\u271d ** simp only [Matrix.toBilin_apply, Matrix.toBilin'_apply, Pi.basisFun_repr] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s h'f : Measurable f \u22a2 Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) = Measure.restrict \u03bc (f '' s) ** apply Measure.ext fun t ht => ?_ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s h'f : Measurable f t : Set E ht : MeasurableSet t \u22a2 \u2191\u2191(Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) t = \u2191\u2191(Measure.restrict \u03bc (f '' s)) t ** 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] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.FunctorToTypes.map_id_apply ** C : Type u inst\u271d : Category.{v, u} C F G H : C \u2964 Type w X Y Z : C \u03c3 : F \u27f6 G \u03c4 : G \u27f6 H a : F.obj X \u22a2 F.map (\ud835\udfd9 X) a = a ** simp [types_id] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Abelian.tfae_epi ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z \u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0] ** tfae_have 3 \u2192 2 ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 \u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0] ** tfae_have 1 \u2192 3 ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 tfae_1_to_3 : Epi f \u2192 Exact f 0 \u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0] ** tfae_have 2 \u2192 1 ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 tfae_1_to_3 : Epi f \u2192 Exact f 0 tfae_2_to_1 : cokernel.\u03c0 f = 0 \u2192 Epi f \u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0] ** tfae_finish ** case tfae_3_to_2 C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z \u22a2 Exact f 0 \u2192 cokernel.\u03c0 f = 0 ** rw [exact_iff] ** case tfae_3_to_2 C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z \u22a2 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0 \u2192 cokernel.\u03c0 f = 0 ** rintro \u27e8-, h\u27e9 ** case tfae_3_to_2.intro C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z h : kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0 \u22a2 cokernel.\u03c0 f = 0 ** exact zero_of_epi_comp _ h ** case tfae_1_to_3 C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 \u22a2 Epi f \u2192 Exact f 0 ** rw [exact_iff] ** case tfae_1_to_3 C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 \u22a2 Epi f \u2192 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0 ** intro ** case tfae_1_to_3 C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 \u271d : Epi f \u22a2 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0 ** exact \u27e8by simp, by simp [cokernel.\u03c0_of_epi]\u27e9 ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 \u271d : Epi f \u22a2 f \u226b 0 = 0 ** simp ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 \u271d : Epi f \u22a2 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0 ** simp [cokernel.\u03c0_of_epi] ** case tfae_2_to_1 C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z tfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0 tfae_1_to_3 : Epi f \u2192 Exact f 0 \u22a2 cokernel.\u03c0 f = 0 \u2192 Epi f ** exact epi_of_cokernel_\u03c0_eq_zero _ ** Qed", + "informal": "" + }, + { + "formal": "Nat.unpair_right_le ** n : \u2115 \u22a2 (unpair n).2 \u2264 n ** simpa using right_le_pair n.unpair.1 n.unpair.2 ** Qed", + "informal": "" + }, + { + "formal": "isEmpty_psigma ** \u03b1\u271d : Sort u_1 \u03b2 : Sort u_2 \u03b3 : Sort u_3 \u03b1 : Sort u_5 E : \u03b1 \u2192 Sort u_4 \u22a2 IsEmpty (PSigma E) \u2194 \u2200 (a : \u03b1), IsEmpty (E a) ** simp only [\u2190 not_nonempty_iff, nonempty_psigma, not_exists] ** Qed", + "informal": "" + }, + { + "formal": "smul_one_mul ** M\u271d : Type u_1 N\u271d : Type u_2 G : Type u_3 A : Type u_4 B : Type u_5 \u03b1 : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 \u03b4 : Type u_9 M : Type u_10 N : Type u_11 inst\u271d\u00b2 : MulOneClass N inst\u271d\u00b9 : SMul M N inst\u271d : IsScalarTower M N N x : M y : N \u22a2 x \u2022 1 * y = x \u2022 y ** rw [smul_mul_assoc, one_mul] ** Qed", + "informal": "" + }, + { + "formal": "TruncatedWittVector.mk_coeff ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 x : TruncatedWittVector p n R \u22a2 (mk p fun i => coeff i x) = x ** ext i ** case h p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 x : TruncatedWittVector p n R i : Fin n \u22a2 coeff i (mk p fun i => coeff i x) = coeff i x ** rw [coeff_mk] ** Qed", + "informal": "" + }, + { + "formal": "Denumerable.prod_ofNat_val ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : Denumerable \u03b1 inst\u271d : Denumerable \u03b2 n : \u2115 \u22a2 ofNat (\u03b1 \u00d7 \u03b2) n = (ofNat \u03b1 (unpair n).1, ofNat \u03b2 (unpair n).2) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Nat.ArithmeticFunction.coe_zeta_smul_apply ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : Semiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M f : ArithmeticFunction M x : \u2115 \u22a2 \u2191(\u2191\u03b6 \u2022 f) x = \u2211 i in divisors x, \u2191f i ** rw [smul_apply] ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : Semiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M f : ArithmeticFunction M x : \u2115 \u22a2 \u2211 x in divisorsAntidiagonal x, \u2191\u2191\u03b6 x.1 \u2022 \u2191f x.2 = \u2211 i in divisors x, \u2191f i ** trans \u2211 i in divisorsAntidiagonal x, f i.snd ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : Semiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M f : ArithmeticFunction M x : \u2115 \u22a2 \u2211 x in divisorsAntidiagonal x, \u2191\u2191\u03b6 x.1 \u2022 \u2191f x.2 = \u2211 i in divisorsAntidiagonal x, \u2191f i.2 ** refine' sum_congr rfl fun i hi => _ ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : Semiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M f : ArithmeticFunction M x : \u2115 i : \u2115 \u00d7 \u2115 hi : i \u2208 divisorsAntidiagonal x \u22a2 \u2191\u2191\u03b6 i.1 \u2022 \u2191f i.2 = \u2191f i.2 ** rcases mem_divisorsAntidiagonal.1 hi with \u27e8rfl, h\u27e9 ** case intro R : Type u_1 M : Type u_2 inst\u271d\u00b2 : Semiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M f : ArithmeticFunction M i : \u2115 \u00d7 \u2115 hi : i \u2208 divisorsAntidiagonal (i.1 * i.2) h : i.1 * i.2 \u2260 0 \u22a2 \u2191\u2191\u03b6 i.1 \u2022 \u2191f i.2 = \u2191f i.2 ** rw [natCoe_apply, zeta_apply_ne (left_ne_zero_of_mul h), cast_one, one_smul] ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : Semiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M f : ArithmeticFunction M x : \u2115 \u22a2 \u2211 i in divisorsAntidiagonal x, \u2191f i.2 = \u2211 i in divisors x, \u2191f i ** rw [\u2190 map_div_left_divisors, sum_map, Function.Embedding.coeFn_mk] ** Qed", + "informal": "" + }, + { + "formal": "IsAdjoinRootMonic.modByMonic_repr_map ** R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f g : R[X] \u22a2 f \u2223 IsAdjoinRoot.repr h.toIsAdjoinRoot (\u2191h.map g) - g ** rw [\u2190 h.mem_ker_map, RingHom.sub_mem_ker_iff, map_repr] ** Qed", + "informal": "" + }, + { + "formal": "Seminorm.ball_zero_eq_preimage_ball ** R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u2076 : SeminormedRing \ud835\udd5c inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module \ud835\udd5c E inst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b2 : AddCommGroup E\u2082 inst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d : RingHomIsometric \u03c3\u2081\u2082 p : Seminorm \ud835\udd5c E r : \u211d \u22a2 ball p 0 r = \u2191p \u207b\u00b9' Metric.ball 0 r ** rw [ball_zero_eq, preimage_metric_ball] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicClosure.AdjoinMonic.exists_root ** k : Type u inst\u271d : Field k f : k[X] hfm : Monic f hfi : Irreducible f \u22a2 Polynomial.eval\u2082 (toAdjoinMonic k) (\u2191(Ideal.Quotient.mk (maxIdeal k)) (MvPolynomial.X { val := f, property := (_ : Monic f \u2227 Irreducible f) })) f = 0 ** erw [toAdjoinMonic, \u2190 hom_eval\u2082, Ideal.Quotient.eq_zero_iff_mem] ** k : Type u inst\u271d : Field k f : k[X] hfm : Monic f hfi : Irreducible f \u22a2 Polynomial.eval\u2082 MvPolynomial.C (MvPolynomial.X { val := f, property := (_ : Monic f \u2227 Irreducible f) }) f \u2208 maxIdeal k ** exact le_maxIdeal k (Ideal.subset_span <| \u27e8_, rfl\u27e9) ** Qed", + "informal": "" + }, + { + "formal": "Bool.eq_true_of_not_eq_false' ** a : Bool \u22a2 (!decide (a = false)) = true \u2192 a = true ** cases a <;> decide ** Qed", + "informal": "" + }, + { + "formal": "inner_eq_sum_norm_sq_div_four ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E \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 ** rw [\u2190 re_add_im \u27eax, y\u27eb, re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four,\n im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E \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 = (\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 ** push_cast ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E \u22a2 (\u2191\u2016x + y\u2016 * \u2191\u2016x + y\u2016 - \u2191\u2016x - y\u2016 * \u2191\u2016x - y\u2016) / 4 + (\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 = (\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 ** simp only [sq, \u2190 mul_div_right_comm, \u2190 add_div] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.hasZeroObject_of_hasInitial_object ** C : Type u inst\u271d\u00b3 : Category.{v, u} C D : Type u' inst\u271d\u00b2 : Category.{v', u'} D inst\u271d\u00b9 : HasZeroMorphisms C inst\u271d : HasInitial C \u22a2 HasZeroObject C ** refine' \u27e8\u27e8\u22a5_ C, fun X => \u27e8\u27e8\u27e80\u27e9, by aesop_cat\u27e9\u27e9, fun X => \u27e8\u27e8\u27e80\u27e9, fun f => _\u27e9\u27e9\u27e9\u27e9 ** C : Type u inst\u271d\u00b3 : Category.{v, u} C D : Type u' inst\u271d\u00b2 : Category.{v', u'} D inst\u271d\u00b9 : HasZeroMorphisms C inst\u271d : HasInitial C X : C f : X \u27f6 \u22a5_ C \u22a2 f = default ** calc\n f = f \u226b \ud835\udfd9 _ := (Category.comp_id _).symm\n _ = f \u226b 0 := by congr!\n _ = 0 := HasZeroMorphisms.comp_zero _ _ ** C : Type u inst\u271d\u00b3 : Category.{v, u} C D : Type u' inst\u271d\u00b2 : Category.{v', u'} D inst\u271d\u00b9 : HasZeroMorphisms C inst\u271d : HasInitial C X : C \u22a2 \u2200 (a : \u22a5_ C \u27f6 X), a = default ** aesop_cat ** C : Type u inst\u271d\u00b3 : Category.{v, u} C D : Type u' inst\u271d\u00b2 : Category.{v', u'} D inst\u271d\u00b9 : HasZeroMorphisms C inst\u271d : HasInitial C X : C f : X \u27f6 \u22a5_ C \u22a2 f \u226b \ud835\udfd9 (\u22a5_ C) = f \u226b 0 ** congr! ** Qed", + "informal": "" + }, + { + "formal": "IntermediateField.adjoin_simple_le_iff ** F : Type u_1 inst\u271d\u00b2 : Field F E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E S : Set E \u03b1 : E K : IntermediateField F E \u22a2 F\u27ee\u03b1\u27ef \u2264 K \u2194 \u03b1 \u2208 K ** simp ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_C_mul_X ** R : Type u S : Type v a b : R n\u271d m : \u2115 inst\u271d : Semiring R p q r : R[X] x : R n : \u2115 \u22a2 coeff (\u2191C x * X) n = if n = 1 then x else 0 ** rw [\u2190 pow_one X, coeff_C_mul_X_pow] ** Qed", + "informal": "" + }, + { + "formal": "LinearEquiv.det_symm_mul_det ** R : Type u_1 inst\u271d\u2078 : CommRing R M : Type u_2 inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M M' : Type u_3 inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : Module R M' \u03b9 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b9 inst\u271d\u00b2 : Fintype \u03b9 e : Basis \u03b9 R M A : Type u_5 inst\u271d\u00b9 : CommRing A inst\u271d : Module A M f : M \u2243\u2097[A] M \u22a2 \u2191LinearMap.det \u2191(symm f) * \u2191LinearMap.det \u2191f = 1 ** simp [\u2190 LinearMap.det_comp] ** Qed", + "informal": "" + }, + { + "formal": "SchwartzMap.isBigO_cocompact_zpow_neg_nat ** \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 D : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F f : \ud835\udce2(E, F) k : \u2115 \u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k) ** obtain \u27e8d, _, hd'\u27e9 := f.decay k 0 ** case intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 D : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F f : \ud835\udce2(E, F) k : \u2115 d : \u211d left\u271d : 0 < d hd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d 0 (\u2191f) x\u2016 \u2264 d \u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k) ** simp only [norm_iteratedFDeriv_zero] at hd' ** case intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 D : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F f : \ud835\udce2(E, F) k : \u2115 d : \u211d left\u271d : 0 < d hd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d \u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k) ** simp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith] ** case intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 D : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F f : \ud835\udce2(E, F) k : \u2115 d : \u211d left\u271d : 0 < d hd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d \u22a2 \u2203 c, \u2200\u1da0 (x : E) in cocompact E, \u2016\u2191f x\u2016 \u2264 c * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016 ** refine' \u27e8d, Filter.Eventually.filter_mono Filter.cocompact_le_cofinite _\u27e9 ** case intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 D : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F f : \ud835\udce2(E, F) k : \u2115 d : \u211d left\u271d : 0 < d hd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d \u22a2 \u2200\u1da0 (x : E) in cofinite, \u2016\u2191f x\u2016 \u2264 d * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016 ** refine' (Filter.eventually_cofinite_ne 0).mono fun x hx => _ ** case intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 D : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F f : \ud835\udce2(E, F) k : \u2115 d : \u211d left\u271d : 0 < d hd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d x : E hx : x \u2260 0 \u22a2 \u2016\u2191f x\u2016 \u2264 d * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016 ** rw [Real.norm_of_nonneg (zpow_nonneg (norm_nonneg _) _), zpow_neg, \u2190 div_eq_mul_inv, le_div_iff'] ** case intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 D : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F f : \ud835\udce2(E, F) k : \u2115 d : \u211d left\u271d : 0 < d hd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d x : E hx : x \u2260 0 \u22a2 \u2016x\u2016 ^ \u2191k * \u2016\u2191f x\u2016 \u2264 d case intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 D : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F f : \ud835\udce2(E, F) k : \u2115 d : \u211d left\u271d : 0 < d hd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d x : E hx : x \u2260 0 \u22a2 0 < \u2016x\u2016 ^ \u2191k ** exacts [hd' x, zpow_pos_of_pos (norm_pos_iff.mpr hx) _] ** Qed", + "informal": "" + }, + { + "formal": "multiplicity.is_greatest ** \u03b1 : Type u_1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : \u03b1 m : \u2115 hm : multiplicity a b < \u2191m h : a ^ m \u2223 b \u22a2 False ** rw [PartENat.lt_coe_iff] at hm ** \u03b1 : Type u_1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : \u03b1 m : \u2115 hm : \u2203 h, Part.get (multiplicity a b) h < m h : a ^ m \u2223 b \u22a2 False ** exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) ** Qed", + "informal": "" + }, + { + "formal": "IsSMulRegular.of_smul_eq_one ** R : Type u_1 S : Type u_2 M : Type u_3 a b : R s : S inst\u271d\u2074 : Monoid S inst\u271d\u00b3 : SMul R M inst\u271d\u00b2 : SMul R S inst\u271d\u00b9 : MulAction S M inst\u271d : IsScalarTower R S M h : a \u2022 s = 1 \u22a2 IsSMulRegular M (a \u2022 s) ** rw [h] ** R : Type u_1 S : Type u_2 M : Type u_3 a b : R s : S inst\u271d\u2074 : Monoid S inst\u271d\u00b3 : SMul R M inst\u271d\u00b2 : SMul R S inst\u271d\u00b9 : MulAction S M inst\u271d : IsScalarTower R S M h : a \u2022 s = 1 \u22a2 IsSMulRegular M 1 ** exact one M ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.AEEqFun.coeFn_comp\u2082Measurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9\u2074 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : PseudoMetrizableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b3 inst\u271d\u2076 : PseudoMetrizableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b4 inst\u271d\u00b9 : OpensMeasurableSpace \u03b4 inst\u271d : SecondCountableTopology \u03b4 g : \u03b2 \u2192 \u03b3 \u2192 \u03b4 hg : Measurable (uncurry g) f\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2 f\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3 \u22a2 \u2191(comp\u2082Measurable g hg f\u2081 f\u2082) =\u1d50[\u03bc] fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a) ** rw [comp\u2082Measurable_eq_mk] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9\u2074 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : PseudoMetrizableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b3 inst\u271d\u2076 : PseudoMetrizableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b4 inst\u271d\u00b9 : OpensMeasurableSpace \u03b4 inst\u271d : SecondCountableTopology \u03b4 g : \u03b2 \u2192 \u03b3 \u2192 \u03b4 hg : Measurable (uncurry g) f\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2 f\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3 \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 => g (\u2191f\u2081 a) (\u2191f\u2082 a) ** apply coeFn_mk ** Qed", + "informal": "" + }, + { + "formal": "Holor.cprankMax_nil ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : AddMonoid \u03b1 x : Holor \u03b1 [] \u22a2 CPRankMax 1 x ** have h := CPRankMax.succ 0 x 0 (CPRankMax1.nil x) CPRankMax.zero ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : AddMonoid \u03b1 x : Holor \u03b1 [] h : CPRankMax (0 + 1) (x + 0) \u22a2 CPRankMax 1 x ** rwa [add_zero x, zero_add] at h ** Qed", + "informal": "" + }, + { + "formal": "eq_of_norm_le_re_inner_eq_norm_sq ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E hle : \u2016x\u2016 \u2264 \u2016y\u2016 h : \u2191re (inner x y) = \u2016y\u2016 ^ 2 \u22a2 x = y ** suffices H : re \u27eax - y, x - y\u27eb \u2264 0 ** case H \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E hle : \u2016x\u2016 \u2264 \u2016y\u2016 h : \u2191re (inner x y) = \u2016y\u2016 ^ 2 \u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0 ** have H\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2 := by gcongr ** case H \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E hle : \u2016x\u2016 \u2264 \u2016y\u2016 h : \u2191re (inner x y) = \u2016y\u2016 ^ 2 H\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2 \u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0 ** have H\u2082 : re \u27eay, x\u27eb = \u2016y\u2016 ^ 2 := by rwa [\u2190 inner_conj_symm, conj_re] ** case H \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E hle : \u2016x\u2016 \u2264 \u2016y\u2016 h : \u2191re (inner x y) = \u2016y\u2016 ^ 2 H\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2 H\u2082 : \u2191re (inner y x) = \u2016y\u2016 ^ 2 \u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0 ** simpa [inner_sub_left, inner_sub_right, \u2190 norm_sq_eq_inner, h, H\u2082] using H\u2081 ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E hle : \u2016x\u2016 \u2264 \u2016y\u2016 h : \u2191re (inner x y) = \u2016y\u2016 ^ 2 H : \u2191re (inner (x - y) (x - y)) \u2264 0 \u22a2 x = y ** rwa [inner_self_nonpos, sub_eq_zero] at H ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E hle : \u2016x\u2016 \u2264 \u2016y\u2016 h : \u2191re (inner x y) = \u2016y\u2016 ^ 2 \u22a2 \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2 ** gcongr ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E hle : \u2016x\u2016 \u2264 \u2016y\u2016 h : \u2191re (inner x y) = \u2016y\u2016 ^ 2 H\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2 \u22a2 \u2191re (inner y x) = \u2016y\u2016 ^ 2 ** rwa [\u2190 inner_conj_symm, conj_re] ** Qed", + "informal": "" + }, + { + "formal": "SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source ** \u03b9 : Type u\u03b9 E : Type uE inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : FiniteDimensional \u211d E F : Type uF inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F H : Type uH inst\u271d\u2075 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u2074 : TopologicalSpace M inst\u271d\u00b3 : ChartedSpace H M inst\u271d\u00b2 : SmoothManifoldWithCorners I M inst\u271d\u00b9 : T2Space M inst\u271d : SigmaCompactSpace M \u22a2 \u2203 f, IsSubordinate f fun x => (chartAt H x).toLocalEquiv.source ** apply exists_isSubordinate _ isClosed_univ _ (fun i \u21a6 (chartAt H _).open_source) (fun x _ \u21a6 ?_) ** \u03b9 : Type u\u03b9 E : Type uE inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : FiniteDimensional \u211d E F : Type uF inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F H : Type uH inst\u271d\u2075 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u2074 : TopologicalSpace M inst\u271d\u00b3 : ChartedSpace H M inst\u271d\u00b2 : SmoothManifoldWithCorners I M inst\u271d\u00b9 : T2Space M inst\u271d : SigmaCompactSpace M x : M x\u271d : x \u2208 univ \u22a2 x \u2208 \u22c3 i, (chartAt H i).toLocalEquiv.source ** exact mem_iUnion_of_mem x (mem_chart_source H x) ** Qed", + "informal": "" + }, + { + "formal": "Submodule.toLinearPMap_graph_eq ** R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 \u22a2 LinearPMap.graph (toLinearPMap g) = g ** ext x ** case h R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x : E \u00d7 F \u22a2 x \u2208 LinearPMap.graph (toLinearPMap g) \u2194 x \u2208 g ** constructor <;> intro hx ** case h.mpr R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x : E \u00d7 F hx : x \u2208 g \u22a2 x \u2208 LinearPMap.graph (toLinearPMap g) ** rw [LinearPMap.mem_graph_iff] ** case h.mpr R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x : E \u00d7 F hx : x \u2208 g \u22a2 \u2203 y, \u2191y = x.1 \u2227 \u2191(toLinearPMap g) y = x.2 ** cases' x with x_fst x_snd ** case h.mpr.mk R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x_fst : E x_snd : F hx : (x_fst, x_snd) \u2208 g \u22a2 \u2203 y, \u2191y = (x_fst, x_snd).1 \u2227 \u2191(toLinearPMap g) y = (x_fst, x_snd).2 ** 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 ** case h.mpr.mk R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x_fst : E x_snd : F hx : (x_fst, x_snd) \u2208 g hx_fst : x_fst \u2208 map (LinearMap.fst R E F) g \u22a2 \u2203 y, \u2191y = (x_fst, x_snd).1 \u2227 \u2191(toLinearPMap g) y = (x_fst, x_snd).2 ** refine' \u27e8\u27e8x_fst, hx_fst\u27e9, Subtype.coe_mk x_fst hx_fst, _\u27e9 ** case h.mpr.mk R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x_fst : E x_snd : F hx : (x_fst, x_snd) \u2208 g hx_fst : x_fst \u2208 map (LinearMap.fst R E F) g \u22a2 \u2191(toLinearPMap g) { val := x_fst, property := hx_fst } = (x_fst, x_snd).2 ** rw [toLinearPMap_apply_aux hg] ** case h.mpr.mk R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x_fst : E x_snd : F hx : (x_fst, x_snd) \u2208 g hx_fst : x_fst \u2208 map (LinearMap.fst R E F) g \u22a2 valFromGraph hg (_ : \u2191{ val := x_fst, property := hx_fst } \u2208 map (LinearMap.fst R E F) g) = (x_fst, x_snd).2 ** exact (existsUnique_from_graph @hg hx_fst).unique (valFromGraph_mem hg hx_fst) hx ** case h.mp R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x : E \u00d7 F hx : x \u2208 LinearPMap.graph (toLinearPMap g) \u22a2 x \u2208 g ** rw [LinearPMap.mem_graph_iff] at hx ** case h.mp R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x : E \u00d7 F hx : \u2203 y, \u2191y = x.1 \u2227 \u2191(toLinearPMap g) y = x.2 \u22a2 x \u2208 g ** rcases hx with \u27e8y, hx1, hx2\u27e9 ** case h.mp.intro.intro R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x : E \u00d7 F y : { x // x \u2208 (toLinearPMap g).domain } hx1 : \u2191y = x.1 hx2 : \u2191(toLinearPMap g) y = x.2 \u22a2 x \u2208 g ** convert g.mem_graph_toLinearPMap hg y using 1 ** case h.e'_4 R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x : E \u00d7 F y : { x // x \u2208 (toLinearPMap g).domain } hx1 : \u2191y = x.1 hx2 : \u2191(toLinearPMap g) y = x.2 \u22a2 x = (\u2191y, \u2191(toLinearPMap g) y) ** exact Prod.ext hx1.symm hx2.symm ** R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x_fst : E x_snd : F hx : (x_fst, x_snd) \u2208 g \u22a2 x_fst \u2208 map (LinearMap.fst R E F) g ** simp only [mem_map, LinearMap.fst_apply, Prod.exists, exists_and_right, exists_eq_right] ** R : Type u_1 inst\u271d\u2076 : Ring R E : Type u_2 inst\u271d\u2075 : AddCommGroup E inst\u271d\u2074 : Module R E F : Type u_3 inst\u271d\u00b3 : AddCommGroup F inst\u271d\u00b2 : Module R F G : Type u_4 inst\u271d\u00b9 : AddCommGroup G inst\u271d : Module R G g : Submodule R (E \u00d7 F) hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0 x_fst : E x_snd : F hx : (x_fst, x_snd) \u2208 g \u22a2 \u2203 x, (x_fst, x) \u2208 g ** exact \u27e8x_snd, hx\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "derivWithin_Ioi_eq_Ici ** \ud835\udd5c : Type u inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F E\u271d : Type w inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d f\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x\u271d : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E x : \u211d \u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x ** by_cases H : DifferentiableWithinAt \u211d f (Ioi x) x ** case pos \ud835\udd5c : Type u inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F E\u271d : Type w inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d f\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x\u271d : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E x : \u211d H : DifferentiableWithinAt \u211d f (Ioi x) x \u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x ** have A := H.hasDerivWithinAt.Ici_of_Ioi ** case pos \ud835\udd5c : Type u inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F E\u271d : Type w inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d f\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x\u271d : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E x : \u211d H : DifferentiableWithinAt \u211d f (Ioi x) x A : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x \u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x ** have B := (differentiableWithinAt_Ioi_iff_Ici.1 H).hasDerivWithinAt ** case pos \ud835\udd5c : Type u inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F E\u271d : Type w inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d f\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x\u271d : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E x : \u211d H : DifferentiableWithinAt \u211d f (Ioi x) x A : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x B : HasDerivWithinAt f (derivWithin f (Ici x) x) (Ici x) x \u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x ** simpa using (uniqueDiffOn_Ici x).eq left_mem_Ici A B ** case neg \ud835\udd5c : Type u inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F E\u271d : Type w inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d f\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x\u271d : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E x : \u211d H : \u00acDifferentiableWithinAt \u211d f (Ioi x) x \u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x ** rw [derivWithin_zero_of_not_differentiableWithinAt H,\n derivWithin_zero_of_not_differentiableWithinAt] ** case neg \ud835\udd5c : Type u inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F E\u271d : Type w inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d f\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x\u271d : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E x : \u211d H : \u00acDifferentiableWithinAt \u211d f (Ioi x) x \u22a2 \u00acDifferentiableWithinAt \u211d f (Ici x) x ** rwa [differentiableWithinAt_Ioi_iff_Ici] at H ** Qed", + "informal": "" + }, + { + "formal": "bernoulli'_eq_bernoulli ** A : Type u_1 inst\u271d\u00b9 : CommRing A inst\u271d : Algebra \u211a A n : \u2115 \u22a2 bernoulli' n = (-1) ^ n * bernoulli n ** simp [bernoulli, \u2190 mul_assoc, \u2190 sq, \u2190 pow_mul, mul_comm n 2, pow_mul] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Mem\u2112p.exists_boundedContinuous_snorm_sub_le ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u2191g) p ** 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) ** case H \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g) ** apply hf.induction_dense hp _ _ _ _ h\u03b5 ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \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) \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2200 (f g : \u03b1 \u2192 E), Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192 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)) \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2200 (f : \u03b1 \u2192 E), Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192 AEStronglyMeasurable f \u03bc ** rotate_left ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \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) ** intro c t ht ht\u03bc \u03b5 h\u03b5 ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \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) ** rcases exists_Lp_half E \u03bc p h\u03b5 with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \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) ** 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 ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \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 case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 \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) ** exact exists_snorm_indicator_le hp c \u03b4pos.ne' ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 \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) ** have h\u03b7_pos' : (0 : \u211d\u22650\u221e) < \u03b7 := ENNReal.coe_pos.2 \u03b7pos ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 \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) ** obtain \u27e8s, st, s_closed, \u03bcs\u27e9 : \u2203 s, s \u2286 t \u2227 IsClosed s \u2227 \u03bc (t \\ s) < \u03b7 ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 \u22a2 \u2203 s, s \u2286 t \u2227 IsClosed s \u2227 \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 \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) ** exact ht.exists_isClosed_diff_lt ht\u03bc.ne h\u03b7_pos'.ne' ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 \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) ** have hs\u03bc : \u03bc s < \u221e := (measure_mono st).trans_lt ht\u03bc ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 \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) ** 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 ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 I1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4 \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) ** 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 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 I1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4 f : \u03b1 \u2192 E f_cont : Continuous f I2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4 f_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016 f_mem : Mem\u2112p f p \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) ** 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] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 I1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4 f : \u03b1 \u2192 E f_cont : Continuous f I2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4 f_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016 f_mem : Mem\u2112p f p I3 : snorm (f - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b5 \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) ** refine' \u27e8f, I3, f_cont, f_mem, _\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 I1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4 f : \u03b1 \u2192 E f_cont : Continuous f I2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4 f_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016 f_mem : Mem\u2112p f p I3 : snorm (f - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b5 \u22a2 IsBounded (range f) ** exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).isBounded_range ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 H : \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g) \u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u2191g) p ** rcases H with \u27e8g, hg, g_cont, g_mem, g_bd\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 g : \u03b1 \u2192 E hg : snorm (f - g) p \u03bc \u2264 \u03b5 g_cont : Continuous g g_mem : Mem\u2112p g p g_bd : IsBounded (range g) \u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u2191g) p ** exact \u27e8\u27e8\u27e8g, g_cont\u27e9, Metric.isBounded_range_iff.1 g_bd\u27e9, hg, g_mem\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2200 (f g : \u03b1 \u2192 E), Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192 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)) ** rintro f g \u27e8f_cont, f_mem, f_bd\u27e9 \u27e8g_cont, g_mem, g_bd\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f g : \u03b1 \u2192 E f_cont : Continuous f f_mem : Mem\u2112p f p f_bd : IsBounded (range f) g_cont : Continuous g g_mem : Mem\u2112p g p g_bd : IsBounded (range g) \u22a2 Continuous (f + g) \u2227 Mem\u2112p (f + g) p \u2227 IsBounded (range (f + g)) ** refine' \u27e8f_cont.add g_cont, f_mem.add g_mem, _\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f g : \u03b1 \u2192 E f_cont : Continuous f f_mem : Mem\u2112p f p f_bd : IsBounded (range f) g_cont : Continuous g g_mem : Mem\u2112p g p g_bd : IsBounded (range g) \u22a2 IsBounded (range (f + g)) ** let f' : \u03b1 \u2192\u1d47 E := \u27e8\u27e8f, f_cont\u27e9, Metric.isBounded_range_iff.1 f_bd\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f g : \u03b1 \u2192 E f_cont : Continuous f f_mem : Mem\u2112p f p f_bd : IsBounded (range f) g_cont : Continuous g g_mem : Mem\u2112p g p g_bd : IsBounded (range g) f' : \u03b1 \u2192\u1d47 E := { toContinuousMap := ContinuousMap.mk f, map_bounded' := (_ : \u2203 C, \u2200 (x y : \u03b1), dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) \u2264 C) } \u22a2 IsBounded (range (f + g)) ** let g' : \u03b1 \u2192\u1d47 E := \u27e8\u27e8g, g_cont\u27e9, Metric.isBounded_range_iff.1 g_bd\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f g : \u03b1 \u2192 E f_cont : Continuous f f_mem : Mem\u2112p f p f_bd : IsBounded (range f) g_cont : Continuous g g_mem : Mem\u2112p g p g_bd : IsBounded (range g) f' : \u03b1 \u2192\u1d47 E := { toContinuousMap := ContinuousMap.mk f, map_bounded' := (_ : \u2203 C, \u2200 (x y : \u03b1), dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) \u2264 C) } g' : \u03b1 \u2192\u1d47 E := { toContinuousMap := ContinuousMap.mk g, map_bounded' := (_ : \u2203 C, \u2200 (x y : \u03b1), dist (ContinuousMap.toFun (ContinuousMap.mk g) x) (ContinuousMap.toFun (ContinuousMap.mk g) y) \u2264 C) } \u22a2 IsBounded (range (f + g)) ** exact (f' + g').isBounded_range ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2200 (f : \u03b1 \u2192 E), Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192 AEStronglyMeasurable f \u03bc ** exact fun f \u27e8_, h, _\u27e9 => h.aestronglyMeasurable ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 \u22a2 snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4 ** rw [\u2190 snorm_neg, neg_sub, \u2190 indicator_diff st] ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 \u22a2 snorm (Set.indicator (t \\ s) fun _y => c) p \u03bc \u2264 \u03b4 ** exact h\u03b7 _ \u03bcs.le ** \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 I1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4 f : \u03b1 \u2192 E f_cont : Continuous f I2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4 f_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016 f_mem : Mem\u2112p f p \u22a2 snorm (f - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b5 ** 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 ** case h.e'_3.h.e'_5 \u03b1 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : T4Space \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : Measure.WeaklyRegular \u03bc hp : p \u2260 \u22a4 f\u271d : \u03b1 \u2192 E hf : Mem\u2112p f\u271d p \u03b5\u271d : \u211d\u22650\u221e h\u03b5\u271d : \u03b5\u271d \u2260 0 c : E t : Set \u03b1 ht : MeasurableSet t ht\u03bc : \u2191\u2191\u03bc t < \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b4 : \u211d\u22650\u221e \u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (f g : \u03b1 \u2192 E), 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 \u03b7 : \u211d\u22650 \u03b7pos : 0 < \u03b7 h\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4 h\u03b7_pos' : 0 < \u2191\u03b7 s : Set \u03b1 st : s \u2286 t s_closed : IsClosed s \u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7 hs\u03bc : \u2191\u2191\u03bc s < \u22a4 I1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4 f : \u03b1 \u2192 E f_cont : Continuous f I2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4 f_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016 f_mem : Mem\u2112p f p \u22a2 (f - Set.indicator t fun _y => c) = (f - Set.indicator s fun x => c) + ((Set.indicator s fun x => c) - Set.indicator t fun x => c) ** simp only [sub_add_sub_cancel] ** Qed", + "informal": "" + }, + { + "formal": "Nat.coprime_self_sub_right ** m n : \u2115 h : m \u2264 n \u22a2 Coprime n (n - m) \u2194 Coprime n m ** rw [Coprime, Coprime, gcd_self_sub_right h] ** Qed", + "informal": "" + }, + { + "formal": "SmoothBumpFunction.support_eq_symm_image ** E : Type uE inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : FiniteDimensional \u211d E H : Type uH inst\u271d\u00b3 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : ChartedSpace H M inst\u271d : SmoothManifoldWithCorners I M c : M f : SmoothBumpFunction I c x : M \u22a2 support \u2191f = \u2191(LocalEquiv.symm (extChartAt I c)) '' (ball (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Set.toFinset (rootSet p \u211d)) + Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ** by_cases hp : p = 0 ** case neg F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Set.toFinset (rootSet p \u211d)) + Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ** have inj : Function.Injective (IsScalarTower.toAlgHom \u211a \u211d \u2102) := (algebraMap \u211d \u2102).injective ** case neg F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Set.toFinset (rootSet p \u211d)) + Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ** rw [\u2190 Finset.card_image_of_injective _ Subtype.coe_injective, \u2190\n Finset.card_image_of_injective _ inj] ** case neg F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) + Finset.card (Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))) ** let a : Finset \u2102 := ?_ ** case neg.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := ?neg.refine_1\u271d \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) + Finset.card (Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))) case neg.refine_1 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) \u22a2 Finset \u2102 ** let b : Finset \u2102 := ?_ ** case neg.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := ?neg.refine_1\u271d b : Finset \u2102 := ?neg.refine_2.refine_1\u271d \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) + Finset.card (Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))) case neg.refine_2.refine_1 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := ?neg.refine_1\u271d \u22a2 Finset \u2102 case neg.refine_1 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) \u22a2 Finset \u2102 ** let c : Finset \u2102 := ?_ ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := ?neg.refine_1\u271d b : Finset \u2102 := ?neg.refine_2.refine_1\u271d c : Finset \u2102 := ?neg.refine_2.refine_2.refine_1\u271d \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) + Finset.card (Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))) case neg.refine_2.refine_2.refine_1 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := ?neg.refine_1\u271d b : Finset \u2102 := ?neg.refine_2.refine_1\u271d \u22a2 Finset \u2102 case neg.refine_2.refine_1 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := ?neg.refine_1\u271d \u22a2 Finset \u2102 case neg.refine_1 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) \u22a2 Finset \u2102 ** suffices a.card = b.card + c.card by exact this ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) \u22a2 Finset.card a = Finset.card b + Finset.card c ** 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] ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u22a2 Finset.card a = Finset.card b + Finset.card c ** 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 ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 hc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u22a2 Finset.card a = Finset.card b + Finset.card c ** rw [\u2190 Finset.card_disjoint_union] ** case pos F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : p = 0 \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Set.toFinset (rootSet p \u211d)) + Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ** haveI : IsEmpty (p.rootSet \u2102) := by rw [hp, rootSet_zero]; infer_instance ** case pos F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : p = 0 this : IsEmpty \u2191(rootSet p \u2102) \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Set.toFinset (rootSet p \u211d)) + Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ** simp_rw [(galActionHom p \u2102 _).support.eq_empty_of_isEmpty, hp, rootSet_zero,\n Set.toFinset_empty, Finset.card_empty] ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : p = 0 \u22a2 IsEmpty \u2191(rootSet p \u2102) ** rw [hp, rootSet_zero] ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : p = 0 \u22a2 IsEmpty \u2191\u2205 ** infer_instance ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := ?neg.refine_1\u271d b : Finset \u2102 := ?neg.refine_2.refine_1\u271d c : Finset \u2102 := ?neg.refine_2.refine_2.refine_1\u271d this : Finset.card a = Finset.card b + Finset.card c \u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) = Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) + Finset.card (Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))) ** exact this ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) \u22a2 \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 ** intro z ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) z : \u2102 \u22a2 z \u2208 a \u2194 \u2191(aeval z) p = 0 ** rw [Set.mem_toFinset, mem_rootSet_of_ne hp] ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 \u22a2 \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 ** intro z ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 \u22a2 z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 ** simp_rw [Finset.mem_image, Set.mem_toFinset, mem_rootSet_of_ne hp] ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 \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 ** constructor ** case mp F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 \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 ** rintro \u27e8w, hw, rfl\u27e9 ** case mp.intro.intro F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 w : \u211d hw : \u2191(aeval w) p = 0 \u22a2 \u2191(aeval (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w)) p = 0 \u2227 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w).im = 0 ** exact \u27e8by rw [aeval_algHom_apply, hw, AlgHom.map_zero], rfl\u27e9 ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 w : \u211d hw : \u2191(aeval w) p = 0 \u22a2 \u2191(aeval (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w)) p = 0 ** rw [aeval_algHom_apply, hw, AlgHom.map_zero] ** case mpr F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 \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 ** rintro \u27e8hz1, hz2\u27e9 ** case mpr.intro F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im = 0 \u22a2 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z ** have key : IsScalarTower.toAlgHom \u211a \u211d \u2102 z.re = z := by ext; rfl; rw [hz2]; rfl ** case mpr.intro F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im = 0 key : \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z \u22a2 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z ** exact \u27e8z.re, inj (by rwa [\u2190 aeval_algHom_apply, key, AlgHom.map_zero]), key\u27e9 ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im = 0 \u22a2 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z ** ext ** case a F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im = 0 \u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).re = z.re case a F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im = 0 \u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = z.im ** rfl ** case a F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im = 0 \u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = z.im ** rw [hz2] ** case a F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im = 0 \u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = 0 ** rfl ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im = 0 key : \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z \u22a2 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) (\u2191(aeval z.re) p) = \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) 0 ** rwa [\u2190 aeval_algHom_apply, key, AlgHom.map_zero] ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u22a2 \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 ** intro w ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 w : \u2191(rootSet p \u2102) \u22a2 \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 ** rw [Subtype.ext_iff, galActionHom_restrict] ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 w : \u2191(rootSet p \u2102) \u22a2 \u2191(AlgEquiv.restrictScalars \u211a Complex.conjAe) \u2191w = \u2191w \u2194 (\u2191w).im = 0 ** exact Complex.conj_eq_iff_im ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 \u22a2 \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 ** intro z ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 z : \u2102 \u22a2 z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 ** simp_rw [Finset.mem_image] ** F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 z : \u2102 \u22a2 (\u2203 a, a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227 \u2191a = z) \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 ** constructor ** case mp F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 z : \u2102 \u22a2 (\u2203 a, a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227 \u2191a = z) \u2192 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 ** rintro \u27e8w, hw, rfl\u27e9 ** case mp.intro.intro F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 w : { x // x \u2208 rootSet p \u2102 } hw : w \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u22a2 \u2191(aeval \u2191w) p = 0 \u2227 (\u2191w).im \u2260 0 ** exact \u27e8(mem_rootSet.mp w.2).2, mt (hc0 w).mpr (Equiv.Perm.mem_support.mp hw)\u27e9 ** case mpr F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 z : \u2102 \u22a2 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u2192 \u2203 a, a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227 \u2191a = z ** rintro \u27e8hz1, hz2\u27e9 ** case mpr.intro F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 z : \u2102 hz1 : \u2191(aeval z) p = 0 hz2 : z.im \u2260 0 \u22a2 \u2203 a, a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227 \u2191a = z ** exact \u27e8\u27e8z, mem_rootSet.mpr \u27e8hp, hz1\u27e9\u27e9, Equiv.Perm.mem_support.mpr (mt (hc0 _).mp hz2), rfl\u27e9 ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 hc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u22a2 Finset.card a = Finset.card (b \u222a c) ** apply congr_arg Finset.card ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 hc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u22a2 a = b \u222a c ** simp_rw [Finset.ext_iff, Finset.mem_union, ha, hb, hc] ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 hc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \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 ** tauto ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 hc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u22a2 Disjoint b c ** rw [Finset.disjoint_left] ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 hc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u22a2 \u2200 \u2983a : \u2102\u2984, a \u2208 b \u2192 \u00aca \u2208 c ** intro z ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 hc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 z : \u2102 \u22a2 z \u2208 b \u2192 \u00acz \u2208 c ** rw [hb, hc] ** case neg.refine_2.refine_2.refine_2 F : Type u_1 inst\u271d\u00b2 : Field F p\u271d q : F[X] E : Type u_2 inst\u271d\u00b9 : Field E inst\u271d : Algebra F E p : \u211a[X] hp : \u00acp = 0 inj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a : Finset \u2102 := Set.toFinset (rootSet p \u2102) b : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d)) c : Finset \u2102 := Finset.image (fun a => \u2191a) (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) ha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0 hb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0 hc0 : \u2200 (w : \u2191(rootSet p \u2102)), \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0 hc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 z : \u2102 \u22a2 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u2192 \u00ac(\u2191(aeval z) p = 0 \u2227 z.im \u2260 0) ** tauto ** Qed", + "informal": "" + }, + { + "formal": "Function.extend_inv ** I : Type u \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f\u271d : I \u2192 Type v\u2081 g\u271d : I \u2192 Type v\u2082 h : I \u2192 Type v\u2083 x y : (i : I) \u2192 f\u271d i i : I inst\u271d : Inv \u03b3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 e : \u03b2 \u2192 \u03b3 \u22a2 extend f g\u207b\u00b9 e\u207b\u00b9 = (extend f g e)\u207b\u00b9 ** classical\nfunext x\nsimp only [not_exists, extend_def, Pi.inv_apply, apply_dite Inv.inv] ** I : Type u \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f\u271d : I \u2192 Type v\u2081 g\u271d : I \u2192 Type v\u2082 h : I \u2192 Type v\u2083 x y : (i : I) \u2192 f\u271d i i : I inst\u271d : Inv \u03b3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 e : \u03b2 \u2192 \u03b3 \u22a2 extend f g\u207b\u00b9 e\u207b\u00b9 = (extend f g e)\u207b\u00b9 ** funext x ** case h I : Type u \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f\u271d : I \u2192 Type v\u2081 g\u271d : I \u2192 Type v\u2082 h : I \u2192 Type v\u2083 x\u271d y : (i : I) \u2192 f\u271d i i : I inst\u271d : Inv \u03b3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 e : \u03b2 \u2192 \u03b3 x : \u03b2 \u22a2 extend f g\u207b\u00b9 e\u207b\u00b9 x = (extend f g e)\u207b\u00b9 x ** simp only [not_exists, extend_def, Pi.inv_apply, apply_dite Inv.inv] ** Qed", + "informal": "" + }, + { + "formal": "WType.depth_pos ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 inst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a) t : WType \u03b2 \u22a2 0 < depth t ** cases t ** case mk \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 inst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a) a\u271d : \u03b1 f\u271d : \u03b2 a\u271d \u2192 WType \u03b2 \u22a2 0 < depth (mk a\u271d f\u271d) ** apply Nat.succ_pos ** Qed", + "informal": "" + }, + { + "formal": "balancedHull.balanced ** \ud835\udd5c : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b2 : SeminormedRing \ud835\udd5c inst\u271d\u00b9 : AddCommGroup E inst\u271d : Module \ud835\udd5c E s\u271d s : Set E \u22a2 Balanced \ud835\udd5c (balancedHull \ud835\udd5c s) ** intro a ha ** \ud835\udd5c : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b2 : SeminormedRing \ud835\udd5c inst\u271d\u00b9 : AddCommGroup E inst\u271d : Module \ud835\udd5c E s\u271d s : Set E a : \ud835\udd5c ha : \u2016a\u2016 \u2264 1 \u22a2 a \u2022 balancedHull \ud835\udd5c s \u2286 balancedHull \ud835\udd5c s ** simp_rw [balancedHull, smul_set_iUnion\u2082, subset_def, mem_iUnion\u2082] ** \ud835\udd5c : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b2 : SeminormedRing \ud835\udd5c inst\u271d\u00b9 : AddCommGroup E inst\u271d : Module \ud835\udd5c E s\u271d s : Set E a : \ud835\udd5c ha : \u2016a\u2016 \u2264 1 \u22a2 \u2200 (x : E), (\u2203 i j, x \u2208 a \u2022 i \u2022 s) \u2192 \u2203 i j, x \u2208 i \u2022 s ** rintro x \u27e8r, hr, hx\u27e9 ** case intro.intro \ud835\udd5c : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b2 : SeminormedRing \ud835\udd5c inst\u271d\u00b9 : AddCommGroup E inst\u271d : Module \ud835\udd5c E s\u271d s : Set E a : \ud835\udd5c ha : \u2016a\u2016 \u2264 1 x : E r : \ud835\udd5c hr : \u2016r\u2016 \u2264 1 hx : x \u2208 a \u2022 r \u2022 s \u22a2 \u2203 i j, x \u2208 i \u2022 s ** rw [\u2190 smul_assoc] at hx ** case intro.intro \ud835\udd5c : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b2 : SeminormedRing \ud835\udd5c inst\u271d\u00b9 : AddCommGroup E inst\u271d : Module \ud835\udd5c E s\u271d s : Set E a : \ud835\udd5c ha : \u2016a\u2016 \u2264 1 x : E r : \ud835\udd5c hr : \u2016r\u2016 \u2264 1 hx\u271d : x \u2208 a \u2022 r \u2022 s hx : x \u2208 (a \u2022 r) \u2022 s \u22a2 \u2203 i j, x \u2208 i \u2022 s ** exact \u27e8a \u2022 r, (SeminormedRing.norm_mul _ _).trans (mul_le_one ha (norm_nonneg r) hr), hx\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearMap.op_norm_bound_of_ball_bound ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E r : \u211d r_pos : 0 < r c : \u211d f : E \u2192L[\ud835\udd5c] \ud835\udd5c h : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c \u22a2 \u2016f\u2016 \u2264 c / r ** apply ContinuousLinearMap.op_norm_le_bound ** case hM \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E r : \u211d r_pos : 0 < r c : \u211d f : E \u2192L[\ud835\udd5c] \ud835\udd5c h : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c \u22a2 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 c / r * \u2016x\u2016 ** apply LinearMap.bound_of_ball_bound' r_pos ** case hM.h \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E r : \u211d r_pos : 0 < r c : \u211d f : E \u2192L[\ud835\udd5c] \ud835\udd5c h : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c \u22a2 \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191\u2191f z\u2016 \u2264 c ** exact fun z hz => h z hz ** case hMp \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E r : \u211d r_pos : 0 < r c : \u211d f : E \u2192L[\ud835\udd5c] \ud835\udd5c h : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c \u22a2 0 \u2264 c / r ** apply div_nonneg _ r_pos.le ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E r : \u211d r_pos : 0 < r c : \u211d f : E \u2192L[\ud835\udd5c] \ud835\udd5c h : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c \u22a2 0 \u2264 c ** exact\n (norm_nonneg _).trans\n (h 0 (by simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le])) ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E r : \u211d r_pos : 0 < r c : \u211d f : E \u2192L[\ud835\udd5c] \ud835\udd5c h : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c \u22a2 0 \u2208 closedBall 0 r ** simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le] ** Qed", + "informal": "" + }, + { + "formal": "iInf_emptyset ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b2 \u2192 \u03b1 \u22a2 \u2a05 x \u2208 \u2205, f x = \u22a4 ** simp ** Qed", + "informal": "" + }, + { + "formal": "List.append_inj ** \u03b1 : Type u_1 a : \u03b1 s\u2081 : List \u03b1 b : \u03b1 s\u2082 t\u2081 t\u2082 : List \u03b1 h : a :: s\u2081 ++ t\u2081 = b :: s\u2082 ++ t\u2082 hl : length (a :: s\u2081) = length (b :: s\u2082) \u22a2 a :: s\u2081 = b :: s\u2082 \u2227 t\u2081 = t\u2082 ** simp [append_inj (cons.inj h).2 (Nat.succ.inj hl)] at h \u22a2 ** \u03b1 : Type u_1 a : \u03b1 s\u2081 : List \u03b1 b : \u03b1 s\u2082 t\u2081 t\u2082 : List \u03b1 hl : length (a :: s\u2081) = length (b :: s\u2082) h : a = b \u22a2 a = b ** exact h ** Qed", + "informal": "" + }, + { + "formal": "orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : InnerProductSpace \u211d F K : Submodule \ud835\udd5c E inst\u271d : HasOrthogonalProjection K v : E hv : v \u2208 K\u15ee \u22a2 \u2191(orthogonalProjection K) v = 0 ** ext ** case a \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : InnerProductSpace \u211d F K : Submodule \ud835\udd5c E inst\u271d : HasOrthogonalProjection K v : E hv : v \u2208 K\u15ee \u22a2 \u2191(\u2191(orthogonalProjection K) v) = \u21910 ** convert eq_orthogonalProjection_of_mem_orthogonal (K := K) _ _ <;> simp [hv] ** Qed", + "informal": "" + }, + { + "formal": "Finset.piecewise_univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : Fintype \u03b1 s t : Finset \u03b1 inst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 univ) \u03b4 : \u03b1 \u2192 Sort u_4 f g : (i : \u03b1) \u2192 \u03b4 i \u22a2 piecewise univ f g = f ** ext i ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : Fintype \u03b1 s t : Finset \u03b1 inst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 univ) \u03b4 : \u03b1 \u2192 Sort u_4 f g : (i : \u03b1) \u2192 \u03b4 i i : \u03b1 \u22a2 piecewise univ f g i = f i ** simp [piecewise] ** Qed", + "informal": "" + }, + { + "formal": "Filter.tendsto_const_mul_atBot_iff_pos ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 inst\u271d\u00b9 : LinearOrderedField \u03b1 l : Filter \u03b2 f : \u03b2 \u2192 \u03b1 r : \u03b1 inst\u271d : NeBot l h : Tendsto f l atBot \u22a2 Tendsto (fun x => r * f x) l atBot \u2194 0 < r ** simp [tendsto_const_mul_atBot_iff, h, h.not_tendsto disjoint_atBot_atTop] ** Qed", + "informal": "" + }, + { + "formal": "VitaliFamily.ae_eventually_measure_pos ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a ** set s := {x | \u00ac\u2200\u1da0 a in v.filterAt x, 0 < \u03bc a} with hs ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a ** simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a ** change \u03bc s = 0 ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} \u22a2 \u2191\u2191\u03bc s = 0 ** let f : \u03b1 \u2192 Set (Set \u03b1) := fun _ => {a | \u03bc a = 0} ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} \u22a2 \u2191\u2191\u03bc s = 0 ** 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 ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 \u2191\u2191\u03bc s = 0 ** refine' le_antisymm _ bot_le ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 \u2191\u2191\u03bc s \u2264 0 ** 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] ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} \u22a2 FineSubfamilyOn v f s ** intro x hx \u03b5 \u03b5pos ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} x : \u03b1 hx : x \u2208 s \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** rw [hs] at hx ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} x : \u03b1 hx : x \u2208 {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} x : \u03b1 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 hx : \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 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} x : \u03b1 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 hx : \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 a : Set \u03b1 a_sets : a \u2208 setsAt v x ax : a \u2286 closedBall x \u03b5 \u03bca : \u2191\u2191\u03bc a = 0 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9 ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = \u2211' (x : \u2191(FineSubfamilyOn.index h)), 0 ** congr ** case e_f \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 (fun x => \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x)) = fun x => 0 ** ext1 x ** case e_f.h \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s x : \u2191(FineSubfamilyOn.index h) \u22a2 \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = 0 ** exact h.covering_mem x.2 ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), 0 = 0 ** simp only [tsum_zero, add_zero] ** Qed", + "informal": "" + }, + { + "formal": "Nat.log_div_mul_self ** b n : \u2115 \u22a2 log b (n / b * b) = log b n ** cases' le_or_lt b 1 with hb hb ** case inr b n : \u2115 hb : 1 < b \u22a2 log b (n / b * b) = log b n ** cases' lt_or_le n b with h h ** case inr.inr b n : \u2115 hb : 1 < b h : b \u2264 n \u22a2 log b (n / b * b) = log b n ** 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)] ** case inl b n : \u2115 hb : b \u2264 1 \u22a2 log b (n / b * b) = log b n ** rw [log_of_left_le_one hb, log_of_left_le_one hb] ** case inr.inl b n : \u2115 hb : 1 < b h : n < b \u22a2 log b (n / b * b) = log b n ** rw [div_eq_of_lt h, zero_mul, log_zero_right, log_of_lt h] ** Qed", + "informal": "" + }, + { + "formal": "Commute.orderOf_mul_dvd_lcm ** G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Monoid G inst\u271d : AddMonoid A x y : G a b : A n m : \u2115 h : Commute x y \u22a2 orderOf (x * y) \u2223 lcm (orderOf x) (orderOf y) ** rw [orderOf, \u2190 comp_mul_left] ** G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Monoid G inst\u271d : AddMonoid A x y : G a b : A n m : \u2115 h : Commute x y \u22a2 minimalPeriod ((fun x_1 => x * x_1) \u2218 fun x => y * x) 1 \u2223 lcm (orderOf x) (orderOf y) ** exact Function.Commute.minimalPeriod_of_comp_dvd_lcm h.function_commute_mul_left ** Qed", + "informal": "" + }, + { + "formal": "PowerBasis.trace_gen_eq_sum_roots ** R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : CommRing S inst\u271d\u00b9\u2070 : CommRing T inst\u271d\u2079 : Algebra R S inst\u271d\u2078 : Algebra R T K : Type u_4 L : Type u_5 inst\u271d\u2077 : Field K inst\u271d\u2076 : Field L inst\u271d\u2075 : Algebra K L \u03b9 \u03ba : Type w inst\u271d\u2074 : Fintype \u03b9 F : Type u_6 inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Algebra K S inst\u271d\u00b9 : Algebra K F inst\u271d : Nontrivial S pb : PowerBasis K S hf : Splits (algebraMap K F) (minpoly K pb.gen) \u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K S) pb.gen) = Multiset.sum (aroots (minpoly K pb.gen) F) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Nat.ModEq.add ** m n a b c d : \u2115 h\u2081 : a \u2261 b [MOD n] h\u2082 : c \u2261 d [MOD n] \u22a2 a + c \u2261 b + d [MOD n] ** rw [modEq_iff_dvd, Int.ofNat_add, Int.ofNat_add, add_sub_add_comm] ** m n a b c d : \u2115 h\u2081 : a \u2261 b [MOD n] h\u2082 : c \u2261 d [MOD n] \u22a2 \u2191n \u2223 \u2191b - \u2191a + (\u2191d - \u2191c) ** exact dvd_add h\u2081.dvd h\u2082.dvd ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Subgraph.edgeSet_sSup ** \u03b9 : Sort u_1 V : Type u W : Type v G : SimpleGraph V G\u2081 G\u2082 : Subgraph G a b : V s : Set (Subgraph G) \u22a2 edgeSet (sSup s) = \u22c3 G' \u2208 s, edgeSet G' ** ext e ** case h \u03b9 : Sort u_1 V : Type u W : Type v G : SimpleGraph V G\u2081 G\u2082 : Subgraph G a b : V s : Set (Subgraph G) e : Sym2 V \u22a2 e \u2208 edgeSet (sSup s) \u2194 e \u2208 \u22c3 G' \u2208 s, edgeSet G' ** induction e using Sym2.ind ** case h.h \u03b9 : Sort u_1 V : Type u W : Type v G : SimpleGraph V G\u2081 G\u2082 : Subgraph G a b : V s : Set (Subgraph G) x\u271d y\u271d : V \u22a2 Quotient.mk (Sym2.Rel.setoid V) (x\u271d, y\u271d) \u2208 edgeSet (sSup s) \u2194 Quotient.mk (Sym2.Rel.setoid V) (x\u271d, y\u271d) \u2208 \u22c3 G' \u2208 s, edgeSet G' ** simp ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Abelian.IsEquivalence.exact_iff ** C : Type u\u2081 inst\u271d\u2074 : Category.{v\u2081, u\u2081} C inst\u271d\u00b3 : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z D : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} D inst\u271d\u00b9 : Abelian D F : C \u2964 D inst\u271d : IsEquivalence F \u22a2 Exact (F.map f) (F.map g) \u2194 Exact f g ** simp only [exact_iff, \u2190 F.map_eq_zero_iff, F.map_comp, Category.assoc, \u2190\n kernelComparison_comp_\u03b9 g F, \u2190 \u03c0_comp_cokernelComparison f F] ** C : Type u\u2081 inst\u271d\u2074 : Category.{v\u2081, u\u2081} C inst\u271d\u00b3 : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z D : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} D inst\u271d\u00b9 : Abelian D F : C \u2964 D inst\u271d : IsEquivalence F \u22a2 F.map f \u226b F.map g = 0 \u2227 kernel.\u03b9 (F.map g) \u226b cokernel.\u03c0 (F.map f) = 0 \u2194 F.map f \u226b F.map g = 0 \u2227 kernelComparison g F \u226b kernel.\u03b9 (F.map g) \u226b cokernel.\u03c0 (F.map f) \u226b cokernelComparison f F = 0 ** rw [IsIso.comp_left_eq_zero (kernelComparison g F), \u2190 Category.assoc,\n IsIso.comp_right_eq_zero _ (cokernelComparison f F)] ** Qed", + "informal": "" + }, + { + "formal": "padicValRat.zero ** p : \u2115 \u22a2 padicValRat p 0 = 0 ** simp [padicValRat] ** Qed", + "informal": "" + }, + { + "formal": "Real.log_le_sub_one_of_pos ** x\u271d y x : \u211d hx : 0 < x \u22a2 log x \u2264 x - 1 ** rw [le_sub_iff_add_le] ** x\u271d y x : \u211d hx : 0 < x \u22a2 log x + 1 \u2264 x ** convert add_one_le_exp (log x) ** case h.e'_4 x\u271d y x : \u211d hx : 0 < x \u22a2 x = rexp (log x) ** rw [exp_log hx] ** Qed", + "informal": "" + }, + { + "formal": "Encodable.decode\u2082_ne_none_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Encodable \u03b1 n : \u2115 \u22a2 decode\u2082 \u03b1 n \u2260 none \u2194 n \u2208 Set.range encode ** simp_rw [Set.range, Set.mem_setOf_eq, Ne.def, Option.eq_none_iff_forall_not_mem,\n Encodable.mem_decode\u2082, not_forall, not_not] ** Qed", + "informal": "" + }, + { + "formal": "wittPolynomial_one ** p : \u2115 R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : DecidableEq R S : Type u_2 inst\u271d : CommRing S \u22a2 W_ R 1 = \u2191C \u2191p * X 1 + X 0 ^ p ** simp only [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, range_one, sum_singleton,\n one_mul, pow_one, C_1, pow_zero, tsub_self, tsub_zero] ** Qed", + "informal": "" + }, + { + "formal": "Set.unop_mem_unop ** \u03b1 : Type u_1 s : Set \u03b1\u1d52\u1d56 a : \u03b1\u1d52\u1d56 \u22a2 a.unop \u2208 Set.unop s \u2194 a \u2208 s ** rfl ** Qed", + "informal": "" + }, + { + "formal": "LocallyConstant.coe_comap ** X : Type u_1 Y : Type u_2 Z : Type u_3 \u03b1 : Type u_4 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : TopologicalSpace Y f : X \u2192 Y g : LocallyConstant Y Z hf : Continuous f \u22a2 \u2191(comap f g) = \u2191g \u2218 f ** rw [comap, dif_pos hf] ** X : Type u_1 Y : Type u_2 Z : Type u_3 \u03b1 : Type u_4 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : TopologicalSpace Y f : X \u2192 Y g : LocallyConstant Y Z hf : Continuous f \u22a2 \u2191{ toFun := \u2191g \u2218 f, isLocallyConstant := (_ : IsLocallyConstant (g.toFun \u2218 f)) } = \u2191g \u2218 f ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Pi.isCompl_iff ** \u03b9 : Type u_1 \u03b1' : \u03b9 \u2192 Type u_2 inst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1' i) inst\u271d : (i : \u03b9) \u2192 BoundedOrder (\u03b1' i) f g : (i : \u03b9) \u2192 \u03b1' i \u22a2 IsCompl f g \u2194 \u2200 (i : \u03b9), IsCompl (f i) (g i) ** simp_rw [_root_.isCompl_iff, disjoint_iff, codisjoint_iff, forall_and] ** Qed", + "informal": "" + }, + { + "formal": "ZMod.cast_eq_val ** n : \u2115 R : Type u_1 inst\u271d\u00b9 : AddGroupWithOne R inst\u271d : NeZero n a : ZMod n \u22a2 \u2191a = \u2191(val a) ** cases n ** case succ R : Type u_1 inst\u271d\u00b9 : AddGroupWithOne R n\u271d : \u2115 inst\u271d : NeZero (Nat.succ n\u271d) a : ZMod (Nat.succ n\u271d) \u22a2 \u2191a = \u2191(val a) ** rfl ** case zero R : Type u_1 inst\u271d\u00b9 : AddGroupWithOne R inst\u271d : NeZero Nat.zero a : ZMod Nat.zero \u22a2 \u2191a = \u2191(val a) ** cases NeZero.ne 0 rfl ** Qed", + "informal": "" + }, + { + "formal": "Ideal.exists_radical_pow_le_of_fg ** R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) \u22a2 \u2203 n, radical I ^ n \u2264 I ** have := le_refl I.radical ** R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) this : radical I \u2264 radical I \u22a2 \u2203 n, radical I ^ n \u2264 I ** revert this ** R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) \u22a2 radical I \u2264 radical I \u2192 \u2203 n, radical I ^ n \u2264 I ** refine' Submodule.fg_induction _ _ (fun J => J \u2264 I.radical \u2192 \u2203 n : \u2115, J ^ n \u2264 I) _ _ _ h ** case refine'_1 R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) \u22a2 \u2200 (x : R), (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) (Submodule.span R {x}) ** intro x hx ** case refine'_1 R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) x : R hx : Submodule.span R {x} \u2264 radical I \u22a2 \u2203 n, Submodule.span R {x} ^ n \u2264 I ** obtain \u27e8n, hn\u27e9 := hx (subset_span (Set.mem_singleton x)) ** case refine'_1.intro R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) x : R hx : Submodule.span R {x} \u2264 radical I n : \u2115 hn : x ^ n \u2208 I \u22a2 \u2203 n, Submodule.span R {x} ^ n \u2264 I ** exact \u27e8n, by rwa [\u2190 Ideal.span, span_singleton_pow, span_le, Set.singleton_subset_iff]\u27e9 ** R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) x : R hx : Submodule.span R {x} \u2264 radical I n : \u2115 hn : x ^ n \u2208 I \u22a2 Submodule.span R {x} ^ n \u2264 I ** rwa [\u2190 Ideal.span, span_singleton_pow, span_le, Set.singleton_subset_iff] ** case refine'_2 R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) \u22a2 \u2200 (M\u2081 M\u2082 : Submodule R R), (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) M\u2081 \u2192 (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) ** intro J K hJ hK hJK ** case refine'_2 R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I \u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I ** obtain \u27e8n, hn\u27e9 := hJ fun x hx => hJK <| Ideal.mem_sup_left hx ** case refine'_2.intro R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I \u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I ** obtain \u27e8m, hm\u27e9 := hK fun x hx => hJK <| Ideal.mem_sup_right hx ** case refine'_2.intro.intro R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I \u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I ** use n + m ** case h R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I \u22a2 (J \u2294 K) ^ (n + m) \u2264 I ** rw [\u2190 Ideal.add_eq_sup, add_pow, Ideal.sum_eq_sup, Finset.sup_le_iff] ** case h R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I \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 ** refine' fun i _ => Ideal.mul_le_right.trans _ ** case h R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I i : \u2115 x\u271d : i \u2208 Finset.range (n + m + 1) \u22a2 J ^ i * K ^ (n + m - i) \u2264 I ** obtain h | h := le_or_lt n i ** case h.inl R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h\u271d : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I i : \u2115 x\u271d : i \u2208 Finset.range (n + m + 1) h : n \u2264 i \u22a2 J ^ i * K ^ (n + m - i) \u2264 I ** apply Ideal.mul_le_right.trans ((Ideal.pow_le_pow h).trans hn) ** case h.inr R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h\u271d : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I i : \u2115 x\u271d : i \u2208 Finset.range (n + m + 1) h : i < n \u22a2 J ^ i * K ^ (n + m - i) \u2264 I ** apply Ideal.mul_le_left.trans ** case h.inr R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h\u271d : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I i : \u2115 x\u271d : i \u2208 Finset.range (n + m + 1) h : i < n \u22a2 K ^ (n + m - i) \u2264 I ** refine' (Ideal.pow_le_pow _).trans hm ** case h.inr R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h\u271d : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I i : \u2115 x\u271d : i \u2208 Finset.range (n + m + 1) h : i < n \u22a2 m \u2264 n + m - i ** rw [add_comm, Nat.add_sub_assoc h.le] ** case h.inr R\u271d : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R\u271d inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R\u271d M R : Type u_3 inst\u271d : CommSemiring R I : Ideal R h\u271d : FG (radical I) J K : Submodule R R hJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I hK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I hJK : J \u2294 K \u2264 radical I n : \u2115 hn : J ^ n \u2264 I m : \u2115 hm : K ^ m \u2264 I i : \u2115 x\u271d : i \u2208 Finset.range (n + m + 1) h : i < n \u22a2 m \u2264 m + (n - i) ** apply Nat.le_add_right ** Qed", + "informal": "" + }, + { + "formal": "Nat.pairwise_one_le_dist ** m n : \u2115 hne : m \u2260 n \u22a2 \u2191m \u2260 \u2191n ** exact_mod_cast hne ** Qed", + "informal": "" + }, + { + "formal": "Filter.isUnit_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 \u03b5 : Type u_6 inst\u271d : DivisionMonoid \u03b1 f g : Filter \u03b1 \u22a2 IsUnit f \u2194 \u2203 a, f = pure a \u2227 IsUnit a ** constructor ** case mp F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 \u03b5 : Type u_6 inst\u271d : DivisionMonoid \u03b1 f g : Filter \u03b1 \u22a2 IsUnit f \u2192 \u2203 a, f = pure a \u2227 IsUnit a ** rintro \u27e8u, rfl\u27e9 ** case mp.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 \u03b5 : Type u_6 inst\u271d : DivisionMonoid \u03b1 g : Filter \u03b1 u : (Filter \u03b1)\u02e3 \u22a2 \u2203 a, \u2191u = pure a \u2227 IsUnit a ** obtain \u27e8a, b, ha, hb, h\u27e9 := Filter.mul_eq_one_iff.1 u.mul_inv ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 \u03b5 : Type u_6 inst\u271d : DivisionMonoid \u03b1 g : Filter \u03b1 u : (Filter \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = pure a hb : \u2191u\u207b\u00b9 = pure b h : a * b = 1 \u22a2 \u2203 a, \u2191u = pure a \u2227 IsUnit a ** refine' \u27e8a, ha, \u27e8a, b, h, pure_injective _\u27e9, rfl\u27e9 ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 \u03b5 : Type u_6 inst\u271d : DivisionMonoid \u03b1 g : Filter \u03b1 u : (Filter \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = pure a hb : \u2191u\u207b\u00b9 = pure b h : a * b = 1 \u22a2 pure (b * a) = pure 1 ** rw [\u2190 pure_mul_pure, \u2190 ha, \u2190 hb] ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 \u03b5 : Type u_6 inst\u271d : DivisionMonoid \u03b1 g : Filter \u03b1 u : (Filter \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = pure a hb : \u2191u\u207b\u00b9 = pure b h : a * b = 1 \u22a2 \u2191u\u207b\u00b9 * \u2191u = pure 1 ** exact u.inv_mul ** case mpr F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 \u03b5 : Type u_6 inst\u271d : DivisionMonoid \u03b1 f g : Filter \u03b1 \u22a2 (\u2203 a, f = pure a \u2227 IsUnit a) \u2192 IsUnit f ** rintro \u27e8a, rfl, ha\u27e9 ** case mpr.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 \u03b5 : Type u_6 inst\u271d : DivisionMonoid \u03b1 g : Filter \u03b1 a : \u03b1 ha : IsUnit a \u22a2 IsUnit (pure a) ** exact ha.filter ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Submartingale.ae_tendsto_limitProcess ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (limitProcess f \u2131 \u03bc \u03c9)) ** 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 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (limitProcess f \u2131 \u03bc \u03c9)) ** 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 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R \u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** 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 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 \u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** have hle : \u2a06 n, \u2131 n \u2264 m0 := sSup_le fun m \u27e8n, hn\u27e9 => hn \u25b8 \u2131.le _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 \u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** 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 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) \u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** 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 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) hg'm : AEStronglyMeasurable g' (Measure.trim \u03bc hle) \u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** obtain \u27e8g, hgm, hae\u27e9 := hg'm ** case intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) g : \u03a9 \u2192 \u211d hgm : StronglyMeasurable g hae : g' =\u1d50[Measure.trim \u03bc hle] g \u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** 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 ** case intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) g : \u03a9 \u2192 \u211d hgm : StronglyMeasurable g hae : g' =\u1d50[Measure.trim \u03bc hle] g hg : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) \u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** exact \u27e8g, hgm, measure_eq_zero_of_trim_eq_zero hle hg\u27e9 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R this : \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (limitProcess f \u2131 \u03bc \u03c9)) ** rw [limitProcess, dif_pos this] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R this : \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (Classical.choose this \u03c9)) ** exact (Classical.choose_spec this).2 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) ** filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with \u03c9 h\u03c9 ** case h \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 \u03c9 : \u03a9 h\u03c9 : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) \u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) ** simp_rw [dif_pos h\u03c9] ** case h \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 \u03c9 : \u03a9 h\u03c9 : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) \u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (Exists.choose h\u03c9)) ** exact h\u03c9.choose_spec ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) g : \u03a9 \u2192 \u211d hgm : StronglyMeasurable g hae : g' =\u1d50[Measure.trim \u03bc hle] g \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** filter_upwards [hae, hg'] with \u03c9 h\u03c9 hg'\u03c9 ** case h \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R g' : \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 hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) g : \u03a9 \u2192 \u211d hgm : StronglyMeasurable g hae : g' =\u1d50[Measure.trim \u03bc hle] g \u03c9 : \u03a9 h\u03c9 : g' \u03c9 = g \u03c9 hg'\u03c9 : Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) \u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) ** exact h\u03c9 \u25b8 hg'\u03c9 ** Qed", + "informal": "" + }, + { + "formal": "HasLineDerivWithinAt.mono_of_mem ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 : E \u2192 F f' : F s t : Set E x v : E L : E \u2192L[\ud835\udd5c] F h : HasLineDerivWithinAt \ud835\udd5c f f' t x v hst : t \u2208 \ud835\udcdd[s] x \u22a2 HasLineDerivWithinAt \ud835\udd5c f f' s x v ** apply HasDerivWithinAt.mono_of_mem h ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 : E \u2192 F f' : F s t : Set E x v : E L : E \u2192L[\ud835\udd5c] F h : HasLineDerivWithinAt \ud835\udd5c f f' t x v hst : t \u2208 \ud835\udcdd[s] x \u22a2 (fun t => x + t \u2022 v) \u207b\u00b9' t \u2208 \ud835\udcdd[(fun t => x + t \u2022 v) \u207b\u00b9' s] 0 ** apply ContinuousWithinAt.preimage_mem_nhdsWithin'' _ hst (by simp) ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 : E \u2192 F f' : F s t : Set E x v : E L : E \u2192L[\ud835\udd5c] F h : HasLineDerivWithinAt \ud835\udd5c f f' t x v hst : t \u2208 \ud835\udcdd[s] x \u22a2 ContinuousWithinAt (fun t => x + t \u2022 v) ((fun t => x + t \u2022 v) \u207b\u00b9' s) 0 ** apply Continuous.continuousWithinAt ** case h \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 : E \u2192 F f' : F s t : Set E x v : E L : E \u2192L[\ud835\udd5c] F h : HasLineDerivWithinAt \ud835\udd5c f f' t x v hst : t \u2208 \ud835\udcdd[s] x \u22a2 Continuous fun t => x + t \u2022 v ** continuity ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 : E \u2192 F f' : F s t : Set E x v : E L : E \u2192L[\ud835\udd5c] F h : HasLineDerivWithinAt \ud835\udd5c f f' t x v hst : t \u2208 \ud835\udcdd[s] x \u22a2 x = x + 0 \u2022 v ** simp ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.X_dvd_X ** \u03c3 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CommSemiring R inst\u271d : Nontrivial R i j : \u03c3 \u22a2 X i \u2223 X j \u2194 i = j ** refine' monomial_one_dvd_monomial_one.trans _ ** \u03c3 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CommSemiring R inst\u271d : Nontrivial R i j : \u03c3 \u22a2 ((fun\u2080 | i => 1) \u2264 fun\u2080 | j => 1) \u2194 i = j ** simp_rw [Finsupp.single_le_iff, Nat.one_le_iff_ne_zero, Finsupp.single_apply_ne_zero,\n and_true_iff] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Bicategory.leftUnitor_comp ** B : Type u inst\u271d : Bicategory B a b c d e : B f : a \u27f6 b g : b \u27f6 c \u22a2 (\u03bb_ (f \u226b g)).hom = (\u03b1_ (\ud835\udfd9 a) f g).inv \u226b (\u03bb_ f).hom \u25b7 g ** simp ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.rpow_eq_top_iff_of_pos ** x : \u211d\u22650\u221e y : \u211d hy : 0 < y \u22a2 x ^ y = \u22a4 \u2194 x = \u22a4 ** simp [rpow_eq_top_iff, hy, asymm hy] ** Qed", + "informal": "" + }, + { + "formal": "RingHom.FiniteType.of_comp_finiteType ** A : Type u_1 B : Type u_2 C : Type u_3 inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : CommRing B inst\u271d : CommRing C f : A \u2192+* B g : B \u2192+* C h : FiniteType (RingHom.comp g f) \u22a2 FiniteType g ** let _ := f.toAlgebra ** A : Type u_1 B : Type u_2 C : Type u_3 inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : CommRing B inst\u271d : CommRing C f : A \u2192+* B g : B \u2192+* C h : FiniteType (RingHom.comp g f) x\u271d : Algebra A B := toAlgebra f \u22a2 FiniteType g ** let _ := g.toAlgebra ** A : Type u_1 B : Type u_2 C : Type u_3 inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : CommRing B inst\u271d : CommRing C f : A \u2192+* B g : B \u2192+* C h : FiniteType (RingHom.comp g f) x\u271d\u00b9 : Algebra A B := toAlgebra f x\u271d : Algebra B C := toAlgebra g \u22a2 FiniteType g ** let _ := (g.comp f).toAlgebra ** A : Type u_1 B : Type u_2 C : Type u_3 inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : CommRing B inst\u271d : CommRing C f : A \u2192+* B g : B \u2192+* C h : FiniteType (RingHom.comp g f) x\u271d\u00b2 : Algebra A B := toAlgebra f x\u271d\u00b9 : Algebra B C := toAlgebra g x\u271d : Algebra A C := toAlgebra (RingHom.comp g f) \u22a2 FiniteType g ** let _ : IsScalarTower A B C := RestrictScalars.isScalarTower A B C ** A : Type u_1 B : Type u_2 C : Type u_3 inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : CommRing B inst\u271d : CommRing C f : A \u2192+* B g : B \u2192+* C h : FiniteType (RingHom.comp g f) x\u271d\u00b3 : Algebra A B := toAlgebra f x\u271d\u00b2 : Algebra B C := toAlgebra g x\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f) x\u271d : IsScalarTower A B C := RestrictScalars.isScalarTower A B C \u22a2 FiniteType g ** let _ : Algebra.FiniteType A C := h ** A : Type u_1 B : Type u_2 C : Type u_3 inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : CommRing B inst\u271d : CommRing C f : A \u2192+* B g : B \u2192+* C h : FiniteType (RingHom.comp g f) x\u271d\u2074 : Algebra A B := toAlgebra f x\u271d\u00b3 : Algebra B C := toAlgebra g x\u271d\u00b2 : Algebra A C := toAlgebra (RingHom.comp g f) x\u271d\u00b9 : IsScalarTower A B C := RestrictScalars.isScalarTower A B C x\u271d : Algebra.FiniteType A C := h \u22a2 FiniteType g ** exact Algebra.FiniteType.of_restrictScalars_finiteType A B C ** Qed", + "informal": "" + }, + { + "formal": "Monotone.map_ciSup_of_continuousAt ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : OrderTopology \u03b1 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : Nonempty \u03b3 f : \u03b1 \u2192 \u03b2 g : \u03b3 \u2192 \u03b1 Cf : ContinuousAt f (\u2a06 i, g i) Mf : Monotone f H : BddAbove (range g) \u22a2 f (\u2a06 i, g i) = \u2a06 i, f (g i) ** rw [iSup, Mf.map_csSup_of_continuousAt Cf (range_nonempty _) H, \u2190 range_comp, iSup] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : OrderTopology \u03b1 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : Nonempty \u03b3 f : \u03b1 \u2192 \u03b2 g : \u03b3 \u2192 \u03b1 Cf : ContinuousAt f (\u2a06 i, g i) Mf : Monotone f H : BddAbove (range g) \u22a2 sSup (range (f \u2218 fun i => g i)) = sSup (range fun i => f (g i)) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.ext_ring ** R : Type u_1 R\u2081 : Type u_2 R\u2082 : Type u_3 R\u2083 : Type u_4 k : Type u_5 S : Type u_6 S\u2083 : Type u_7 T : Type u_8 M : Type u_9 M\u2081 : Type u_10 M\u2082 : Type u_11 M\u2083 : Type u_12 N\u2081 : Type u_13 N\u2082 : Type u_14 N\u2083 : Type u_15 \u03b9 : Type u_16 inst\u271d\u00b9\u00b9 : Semiring R inst\u271d\u00b9\u2070 : Semiring S inst\u271d\u2079 : AddCommMonoid M inst\u271d\u2078 : AddCommMonoid M\u2081 inst\u271d\u2077 : AddCommMonoid M\u2082 inst\u271d\u2076 : AddCommMonoid M\u2083 inst\u271d\u2075 : AddCommMonoid N\u2081 inst\u271d\u2074 : AddCommMonoid N\u2082 inst\u271d\u00b3 : AddCommMonoid N\u2083 inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M\u2082 inst\u271d : Module S M\u2083 \u03c3 : R \u2192+* S f\u2097 g\u2097 : M \u2192\u2097[R] M\u2082 f\u271d g\u271d : M \u2192\u209b\u2097[\u03c3] M\u2083 f g : R \u2192\u209b\u2097[\u03c3] M\u2083 h : \u2191f 1 = \u2191g 1 x : R \u22a2 \u2191f x = \u2191g x ** rw [\u2190 mul_one x, \u2190 smul_eq_mul, f.map_smul\u209b\u2097, g.map_smul\u209b\u2097, h] ** Qed", + "informal": "" + }, + { + "formal": "HNNExtension.NormalWord.prod_smul_empty ** case ofGroup G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g\u271d : G \u22a2 ReducedWord.prod \u03c6 (ofGroup g\u271d).toReducedWord \u2022 empty = ofGroup g\u271d ** simp [ofGroup, ReducedWord.prod, of_smul_eq_smul, group_smul_def] ** case cons G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g : G u : \u2124\u02e3 w : NormalWord d h1 : w.head \u2208 TransversalPair.set d u h2 : \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' ih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w \u22a2 ReducedWord.prod \u03c6 (cons g u w h1 h2).toReducedWord \u2022 empty = cons g u w h1 h2 ** rw [prod_cons, \u2190 mul_assoc, mul_smul, ih, mul_smul, t_pow_smul_eq_unitsSMul,\n of_smul_eq_smul, unitsSMul] ** case cons G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g : G u : \u2124\u02e3 w : NormalWord d h1 : w.head \u2208 TransversalPair.set d u h2 : \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' ih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w \u22a2 (g \u2022 if h : Cancels u w then unitsSMulWithCancel \u03c6 u w h else let g' := unitsSMulGroup \u03c6 d u w.head; cons (\u2191g'.1) u ((\u2191g'.2 * w.head\u207b\u00b9) \u2022 w) (_ : \u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9 * w.head \u2208 TransversalPair.set d u) (_ : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).toList) \u2192 ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 toSubgroup A B u \u2192 u = u')) = cons g u w h1 h2 ** rw [dif_neg (not_cancels_of_cons_hyp u w h2)] ** case cons G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g : G u : \u2124\u02e3 w : NormalWord d h1 : w.head \u2208 TransversalPair.set d u h2 : \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' ih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w \u22a2 (g \u2022 let g' := unitsSMulGroup \u03c6 d u w.head; cons (\u2191g'.1) u ((\u2191g'.2 * w.head\u207b\u00b9) \u2022 w) (_ : \u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9 * w.head \u2208 TransversalPair.set d u) (_ : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).toList) \u2192 ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 toSubgroup A B u \u2192 u = u')) = cons g u w h1 h2 ** simp only [unitsSMulGroup] ** case cons G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g : G u : \u2124\u02e3 w : NormalWord d h1 : w.head \u2208 TransversalPair.set d u h2 : \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' ih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w \u22a2 g \u2022 cons (\u2191(\u2191(toSubgroupEquiv \u03c6 u) (\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).1)) u ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 * w.head\u207b\u00b9) \u2022 w) (_ : ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 TransversalPair.set d u) (_ : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 * w.head\u207b\u00b9) \u2022 w).toList) \u2192 ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 toSubgroup A B u \u2192 u = u') = cons g u w h1 h2 ** erw [(d.compl _).equiv_fst_eq_one_of_mem_of_one_mem (one_mem _) h1] ** case cons G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g : G u : \u2124\u02e3 w : NormalWord d h1 : w.head \u2208 TransversalPair.set d u h2 : \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' ih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w \u22a2 g \u2022 cons (\u2191(\u2191(toSubgroupEquiv \u03c6 u) { val := 1, property := (_ : 1 \u2208 toSubgroup A B u) })) u ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 * w.head\u207b\u00b9) \u2022 w) (_ : ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 TransversalPair.set d u) (_ : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 * w.head\u207b\u00b9) \u2022 w).toList) \u2192 ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 toSubgroup A B u \u2192 u = u') = cons g u w h1 h2 ** ext <;> simp ** case cons.h2.a.a G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g : G u : \u2124\u02e3 w : NormalWord d h1 : w.head \u2208 TransversalPair.set d u h2 : \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' ih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w n\u271d : \u2115 a\u271d : \u2124\u02e3 \u00d7 G \u22a2 List.get? ((u, \u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2) :: w.toList) n\u271d = some a\u271d \u2194 List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d ** erw [(d.compl _).equiv_snd_eq_inv_mul] ** case cons.h2.a.a G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g : G u : \u2124\u02e3 w : NormalWord d h1 : w.head \u2208 TransversalPair.set d u h2 : \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' ih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w n\u271d : \u2115 a\u271d : \u2124\u02e3 \u00d7 G \u22a2 List.get? ((u, (\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).1)\u207b\u00b9 * w.head) :: w.toList) n\u271d = some a\u271d \u2194 List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d ** erw [(d.compl _).equiv_fst_eq_one_of_mem_of_one_mem (one_mem _) h1] ** case cons.h2.a.a G : Type u_1 inst\u271d\u00b3 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b2 : Group H M : Type u_3 inst\u271d\u00b9 : Monoid M d : TransversalPair G A B inst\u271d : DecidableEq G g : G u : \u2124\u02e3 w : NormalWord d h1 : w.head \u2208 TransversalPair.set d u h2 : \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' ih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w n\u271d : \u2115 a\u271d : \u2124\u02e3 \u00d7 G \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 List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d ** simp ** Qed", + "informal": "" + }, + { + "formal": "Vector.mapAccumr\u2082_mapAccumr_right ** \u03b1 : Type n : \u2115 \u03b2 : Type xs : Vector \u03b1 n ys : Vector \u03b2 n \u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type s\u2082 : \u03c3\u2082 s\u2081 : \u03c3\u2081 f\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6 f\u2082 : \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 \u22a2 mapAccumr\u2082 f\u2081 xs (mapAccumr f\u2082 ys s\u2082).2 s\u2081 = let m := mapAccumr\u2082 (fun x y s => let r\u2082 := f\u2082 y s.2; let r\u2081 := f\u2081 x r\u2082.2 s.1; ((r\u2081.1, r\u2082.1), r\u2081.2)) xs ys (s\u2081, s\u2082); (m.1.1, m.2) ** induction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082 <;> simp_all ** Qed", + "informal": "" + }, + { + "formal": "PiNat.exists_disjoint_cylinder ** E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) s : Set ((n : \u2115) \u2192 E n) hs : IsClosed s x : (n : \u2115) \u2192 E n hx : \u00acx \u2208 s \u22a2 \u2203 n, Disjoint s (cylinder x n) ** rcases eq_empty_or_nonempty s with (rfl | hne) ** case inr E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) s : Set ((n : \u2115) \u2192 E n) hs : IsClosed s x : (n : \u2115) \u2192 E n hx : \u00acx \u2208 s hne : Set.Nonempty s \u22a2 \u2203 n, Disjoint s (cylinder x n) ** have A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx ** case inr E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) s : Set ((n : \u2115) \u2192 E n) hs : IsClosed s x : (n : \u2115) \u2192 E n hx : \u00acx \u2208 s hne : Set.Nonempty s A : 0 < infDist x s \u22a2 \u2203 n, Disjoint s (cylinder x n) ** obtain \u27e8n, hn\u27e9 : \u2203 n, (1 / 2 : \u211d) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one ** case inr.intro E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) s : Set ((n : \u2115) \u2192 E n) hs : IsClosed s x : (n : \u2115) \u2192 E n hx : \u00acx \u2208 s hne : Set.Nonempty s A : 0 < infDist x s n : \u2115 hn : (1 / 2) ^ n < infDist x s \u22a2 \u2203 n, Disjoint s (cylinder x n) ** refine' \u27e8n, disjoint_left.2 fun y ys hy => ?_\u27e9 ** case inr.intro E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) s : Set ((n : \u2115) \u2192 E n) hs : IsClosed s x : (n : \u2115) \u2192 E n hx : \u00acx \u2208 s hne : Set.Nonempty s A : 0 < infDist x s n : \u2115 hn : (1 / 2) ^ n < infDist x s y : (n : \u2115) \u2192 E n ys : y \u2208 s hy : y \u2208 cylinder x n \u22a2 False ** apply lt_irrefl (infDist x s) ** case inr.intro E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) s : Set ((n : \u2115) \u2192 E n) hs : IsClosed s x : (n : \u2115) \u2192 E n hx : \u00acx \u2208 s hne : Set.Nonempty s A : 0 < infDist x s n : \u2115 hn : (1 / 2) ^ n < infDist x s y : (n : \u2115) \u2192 E n ys : y \u2208 s hy : y \u2208 cylinder x n \u22a2 infDist x s < infDist x s ** 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 ** case inl E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) x : (n : \u2115) \u2192 E n hs : IsClosed \u2205 hx : \u00acx \u2208 \u2205 \u22a2 \u2203 n, Disjoint \u2205 (cylinder x n) ** exact \u27e80, by simp\u27e9 ** E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) x : (n : \u2115) \u2192 E n hs : IsClosed \u2205 hx : \u00acx \u2208 \u2205 \u22a2 Disjoint \u2205 (cylinder x 0) ** simp ** E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) s : Set ((n : \u2115) \u2192 E n) hs : IsClosed s x : (n : \u2115) \u2192 E n hx : \u00acx \u2208 s hne : Set.Nonempty s A : 0 < infDist x s n : \u2115 hn : (1 / 2) ^ n < infDist x s y : (n : \u2115) \u2192 E n ys : y \u2208 s hy : y \u2208 cylinder x n \u22a2 dist x y \u2264 (1 / 2) ^ n ** rw [mem_cylinder_comm] at hy ** E : \u2115 \u2192 Type u_1 inst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n) inst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n) s : Set ((n : \u2115) \u2192 E n) hs : IsClosed s x : (n : \u2115) \u2192 E n hx : \u00acx \u2208 s hne : Set.Nonempty s A : 0 < infDist x s n : \u2115 hn : (1 / 2) ^ n < infDist x s y : (n : \u2115) \u2192 E n ys : y \u2208 s hy : x \u2208 cylinder y n \u22a2 dist x y \u2264 (1 / 2) ^ n ** exact mem_cylinder_iff_dist_le.1 hy ** Qed", + "informal": "" + }, + { + "formal": "Matrix.adjugate_mul ** m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 \u22a2 adjugate A * A = (A\u1d40 * adjugate A\u1d40)\u1d40 ** rw [\u2190 adjugate_transpose, \u2190 transpose_mul, transpose_transpose] ** m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 \u22a2 (A\u1d40 * adjugate A\u1d40)\u1d40 = det A \u2022 1 ** rw [mul_adjugate A\u1d40, det_transpose, transpose_smul, transpose_one] ** Qed", + "informal": "" + }, + { + "formal": "lucas_primality ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 \u22a2 Nat.Prime p ** have h0 : p \u2260 0 := by\n rintro \u27e8\u27e9\n exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _) ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 \u22a2 Nat.Prime p ** have h1 : p \u2260 1 := by\n rintro \u27e8\u27e9\n exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _) ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 \u22a2 Nat.Prime p ** have hp1 : 1 < p := lt_of_le_of_ne h0.bot_lt h1.symm ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p \u22a2 Nat.Prime p ** 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 ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p order_of_a : orderOf a = p - 1 \u22a2 Nat.Prime p ** haveI : NeZero p := \u27e8h0\u27e9 ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p order_of_a : orderOf a = p - 1 this : NeZero p \u22a2 Nat.Prime p ** rw [Nat.prime_iff_card_units] ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p order_of_a : orderOf a = p - 1 this : NeZero p \u22a2 Fintype.card (ZMod p)\u02e3 = p - 1 ** refine' le_antisymm (Nat.card_units_zmod_lt_sub_one hp1) _ ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p order_of_a : orderOf a = p - 1 this : NeZero p \u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3 ** have hp' : p - 2 + 1 = p - 1 := tsub_add_eq_add_tsub hp1 ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p order_of_a : orderOf a = p - 1 this : NeZero p hp' : p - 2 + 1 = p - 1 \u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3 ** let a' : (ZMod p)\u02e3 := Units.mkOfMulEqOne a (a ^ (p - 2)) (by rw [\u2190 pow_succ, hp', ha]) ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p order_of_a : orderOf a = p - 1 this : NeZero p hp' : p - 2 + 1 = p - 1 a' : (ZMod p)\u02e3 := Units.mkOfMulEqOne a (a ^ (p - 2)) (_ : a * a ^ (p - 2) = 1) \u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3 ** 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 ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 \u22a2 p \u2260 0 ** rintro \u27e8\u27e9 ** case refl a : ZMod 0 ha : a ^ (0 - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 0 - 1 \u2192 a ^ ((0 - 1) / q) \u2260 1 \u22a2 False ** exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _) ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 \u22a2 p \u2260 1 ** rintro \u27e8\u27e9 ** case refl a : ZMod 1 ha : a ^ (1 - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 1 - 1 \u2192 a ^ ((1 - 1) / q) \u2260 1 h0 : 1 \u2260 0 \u22a2 False ** exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _) ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p \u22a2 orderOf a = p - 1 ** apply orderOf_eq_of_pow_and_pow_div_prime _ ha hd ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p \u22a2 0 < p - 1 ** exact tsub_pos_of_lt hp1 ** p : \u2115 a : ZMod p ha : a ^ (p - 1) = 1 hd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1 h0 : p \u2260 0 h1 : p \u2260 1 hp1 : 1 < p order_of_a : orderOf a = p - 1 this : NeZero p hp' : p - 2 + 1 = p - 1 \u22a2 a * a ^ (p - 2) = 1 ** rw [\u2190 pow_succ, hp', ha] ** Qed", + "informal": "" + }, + { + "formal": "InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x y : V h : inner x y = 0 h0 : x \u2260 0 \u22a2 angle x (x + y) = Real.arctan (\u2016y\u2016 / \u2016x\u2016) ** 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] ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x y : V h : inner x y = 0 h0 : x \u2260 0 \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))) ** nth_rw 3 [\u2190 Real.sqrt_sq (norm_nonneg x)] ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x y : V h : inner x y = 0 h0 : x \u2260 0 \u22a2 Real.arcsin (\u2016y\u2016 / Real.sqrt (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016)) = Real.arcsin (\u2016y\u2016 / (Real.sqrt (\u2016x\u2016 ^ 2) * Real.sqrt (\u21911 + (\u2016y\u2016 / \u2016x\u2016) ^ 2))) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "wbtw_smul_vadd_smul_vadd_of_nonneg_of_le ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x : P v : V r\u2081 r\u2082 : R hr\u2081 : 0 \u2264 r\u2081 hr\u2082 : r\u2081 \u2264 r\u2082 \u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) ** 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 ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x : P v : V r\u2081 r\u2082 : R hr\u2081 : 0 \u2264 r\u2081 hr\u2082 : r\u2081 \u2264 r\u2082 \u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x ** by_cases h : r\u2081 = 0 ** case neg R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x : P v : V r\u2081 r\u2082 : R hr\u2081 : 0 \u2264 r\u2081 hr\u2082 : r\u2081 \u2264 r\u2082 h : \u00acr\u2081 = 0 \u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x ** simp [lineMap_apply, smul_smul, ((hr\u2081.lt_of_ne' h).trans_le hr\u2082).ne.symm] ** case pos R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x : P v : V r\u2081 r\u2082 : R hr\u2081 : 0 \u2264 r\u2081 hr\u2082 : r\u2081 \u2264 r\u2082 h : r\u2081 = 0 \u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x ** simp [h] ** Qed", + "informal": "" + }, + { + "formal": "PrimeSpectrum.basicOpen_pow ** R : Type u S : Type v inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S f : R n : \u2115 hn : 0 < n \u22a2 \u2191(basicOpen (f ^ n)) = \u2191(basicOpen f) ** simpa using zeroLocus_singleton_pow f n hn ** Qed", + "informal": "" + }, + { + "formal": "Equiv.mul_swap_eq_swap_mul ** \u03b1 : Type u \u03b2 : Type v inst\u271d : DecidableEq \u03b1 f : Perm \u03b1 x y : \u03b1 \u22a2 f * swap x y = swap (\u2191f x) (\u2191f y) * f ** rw [swap_mul_eq_mul_swap, Perm.inv_apply_self, Perm.inv_apply_self] ** Qed", + "informal": "" + }, + { + "formal": "VonNeumannAlgebra.commutant_commutant ** H : Type u inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u2102 H inst\u271d : CompleteSpace H S : VonNeumannAlgebra H \u22a2 \u2191(commutant (commutant S)) = \u2191S ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.measure_union_add_inter' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 hs : MeasurableSet s t : Set \u03b1 \u22a2 \u2191\u2191\u03bc (s \u222a t) + \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t ** rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.cyclotomic'_eq_X_pow_sub_one_div ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 cyclotomic' n K = (X ^ n - 1) /\u2098 \u220f i in Nat.properDivisors n, cyclotomic' i K ** rw [\u2190 prod_cyclotomic'_eq_X_pow_sub_one hpos h, \u2190 Nat.cons_self_properDivisors hpos.ne',\n Finset.prod_cons] ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 cyclotomic' n K = (cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K) /\u2098 \u220f i in Nat.properDivisors n, cyclotomic' i K ** 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 ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n prod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u22a2 cyclotomic' n K = (cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K) /\u2098 \u220f i in Nat.properDivisors n, cyclotomic' i K ** rw [(div_modByMonic_unique (cyclotomic' n K) 0 prod_monic _).1] ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n prod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u22a2 0 + (\u220f i in Nat.properDivisors n, cyclotomic' i K) * cyclotomic' n K = cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K \u2227 degree 0 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) ** simp only [degree_zero, zero_add] ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n prod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u22a2 (\u220f i in Nat.properDivisors n, cyclotomic' i K) * cyclotomic' n K = cyclotomic' n K * \u220f i in Nat.properDivisors n, cyclotomic' i K \u2227 \u22a5 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) ** refine' \u27e8by rw [mul_comm], _\u27e9 ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n prod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u22a2 \u22a5 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) ** rw [bot_lt_iff_ne_bot] ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n prod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u22a2 degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u2260 \u22a5 ** intro h ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h\u271d : IsPrimitiveRoot \u03b6 n prod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K) h : degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) = \u22a5 \u22a2 False ** exact Monic.ne_zero prod_monic (degree_eq_bot.1 h) ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K) ** apply monic_prod_of_monic ** case hs K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 \u2200 (i : \u2115), i \u2208 Nat.properDivisors n \u2192 Monic (cyclotomic' i K) ** intro i _ ** case hs K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n i : \u2115 a\u271d : i \u2208 Nat.properDivisors n \u22a2 Monic (cyclotomic' i K) ** exact cyclotomic'.monic i K ** K\u271d : Type u_1 inst\u271d\u00b2 : Field K\u271d K : Type u_2 inst\u271d\u00b9 : CommRing K inst\u271d : IsDomain K \u03b6 : K n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n prod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u22a2 (\u220f i in Nat.properDivisors n, cyclotomic' i K) * cyclotomic' n K = cyclotomic' n K * \u220f i in Nat.properDivisors n, cyclotomic' i K ** rw [mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "Filter.limsup_bot ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 inst\u271d : CompleteLattice \u03b1 f : \u03b2 \u2192 \u03b1 \u22a2 limsup f \u22a5 = \u22a5 ** simp [limsup] ** Qed", + "informal": "" + }, + { + "formal": "hasSum_one_div_nat_pow_mul_cos ** k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) \u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x)) ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! * Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) ** have ofReal_two : ((2 : \u211d) : \u2102) = 2 := by norm_cast ** k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 \u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x)) ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! * Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) ** convert ((hasSum_iff _ _).mp (this.div_const 2)).1 with n ** k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 \u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ** 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 ** k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 \u22a2 2 \u2264 2 * k ** linarith [Nat.one_le_iff_ne_zero.mpr hk] ** case h.e'_5.h.h.e'_6 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 x\u271d : \u2115 \u22a2 \u2191(fourier \u2191x\u271d) \u2191x + \u2191(fourier (-\u2191x\u271d)) \u2191x = \u2191(fourier \u2191x\u271d) \u2191x + (-1) ^ (2 * k) * \u2191(fourier (-\u2191x\u271d)) \u2191x ** rw [pow_mul (-1 : \u2102), neg_one_sq, one_pow, one_mul] ** case h.e'_6.h.e'_5.h.e'_5 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 \u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k) ** rw [pow_add, pow_one] ** case h.e'_6.h.e'_5.h.e'_5 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 \u22a2 (-1) ^ k * -1 * (2 * \u2191\u03c0) ^ (2 * k) = -((2 * \u2191\u03c0) ^ (2 * k) * (-1) ^ k) ** ring ** k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) \u22a2 \u21912 = 2 ** norm_cast ** case h.e'_5.h k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \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 ** convert (ofReal_re _).symm ** case h.e'_3.h.e'_1 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \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)) ** rw [ofReal_mul] ** case h.e'_3.h.e'_1 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \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)) ** rw [\u2190 mul_div] ** case h.e'_3.h.e'_1 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \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)) ** congr ** case h.e'_3.h.e'_1.e_a k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \u22a2 1 / \u2191n ^ (2 * k) = \u2191(1 / \u2191n ^ (2 * k)) ** rw [ofReal_div, ofReal_one, ofReal_pow] ** case h.e'_3.h.e'_1.e_a k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \u22a2 1 / \u2191n ^ (2 * k) = 1 / \u2191\u2191n ^ (2 * k) ** rfl ** case h.e'_3.h.e'_1.e_a k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \u22a2 (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(Real.cos (2 * \u03c0 * \u2191n * x)) ** 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] ** case h.e'_3.h.e'_1.e_a k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \u22a2 (cexp (2 * \u2191\u03c0 * I * \u2191n * \u2191x) + cexp (2 * \u2191\u03c0 * I * -\u2191n * \u2191x)) / 2 = (cexp (2 * \u2191\u03c0 * \u2191n * \u2191x * I) + cexp (-(2 * \u2191\u03c0 * \u2191n * \u2191x) * I)) / 2 ** congr 3 ** case h.e'_3.h.e'_1.e_a.e_a.e_a.e_z k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \u22a2 2 * \u2191\u03c0 * I * \u2191n * \u2191x = 2 * \u2191\u03c0 * \u2191n * \u2191x * I ** ring ** case h.e'_3.h.e'_1.e_a.e_a.e_a.e_z k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 n : \u2115 \u22a2 2 * \u2191\u03c0 * I * -\u2191n * \u2191x = -(2 * \u2191\u03c0 * \u2191n * \u2191x) * I ** ring ** case h.e'_6 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 \u22a2 (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! * Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))) = ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2).re ** convert (ofReal_re _).symm ** case h.e'_3.h.e'_1 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 \u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 = \u2191((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! * Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) ** 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] ** case h.e'_3.h.e'_1 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 \u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 = (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! * \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) ** rw [bernoulliFun] ** case h.e'_3.h.e'_1 k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 this : HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x)) ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x)) ofReal_two : \u21912 = 2 \u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) / 2 = (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! * \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) ** ring ** Qed", + "informal": "" + }, + { + "formal": "NormedAddGroupHom.comp_assoc ** V : Type u_1 V\u2081 : Type u_2 V\u2082 : Type u_3 V\u2083 : Type u_4 inst\u271d\u2074 : SeminormedAddCommGroup V inst\u271d\u00b3 : SeminormedAddCommGroup V\u2081 inst\u271d\u00b2 : SeminormedAddCommGroup V\u2082 inst\u271d\u00b9 : SeminormedAddCommGroup V\u2083 f\u271d g\u271d : NormedAddGroupHom V\u2081 V\u2082 V\u2084 : Type u_5 inst\u271d : SeminormedAddCommGroup V\u2084 h : NormedAddGroupHom V\u2083 V\u2084 g : NormedAddGroupHom V\u2082 V\u2083 f : NormedAddGroupHom V\u2081 V\u2082 \u22a2 NormedAddGroupHom.comp (NormedAddGroupHom.comp h g) f = NormedAddGroupHom.comp h (NormedAddGroupHom.comp g f) ** ext ** case H V : Type u_1 V\u2081 : Type u_2 V\u2082 : Type u_3 V\u2083 : Type u_4 inst\u271d\u2074 : SeminormedAddCommGroup V inst\u271d\u00b3 : SeminormedAddCommGroup V\u2081 inst\u271d\u00b2 : SeminormedAddCommGroup V\u2082 inst\u271d\u00b9 : SeminormedAddCommGroup V\u2083 f\u271d g\u271d : NormedAddGroupHom V\u2081 V\u2082 V\u2084 : Type u_5 inst\u271d : SeminormedAddCommGroup V\u2084 h : NormedAddGroupHom V\u2083 V\u2084 g : NormedAddGroupHom V\u2082 V\u2083 f : NormedAddGroupHom V\u2081 V\u2082 x\u271d : V\u2081 \u22a2 \u2191(NormedAddGroupHom.comp (NormedAddGroupHom.comp h g) f) x\u271d = \u2191(NormedAddGroupHom.comp h (NormedAddGroupHom.comp g f)) x\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.erase_ne ** R : Type u a b : R m n\u271d : \u2115 inst\u271d : Semiring R p\u271d q p : R[X] n i : \u2115 h : i \u2260 n \u22a2 coeff (erase n p) i = coeff p i ** simp [coeff_erase, h] ** Qed", + "informal": "" + }, + { + "formal": "List.ext ** \u03b1 : Type u_1 a : \u03b1 l\u2081 : List \u03b1 a' : \u03b1 l\u2082 : List \u03b1 h : \u2200 (n : Nat), get? (a :: l\u2081) n = get? (a' :: l\u2082) n \u22a2 a :: l\u2081 = a' :: l\u2082 ** have h0 : some a = some a' := h 0 ** \u03b1 : Type u_1 a : \u03b1 l\u2081 : List \u03b1 a' : \u03b1 l\u2082 : List \u03b1 h : \u2200 (n : Nat), get? (a :: l\u2081) n = get? (a' :: l\u2082) n h0 : some a = some a' \u22a2 a :: l\u2081 = a' :: l\u2082 ** injection h0 with aa ** \u03b1 : Type u_1 a : \u03b1 l\u2081 : List \u03b1 a' : \u03b1 l\u2082 : List \u03b1 h : \u2200 (n : Nat), get? (a :: l\u2081) n = get? (a' :: l\u2082) n aa : a = a' \u22a2 a :: l\u2081 = a' :: l\u2082 ** simp only [aa, ext fun n => h (n+1)] ** Qed", + "informal": "" + }, + { + "formal": "Int.abs_lt_one_iff ** a : \u2124 a0 : |a| < 1 \u22a2 a = 0 ** let \u27e8hn, hp\u27e9 := abs_lt.mp a0 ** a : \u2124 a0 : |a| < 1 hn : -1 < a hp : a < 1 \u22a2 a = 0 ** rw [\u2190zero_add 1, lt_add_one_iff] at hp ** a : \u2124 a0 : |a| < 1 hn : -1 < a hp\u271d : a < 1 hp : a \u2264 0 \u22a2 a = 0 ** exact hp.antisymm hn ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.isSeparator_coprod_of_isSeparator_left ** C : Type u\u2081 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b2 : Category.{v\u2082, u\u2082} D inst\u271d\u00b9 : HasZeroMorphisms C G H : C inst\u271d : HasBinaryCoproduct G H hG : IsSeparator G \u22a2 {G} \u2286 {G, H} ** simp ** Qed", + "informal": "" + }, + { + "formal": "Set.OrdConnected.image_real_toNNReal ** s : Set \u211d t : Set \u211d\u22650 h : OrdConnected s \u22a2 OrdConnected (toNNReal '' s) ** refine' \u27e8ball_image_iff.2 fun x hx => ball_image_iff.2 fun y hy z hz => _\u27e9 ** s : Set \u211d t : Set \u211d\u22650 h : OrdConnected s x : \u211d hx : x \u2208 s y : \u211d hy : y \u2208 s z : \u211d\u22650 hz : z \u2208 Icc (toNNReal x) (toNNReal y) \u22a2 z \u2208 toNNReal '' s ** cases' le_total y 0 with hy\u2080 hy\u2080 ** case inl s : Set \u211d t : Set \u211d\u22650 h : OrdConnected s x : \u211d hx : x \u2208 s y : \u211d hy : y \u2208 s z : \u211d\u22650 hz : z \u2208 Icc (toNNReal x) (toNNReal y) hy\u2080 : y \u2264 0 \u22a2 z \u2208 toNNReal '' s ** rw [mem_Icc, Real.toNNReal_of_nonpos hy\u2080, nonpos_iff_eq_zero] at hz ** case inl s : Set \u211d t : Set \u211d\u22650 h : OrdConnected s x : \u211d hx : x \u2208 s y : \u211d hy : y \u2208 s z : \u211d\u22650 hz : toNNReal x \u2264 z \u2227 z = 0 hy\u2080 : y \u2264 0 \u22a2 z \u2208 toNNReal '' s ** exact \u27e8y, hy, (toNNReal_of_nonpos hy\u2080).trans hz.2.symm\u27e9 ** case inr s : Set \u211d t : Set \u211d\u22650 h : OrdConnected s x : \u211d hx : x \u2208 s y : \u211d hy : y \u2208 s z : \u211d\u22650 hz : z \u2208 Icc (toNNReal x) (toNNReal y) hy\u2080 : 0 \u2264 y \u22a2 z \u2208 toNNReal '' s ** lift y to \u211d\u22650 using hy\u2080 ** case inr.intro s : Set \u211d t : Set \u211d\u22650 h : OrdConnected s x : \u211d hx : x \u2208 s z y : \u211d\u22650 hy : \u2191y \u2208 s hz : z \u2208 Icc (toNNReal x) (toNNReal \u2191y) \u22a2 z \u2208 toNNReal '' s ** rw [toNNReal_coe] at hz ** case inr.intro s : Set \u211d t : Set \u211d\u22650 h : OrdConnected s x : \u211d hx : x \u2208 s z y : \u211d\u22650 hy : \u2191y \u2208 s hz : z \u2208 Icc (toNNReal x) y \u22a2 z \u2208 toNNReal '' s ** exact \u27e8z, h.out hx hy \u27e8toNNReal_le_iff_le_coe.1 hz.1, hz.2\u27e9, toNNReal_coe\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "not_irreducible_one ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : Monoid \u03b1 \u22a2 \u00acIrreducible 1 ** simp [irreducible_iff] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.pow_count ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b9 : CommMonoid \u03b1 s t : Multiset \u03b1 a\u271d : \u03b1 m : Multiset \u03b9 f g : \u03b9 \u2192 \u03b1 inst\u271d : DecidableEq \u03b1 a : \u03b1 \u22a2 a ^ count a s = prod (filter (Eq a) s) ** rw [filter_eq, prod_replicate] ** Qed", + "informal": "" + }, + { + "formal": "Set.Countable.ae_not_mem ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d inst\u271d\u00b9 : NoAtoms \u03bc\u271d \u03b1 : Type u_8 m : MeasurableSpace \u03b1 s : Set \u03b1 h : Set.Countable s \u03bc : Measure \u03b1 inst\u271d : NoAtoms \u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s ** simpa only [ae_iff, Classical.not_not] using h.measure_zero \u03bc ** Qed", + "informal": "" + }, + { + "formal": "ne\u2082\u2083_of_not_collinear ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : DivisionRing k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P p\u2081 p\u2082 p\u2083 : P h : \u00acCollinear k {p\u2081, p\u2082, p\u2083} \u22a2 p\u2082 \u2260 p\u2083 ** rintro rfl ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : DivisionRing k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P p\u2081 p\u2082 : P h : \u00acCollinear k {p\u2081, p\u2082, p\u2082} \u22a2 False ** simp [collinear_pair] at h ** Qed", + "informal": "" + }, + { + "formal": "imageToKernel'_kernelSubobjectIso ** \u03b9 : Type u_1 V : Type u inst\u271d\u00b3 : Category.{v, u} V inst\u271d\u00b2 : HasZeroMorphisms V A B C : V f : A \u27f6 B g : B \u27f6 C w\u271d : f \u226b g = 0 inst\u271d\u00b9 : HasKernels V inst\u271d : HasImages V w : f \u226b g = 0 \u22a2 imageToKernel' f g w \u226b (kernelSubobjectIso g).inv = (imageSubobjectIso f).inv \u226b imageToKernel f g w ** ext ** case h \u03b9 : Type u_1 V : Type u inst\u271d\u00b3 : Category.{v, u} V inst\u271d\u00b2 : HasZeroMorphisms V A B C : V f : A \u27f6 B g : B \u27f6 C w\u271d : f \u226b g = 0 inst\u271d\u00b9 : HasKernels V inst\u271d : HasImages V w : f \u226b g = 0 \u22a2 (imageToKernel' f g w \u226b (kernelSubobjectIso g).inv) \u226b Subobject.arrow (kernelSubobject g) = ((imageSubobjectIso f).inv \u226b imageToKernel f g w) \u226b Subobject.arrow (kernelSubobject g) ** simp [imageToKernel'] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_right ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2079 : IsROrC \ud835\udd5c inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : InnerProductSpace \ud835\udd5c E inst\u271d\u2074 : InnerProductSpace \ud835\udd5c F inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace G inst\u271d : CompleteSpace F A : E \u2192L[\ud835\udd5c] E x : E \u22a2 \u2016\u2191A x\u2016 ^ 2 = \u2191re (inner x (\u2191(\u2191adjoint A * A) x)) ** have h : \u27eax, (A\u2020 * A) x\u27eb = \u27eaA x, A x\u27eb := by rw [\u2190 adjoint_inner_right]; rfl ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2079 : IsROrC \ud835\udd5c inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : InnerProductSpace \ud835\udd5c E inst\u271d\u2074 : InnerProductSpace \ud835\udd5c F inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace G inst\u271d : CompleteSpace F A : E \u2192L[\ud835\udd5c] E x : E h : inner x (\u2191(\u2191adjoint A * A) x) = inner (\u2191A x) (\u2191A x) \u22a2 \u2016\u2191A x\u2016 ^ 2 = \u2191re (inner x (\u2191(\u2191adjoint A * A) x)) ** rw [h, \u2190 inner_self_eq_norm_sq (\ud835\udd5c := \ud835\udd5c) _] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2079 : IsROrC \ud835\udd5c inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : InnerProductSpace \ud835\udd5c E inst\u271d\u2074 : InnerProductSpace \ud835\udd5c F inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace G inst\u271d : CompleteSpace F A : E \u2192L[\ud835\udd5c] E x : E \u22a2 inner x (\u2191(\u2191adjoint A * A) x) = inner (\u2191A x) (\u2191A x) ** rw [\u2190 adjoint_inner_right] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2079 : IsROrC \ud835\udd5c inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : InnerProductSpace \ud835\udd5c E inst\u271d\u2074 : InnerProductSpace \ud835\udd5c F inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace G inst\u271d : CompleteSpace F A : E \u2192L[\ud835\udd5c] E x : E \u22a2 inner x (\u2191(\u2191adjoint A * A) x) = inner x (\u2191(\u2191adjoint A) (\u2191A x)) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Filter.HasBasis.frequently_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 l l' : Filter \u03b1 p : \u03b9 \u2192 Prop s : \u03b9 \u2192 Set \u03b1 t : Set \u03b1 i : \u03b9 p' : \u03b9' \u2192 Prop s' : \u03b9' \u2192 Set \u03b1 i' : \u03b9' hl : HasBasis l p s q : \u03b1 \u2192 Prop \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 ** simp only [Filter.Frequently, hl.eventually_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 l l' : Filter \u03b1 p : \u03b9 \u2192 Prop s : \u03b9 \u2192 Set \u03b1 t : Set \u03b1 i : \u03b9 p' : \u03b9' \u2192 Prop s' : \u03b9' \u2192 Set \u03b1 i' : \u03b9' hl : HasBasis l p s q : \u03b1 \u2192 Prop \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 ** push_neg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 l l' : Filter \u03b1 p : \u03b9 \u2192 Prop s : \u03b9 \u2192 Set \u03b1 t : Set \u03b1 i : \u03b9 p' : \u03b9' \u2192 Prop s' : \u03b9' \u2192 Set \u03b1 i' : \u03b9' hl : HasBasis l p s q : \u03b1 \u2192 Prop \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 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.zipWith_neLocus_eq_right ** \u03b1 : Type u_1 N : \u03b1 \u2192 Type u_2 inst\u271d\u2075 : DecidableEq \u03b1 M : \u03b1 \u2192 Type u_3 P : \u03b1 \u2192 Type u_4 inst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a) inst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a) inst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a) inst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (M a) inst\u271d : (a : \u03b1) \u2192 DecidableEq (P a) F : (a : \u03b1) \u2192 M a \u2192 N a \u2192 P a F0 : \u2200 (a : \u03b1), F a 0 0 = 0 f\u2081 f\u2082 : \u03a0\u2080 (a : \u03b1), M a g : \u03a0\u2080 (a : \u03b1), N a hF : \u2200 (a : \u03b1) (g : N a), Function.Injective fun f => F a f g \u22a2 neLocus (zipWith F F0 f\u2081 g) (zipWith F F0 f\u2082 g) = neLocus f\u2081 f\u2082 ** ext a ** case a \u03b1 : Type u_1 N : \u03b1 \u2192 Type u_2 inst\u271d\u2075 : DecidableEq \u03b1 M : \u03b1 \u2192 Type u_3 P : \u03b1 \u2192 Type u_4 inst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a) inst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a) inst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a) inst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (M a) inst\u271d : (a : \u03b1) \u2192 DecidableEq (P a) F : (a : \u03b1) \u2192 M a \u2192 N a \u2192 P a F0 : \u2200 (a : \u03b1), F a 0 0 = 0 f\u2081 f\u2082 : \u03a0\u2080 (a : \u03b1), M a g : \u03a0\u2080 (a : \u03b1), N a hF : \u2200 (a : \u03b1) (g : N a), Function.Injective fun f => F a f g a : \u03b1 \u22a2 a \u2208 neLocus (zipWith F F0 f\u2081 g) (zipWith F F0 f\u2082 g) \u2194 a \u2208 neLocus f\u2081 f\u2082 ** simpa only [mem_neLocus] using (hF a _).ne_iff ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.X_pow_sub_one_eq_prod ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 X ^ n - 1 = \u220f \u03b6 in nthRootsFinset n R, (X - \u2191C \u03b6) ** 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 ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 X ^ n - 1 = \u220f \u03b6 in nthRootsFinset n R, (X - \u2191C \u03b6) ** rw [nthRootsFinset, \u2190 Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)] ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 X ^ n - 1 = \u220f \u03b6 in { val := nthRoots n 1, nodup := (_ : Multiset.Nodup (nthRoots n 1)) }, (X - \u2191C \u03b6) ** simp only [Finset.prod_mk, RingHom.map_one] ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (nthRoots n 1)) ** rw [nthRoots] ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n \u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1))) ** have hmonic : (X ^ n - C (1 : R)).Monic := monic_X_pow_sub_C (1 : R) (ne_of_lt hpos).symm ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n hmonic : Monic (X ^ n - \u2191C 1) \u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1))) ** symm ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n hmonic : Monic (X ^ n - \u2191C 1) \u22a2 Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1))) = X ^ n - 1 ** apply prod_multiset_X_sub_C_of_monic_of_roots_card_eq hmonic ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n hmonic : Monic (X ^ n - \u2191C 1) \u22a2 \u2191Multiset.card (roots (X ^ n - \u2191C 1)) = natDegree (X ^ n - \u2191C 1) ** rw [@natDegree_X_pow_sub_C R _ _ n 1, \u2190 nthRoots] ** R : Type u_1 inst\u271d\u00b9 : CommRing R inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : 0 < n h : IsPrimitiveRoot \u03b6 n hmonic : Monic (X ^ n - \u2191C 1) \u22a2 \u2191Multiset.card (nthRoots n 1) = n ** exact IsPrimitiveRoot.card_nthRoots h ** Qed", + "informal": "" + }, + { + "formal": "Finset.mem_image\u2082 ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 \u22a2 c \u2208 image\u2082 f s t \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 t \u2227 f a b = c ** simp [image\u2082, and_assoc] ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Perm.cycleOf_apply_apply_self ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 f\u271d g : Perm \u03b1 x\u271d y : \u03b1 f : Perm \u03b1 x : \u03b1 \u22a2 \u2191(cycleOf f x) (\u2191f x) = \u2191f (\u2191f x) ** convert cycleOf_apply_apply_pow_self f x 1 using 1 ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.condexp_undef ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f \u22a2 \u03bc[f|m] = 0 ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : m \u2264 m0 \u22a2 \u03bc[f|m] = 0 case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] = 0 ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : m \u2264 m0 \u22a2 \u03bc[f|m] = 0 ** by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm) ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] = 0 case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] = 0 ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] = 0 ** haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] = 0 ** rw [condexp_of_sigmaFinite, if_neg hf] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] = 0 ** rw [condexp_of_not_le hm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hf : \u00acIntegrable f hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] = 0 ** rw [condexp_of_not_sigmaFinite hm h\u03bcm] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.Chebyshev.T_mul ** R : Type u_1 S : Type u_2 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S \u22a2 \u2200 (n : \u2115), T R (0 * n) = comp (T R 0) (T R n) ** simp ** R : Type u_1 S : Type u_2 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S \u22a2 \u2200 (n : \u2115), T R (1 * n) = comp (T R 1) (T R n) ** simp ** R : Type u_1 S : Type u_2 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S m : \u2115 \u22a2 \u2200 (n : \u2115), T R ((m + 2) * n) = comp (T R (m + 2)) (T R n) ** intro n ** R : Type u_1 S : Type u_2 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S m n : \u2115 \u22a2 T R ((m + 2) * n) = comp (T R (m + 2)) (T R n) ** 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 ** R : Type u_1 S : Type u_2 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S m n : \u2115 this : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) \u22a2 T R ((m + 2) * n) = comp (T R (m + 2)) (T R n) ** simp [this, T_mul m, \u2190 T_mul (m + 1)] ** R : Type u_1 S : Type u_2 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S m n : \u2115 \u22a2 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) ** convert mul_T R n (m * n) using 1 <;> ring_nf ** Qed", + "informal": "" + }, + { + "formal": "Finset.not_mem_of_mem_powerset_of_not_mem ** \u03b1 : Type u_1 s\u271d t\u271d s t : Finset \u03b1 a : \u03b1 ht : t \u2208 powerset s h : \u00aca \u2208 s \u22a2 \u00aca \u2208 t ** apply mt _ h ** \u03b1 : Type u_1 s\u271d t\u271d s t : Finset \u03b1 a : \u03b1 ht : t \u2208 powerset s h : \u00aca \u2208 s \u22a2 a \u2208 t \u2192 a \u2208 s ** apply mem_powerset.1 ht ** Qed", + "informal": "" + }, + { + "formal": "Finset.sup'_le ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : SemilatticeSup \u03b1 s : Finset \u03b2 H : Finset.Nonempty s f : \u03b2 \u2192 \u03b1 a : \u03b1 hs : \u2200 (b : \u03b2), b \u2208 s \u2192 f b \u2264 a \u22a2 sup' s H f \u2264 a ** rw [\u2190 WithBot.coe_le_coe, coe_sup'] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : SemilatticeSup \u03b1 s : Finset \u03b2 H : Finset.Nonempty s f : \u03b2 \u2192 \u03b1 a : \u03b1 hs : \u2200 (b : \u03b2), b \u2208 s \u2192 f b \u2264 a \u22a2 sup s (WithBot.some \u2218 f) \u2264 \u2191a ** exact Finset.sup_le fun b h => WithBot.coe_le_coe.2 <| hs b h ** Qed", + "informal": "" + }, + { + "formal": "Set.image_prod_mk_subset_prod_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 a : \u03b1 b : \u03b2 ha : a \u2208 s \u22a2 Prod.mk a '' t \u2286 s \u00d7\u02e2 t ** rintro _ \u27e8b, hb, rfl\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 a : \u03b1 b\u271d : \u03b2 ha : a \u2208 s b : \u03b2 hb : b \u2208 t \u22a2 (a, b) \u2208 s \u00d7\u02e2 t ** exact \u27e8ha, hb\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Vector.ne_cons_iff ** n : \u2115 \u03b1 : Type u_1 a : \u03b1 v : Vector \u03b1 (Nat.succ n) v' : Vector \u03b1 n \u22a2 v \u2260 a ::\u1d65 v' \u2194 head v \u2260 a \u2228 tail v \u2260 v' ** rw [Ne.def, eq_cons_iff a v v', not_and_or] ** Qed", + "informal": "" + }, + { + "formal": "Filter.lift_principal2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 f\u271d f\u2081 f\u2082 : Filter \u03b1 g g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2 f : Filter \u03b1 s : Set \u03b1 hs : s \u2208 f \u22a2 f \u2264 \ud835\udcdf s ** simp only [hs, le_principal_iff] ** Qed", + "informal": "" + }, + { + "formal": "strictMono_iff_map_pos ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 inst\u271d\u00b3 : OrderedAddCommGroup \u03b1 inst\u271d\u00b2 : OrderedAddCommMonoid \u03b2 inst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2 f : F inst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1 \u22a2 StrictMono \u2191f \u2194 \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a ** refine \u27e8fun h a => ?_, fun h a b hl => ?_\u27e9 ** case refine_1 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 inst\u271d\u00b3 : OrderedAddCommGroup \u03b1 inst\u271d\u00b2 : OrderedAddCommMonoid \u03b2 inst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2 f : F inst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1 h : StrictMono \u2191f a : \u03b1 \u22a2 0 < a \u2192 0 < \u2191f a ** rw [\u2190 map_zero f] ** case refine_1 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 inst\u271d\u00b3 : OrderedAddCommGroup \u03b1 inst\u271d\u00b2 : OrderedAddCommMonoid \u03b2 inst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2 f : F inst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1 h : StrictMono \u2191f a : \u03b1 \u22a2 0 < a \u2192 \u2191f 0 < \u2191f a ** apply h ** case refine_2 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 inst\u271d\u00b3 : OrderedAddCommGroup \u03b1 inst\u271d\u00b2 : OrderedAddCommMonoid \u03b2 inst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2 f : F inst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1 h : \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a a b : \u03b1 hl : a < b \u22a2 \u2191f a < \u2191f b ** rw [\u2190 sub_add_cancel b a, map_add f] ** case refine_2 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 inst\u271d\u00b3 : OrderedAddCommGroup \u03b1 inst\u271d\u00b2 : OrderedAddCommMonoid \u03b2 inst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2 f : F inst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1 h : \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a a b : \u03b1 hl : a < b \u22a2 \u2191f a < \u2191f (b - a) + \u2191f a ** exact lt_add_of_pos_left _ (h _ <| sub_pos.2 hl) ** Qed", + "informal": "" + }, + { + "formal": "exists_dual_vector' ** \ud835\udd5c : Type v inst\u271d\u00b3 : IsROrC \ud835\udd5c E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : Nontrivial E x : E \u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016 ** by_cases hx : x = 0 ** case pos \ud835\udd5c : Type v inst\u271d\u00b3 : IsROrC \ud835\udd5c E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : Nontrivial E x : E hx : x = 0 \u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016 ** obtain \u27e8y, hy\u27e9 := exists_ne (0 : E) ** case pos.intro \ud835\udd5c : Type v inst\u271d\u00b3 : IsROrC \ud835\udd5c E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : Nontrivial E x : E hx : x = 0 y : E hy : y \u2260 0 \u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016 ** 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 ** case pos.intro.intro \ud835\udd5c : Type v inst\u271d\u00b3 : IsROrC \ud835\udd5c E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : Nontrivial E x : E hx : x = 0 y : E hy : y \u2260 0 g : E \u2192L[\ud835\udd5c] \ud835\udd5c hg : \u2016g\u2016 = 1 \u2227 \u2191g y = \u2191\u2016y\u2016 \u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016 ** refine' \u27e8g, hg.left, _\u27e9 ** case pos.intro.intro \ud835\udd5c : Type v inst\u271d\u00b3 : IsROrC \ud835\udd5c E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : Nontrivial E x : E hx : x = 0 y : E hy : y \u2260 0 g : E \u2192L[\ud835\udd5c] \ud835\udd5c hg : \u2016g\u2016 = 1 \u2227 \u2191g y = \u2191\u2016y\u2016 \u22a2 \u2191g x = \u2191\u2016x\u2016 ** simp [hx] ** case neg \ud835\udd5c : Type v inst\u271d\u00b3 : IsROrC \ud835\udd5c E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : Nontrivial E x : E hx : \u00acx = 0 \u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016 ** exact exists_dual_vector \ud835\udd5c x hx ** Qed", + "informal": "" + }, + { + "formal": "div_lt_self_iff ** \u03b1 : Type u inst\u271d\u00b3 : Group \u03b1 inst\u271d\u00b2 : LT \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1 a\u271d b\u271d c d a b : \u03b1 \u22a2 a / b < a \u2194 1 < b ** simp [div_eq_mul_inv] ** Qed", + "informal": "" + }, + { + "formal": "sup_sdiff_symmDiff ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03c0 : \u03b9 \u2192 Type u_4 inst\u271d : GeneralizedBooleanAlgebra \u03b1 a b c d : \u03b1 \u22a2 (a \u2294 b) \\ (a \u2293 b) = a \u2206 b ** rw [symmDiff_eq_sup_sdiff_inf] ** Qed", + "informal": "" + }, + { + "formal": "FiniteDimensional.lt_rank_of_lt_finrank ** K : Type u V : Type v inst\u271d\u2074 : Ring K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V V\u2082 : Type v' inst\u271d\u00b9 : AddCommGroup V\u2082 inst\u271d : Module K V\u2082 n : \u2115 h : n < finrank K V \u22a2 \u2191n < Module.rank K V ** rwa [\u2190 Cardinal.toNat_lt_iff_lt_of_lt_aleph0, toNat_cast] ** case hc K : Type u V : Type v inst\u271d\u2074 : Ring K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V V\u2082 : Type v' inst\u271d\u00b9 : AddCommGroup V\u2082 inst\u271d : Module K V\u2082 n : \u2115 h : n < finrank K V \u22a2 \u2191n < \u2135\u2080 ** exact nat_lt_aleph0 n ** case hd K : Type u V : Type v inst\u271d\u2074 : Ring K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V V\u2082 : Type v' inst\u271d\u00b9 : AddCommGroup V\u2082 inst\u271d : Module K V\u2082 n : \u2115 h : n < finrank K V \u22a2 Module.rank K V < \u2135\u2080 ** contrapose! h ** case hd K : Type u V : Type v inst\u271d\u2074 : Ring K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V V\u2082 : Type v' inst\u271d\u00b9 : AddCommGroup V\u2082 inst\u271d : Module K V\u2082 n : \u2115 h : \u2135\u2080 \u2264 Module.rank K V \u22a2 finrank K V \u2264 n ** rw [finrank, Cardinal.toNat_apply_of_aleph0_le h] ** case hd K : Type u V : Type v inst\u271d\u2074 : Ring K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V V\u2082 : Type v' inst\u271d\u00b9 : AddCommGroup V\u2082 inst\u271d : Module K V\u2082 n : \u2115 h : \u2135\u2080 \u2264 Module.rank K V \u22a2 0 \u2264 n ** exact n.zero_le ** Qed", + "informal": "" + }, + { + "formal": "ciSup_mono ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 inst\u271d : ConditionallyCompleteLattice \u03b1 s t : Set \u03b1 a b : \u03b1 f g : \u03b9 \u2192 \u03b1 B : BddAbove (range g) H : \u2200 (x : \u03b9), f x \u2264 g x \u22a2 iSup f \u2264 iSup g ** cases isEmpty_or_nonempty \u03b9 ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 inst\u271d : ConditionallyCompleteLattice \u03b1 s t : Set \u03b1 a b : \u03b1 f g : \u03b9 \u2192 \u03b1 B : BddAbove (range g) H : \u2200 (x : \u03b9), f x \u2264 g x h\u271d : IsEmpty \u03b9 \u22a2 iSup f \u2264 iSup g ** rw [iSup_of_empty', iSup_of_empty'] ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 inst\u271d : ConditionallyCompleteLattice \u03b1 s t : Set \u03b1 a b : \u03b1 f g : \u03b9 \u2192 \u03b1 B : BddAbove (range g) H : \u2200 (x : \u03b9), f x \u2264 g x h\u271d : Nonempty \u03b9 \u22a2 iSup f \u2264 iSup g ** exact ciSup_le fun x => le_ciSup_of_le B x (H x) ** Qed", + "informal": "" + }, + { + "formal": "le_nhds_of_cauchy_adhp_aux ** \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 f : Filter \u03b1 x : \u03b1 adhs : \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 \u22a2 f \u2264 \ud835\udcdd x ** intro s hs ** \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 f : Filter \u03b1 x : \u03b1 adhs : \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 s : Set \u03b1 hs : s \u2208 \ud835\udcdd x \u22a2 s \u2208 f ** rcases comp_mem_uniformity_sets (mem_nhds_uniformity_iff_right.1 hs) with \u27e8U, U_mem, hU\u27e9 ** case intro.intro \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 f : Filter \u03b1 x : \u03b1 adhs : \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 s : Set \u03b1 hs : s \u2208 \ud835\udcdd x U : Set (\u03b1 \u00d7 \u03b1) U_mem : U \u2208 \ud835\udce4 \u03b1 hU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s} \u22a2 s \u2208 f ** rcases adhs U U_mem with \u27e8t, t_mem, ht, y, hxy, hy\u27e9 ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 f : Filter \u03b1 x : \u03b1 adhs : \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 s : Set \u03b1 hs : s \u2208 \ud835\udcdd x U : Set (\u03b1 \u00d7 \u03b1) U_mem : U \u2208 \ud835\udce4 \u03b1 hU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s} t : Set \u03b1 t_mem : t \u2208 f ht : t \u00d7\u02e2 t \u2286 U y : \u03b1 hxy : (x, y) \u2208 U hy : y \u2208 t \u22a2 s \u2208 f ** apply mem_of_superset t_mem ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v uniformSpace : UniformSpace \u03b1 f : Filter \u03b1 x : \u03b1 adhs : \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 s : Set \u03b1 hs : s \u2208 \ud835\udcdd x U : Set (\u03b1 \u00d7 \u03b1) U_mem : U \u2208 \ud835\udce4 \u03b1 hU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s} t : Set \u03b1 t_mem : t \u2208 f ht : t \u00d7\u02e2 t \u2286 U y : \u03b1 hxy : (x, y) \u2208 U hy : y \u2208 t \u22a2 t \u2286 s ** exact fun z hz => hU (prod_mk_mem_compRel hxy (ht <| mk_mem_prod hy hz)) rfl ** Qed", + "informal": "" + }, + { + "formal": "Traversable.foldrm_toList ** \u03b1 \u03b2 \u03b3 : Type u t : Type u \u2192 Type u inst\u271d\u00b3 : Traversable t inst\u271d\u00b2 : LawfulTraversable t m : Type u \u2192 Type u inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2 x : \u03b2 xs : t \u03b1 \u22a2 foldrm f x xs = List.foldrM f x (toList xs) ** change _ = foldrM.ofFreeMonoid f (FreeMonoid.ofList <| toList xs) x ** \u03b1 \u03b2 \u03b3 : Type u t : Type u \u2192 Type u inst\u271d\u00b3 : Traversable t inst\u271d\u00b2 : LawfulTraversable t m : Type u \u2192 Type u inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2 x : \u03b2 xs : t \u03b1 \u22a2 foldrm f x xs = \u2191(foldrM.ofFreeMonoid f) (\u2191FreeMonoid.ofList (toList xs)) x ** simp only [foldrm, toList_spec, foldMap_hom_free (foldrM.ofFreeMonoid f),\n foldrm.ofFreeMonoid_comp_of, foldrM.get, FreeMonoid.ofList_toList] ** Qed", + "informal": "" + }, + { + "formal": "Nat.clog_anti_left ** b c n : \u2115 hc : 1 < c hb : c \u2264 b \u22a2 clog b n \u2264 clog c n ** rw [\u2190 le_pow_iff_clog_le (lt_of_lt_of_le hc hb)] ** b c n : \u2115 hc : 1 < c hb : c \u2264 b \u22a2 n \u2264 b ^ clog c n ** 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 _ ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.card_le_iff ** \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop o : Ordinal.{u_3} c : Cardinal.{u_3} \u22a2 card o \u2264 c \u2194 o < ord (succ c) ** rw [lt_ord, lt_succ_iff] ** Qed", + "informal": "" + }, + { + "formal": "FreeGroup.lift.range_le ** \u03b1 : Type u L L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) \u03b2 : Type v inst\u271d : Group \u03b2 f : \u03b1 \u2192 \u03b2 x y : FreeGroup \u03b1 s : Subgroup \u03b2 H : Set.range f \u2286 \u2191s \u22a2 MonoidHom.range (\u2191lift f) \u2264 s ** rintro _ \u27e8\u27e8L\u27e9, rfl\u27e9 ** case intro.mk \u03b1 : Type u L\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) \u03b2 : Type v inst\u271d : Group \u03b2 f : \u03b1 \u2192 \u03b2 x y : FreeGroup \u03b1 s : Subgroup \u03b2 H : Set.range f \u2286 \u2191s w\u271d : FreeGroup \u03b1 L : List (\u03b1 \u00d7 Bool) \u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step L) \u2208 s ** 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) ** \u03b1 : Type u L\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) \u03b2 : Type v inst\u271d : Group \u03b2 f : \u03b1 \u2192 \u03b2 x\u271d\u00b9 y : FreeGroup \u03b1 s : Subgroup \u03b2 H : Set.range f \u2286 \u2191s w\u271d : FreeGroup \u03b1 L : List (\u03b1 \u00d7 Bool) x\u271d : \u03b1 \u00d7 Bool tl : List (\u03b1 \u00d7 Bool) ih : \u2191(\u2191lift f) (Quot.mk Red.Step tl) \u2208 s x : \u03b1 b : Bool \u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step ((x, false) :: tl)) \u2208 s ** simp at ih \u22a2 ** \u03b1 : Type u L\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) \u03b2 : Type v inst\u271d : Group \u03b2 f : \u03b1 \u2192 \u03b2 x\u271d\u00b9 y : FreeGroup \u03b1 s : Subgroup \u03b2 H : Set.range f \u2286 \u2191s w\u271d : FreeGroup \u03b1 L : List (\u03b1 \u00d7 Bool) x\u271d : \u03b1 \u00d7 Bool tl : List (\u03b1 \u00d7 Bool) ih : List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s x : \u03b1 b : Bool \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 ** exact s.mul_mem (s.inv_mem <| H \u27e8x, rfl\u27e9) ih ** \u03b1 : Type u L\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) \u03b2 : Type v inst\u271d : Group \u03b2 f : \u03b1 \u2192 \u03b2 x\u271d\u00b9 y : FreeGroup \u03b1 s : Subgroup \u03b2 H : Set.range f \u2286 \u2191s w\u271d : FreeGroup \u03b1 L : List (\u03b1 \u00d7 Bool) x\u271d : \u03b1 \u00d7 Bool tl : List (\u03b1 \u00d7 Bool) ih : \u2191(\u2191lift f) (Quot.mk Red.Step tl) \u2208 s x : \u03b1 b : Bool \u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step ((x, true) :: tl)) \u2208 s ** simp at ih \u22a2 ** \u03b1 : Type u L\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) \u03b2 : Type v inst\u271d : Group \u03b2 f : \u03b1 \u2192 \u03b2 x\u271d\u00b9 y : FreeGroup \u03b1 s : Subgroup \u03b2 H : Set.range f \u2286 \u2191s w\u271d : FreeGroup \u03b1 L : List (\u03b1 \u00d7 Bool) x\u271d : \u03b1 \u00d7 Bool tl : List (\u03b1 \u00d7 Bool) ih : List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s x : \u03b1 b : Bool \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 ** exact s.mul_mem (H \u27e8x, rfl\u27e9) ih ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.aeval_algebraMap_apply ** R : Type u_1 A : Type u_2 B : Type u_3 inst\u271d\u2076 : CommSemiring R inst\u271d\u2075 : CommSemiring A inst\u271d\u2074 : Semiring B inst\u271d\u00b3 : Algebra R A inst\u271d\u00b2 : Algebra A B inst\u271d\u00b9 : Algebra R B inst\u271d : IsScalarTower R A B x : A p : R[X] \u22a2 \u2191(aeval (\u2191(algebraMap A B) x)) p = \u2191(algebraMap A B) (\u2191(aeval x) p) ** rw [aeval_def, aeval_def, hom_eval\u2082, \u2190 IsScalarTower.algebraMap_eq] ** Qed", + "informal": "" + }, + { + "formal": "WithLp.prod_antilipschitzWith_equiv_aux ** p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 \u22a2 AntilipschitzWith (2 ^ ENNReal.toReal (1 / p)) \u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) ** intro x y ** p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) \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) ** rcases p.dichotomy with (rfl | h) ** case inl \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 hp : Fact (1 \u2264 \u22a4) x y : WithLp \u22a4 (\u03b1 \u00d7 \u03b2) \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) ** simp [edist] ** case inr p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p \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) ** have pos : 0 < p.toReal := by positivity ** case inr p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p \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) ** have nonneg : 0 \u2264 1 / p.toReal := by positivity ** case inr p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p \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) ** have cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel' 1 (ne_of_gt pos) ** case inr p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p cancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1 \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) ** rw [prod_edist_eq_add pos, ENNReal.toReal_div 1 p] ** case inr p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p cancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1 \u22a2 (edist x.1 y.1 ^ ENNReal.toReal p + edist x.2 y.2 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264 \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) ** simp only [edist, \u2190 one_div, ENNReal.one_toReal] ** p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p \u22a2 0 < ENNReal.toReal p ** positivity ** p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p \u22a2 0 \u2264 1 / ENNReal.toReal p ** positivity ** p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p cancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1 \u22a2 (edist x.1 y.1 ^ ENNReal.toReal p + edist x.2 y.2 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264 (edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p + edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) ** refine ENNReal.rpow_le_rpow (add_le_add ?_ ?_) nonneg ** case refine_1 p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p cancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1 \u22a2 edist x.1 y.1 ^ ENNReal.toReal p \u2264 edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p ** refine ENNReal.rpow_le_rpow ?_ (le_of_lt pos) ** case refine_1 p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p cancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1 \u22a2 edist x.1 y.1 \u2264 edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ** simp [edist] ** case refine_2 p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p cancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1 \u22a2 edist x.2 y.2 ^ ENNReal.toReal p \u2264 edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p ** refine ENNReal.rpow_le_rpow ?_ (le_of_lt pos) ** case refine_2 p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p cancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1 \u22a2 edist x.2 y.2 \u2264 edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ** simp [edist] ** p : \u211d\u22650\u221e \ud835\udd5c : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 hp : Fact (1 \u2264 p) inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 inst\u271d : PseudoEMetricSpace \u03b2 x y : WithLp p (\u03b1 \u00d7 \u03b2) h : 1 \u2264 ENNReal.toReal p pos : 0 < ENNReal.toReal p nonneg : 0 \u2264 1 / ENNReal.toReal p cancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1 \u22a2 (edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p + edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) = \u2191(2 ^ (1 / ENNReal.toReal p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "sdiff_sdiff_eq_self ** \u03b1 : Type u \u03b2 : Type u_1 w x y z : \u03b1 inst\u271d : GeneralizedBooleanAlgebra \u03b1 h : y \u2264 x \u22a2 x \\ (x \\ y) = y ** rw [sdiff_sdiff_right_self, inf_of_le_right h] ** Qed", + "informal": "" + }, + { + "formal": "LinearPMap.IsFormalAdjoint.le_adjoint ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F T : E \u2192\u2097.[\ud835\udd5c] F S : F \u2192\u2097.[\ud835\udd5c] E hT : Dense \u2191T.domain inst\u271d : CompleteSpace E h : IsFormalAdjoint T S x\u271d\u00b2 : { x // x \u2208 S.domain } x\u271d\u00b9 : { x // x \u2208 T\u2020.domain } hxy : \u2191x\u271d\u00b2 = \u2191x\u271d\u00b9 x\u271d : { x // x \u2208 T.domain } \u22a2 inner (\u2191S x\u271d\u00b2) \u2191x\u271d = inner (\u2191x\u271d\u00b9) (\u2191T x\u271d) ** rw [h.symm, hxy] ** Qed", + "informal": "" + }, + { + "formal": "EuclideanDomain.gcd_val ** R : Type u inst\u271d\u00b9 : EuclideanDomain R inst\u271d : DecidableEq R a b : R \u22a2 gcd a b = gcd (b % a) a ** rw [gcd] ** R : Type u inst\u271d\u00b9 : EuclideanDomain R inst\u271d : DecidableEq R a b : R \u22a2 (if a0 : a = 0 then b else let_fun x := (_ : EuclideanDomain.r (b % a) a); gcd (b % a) a) = gcd (b % a) a ** split_ifs with h <;> [simp only [h, mod_zero, gcd_zero_right]; rfl] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.lt_ofReal_iff_toReal_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 a\u271d b\u271d c d : \u211d\u22650\u221e r p q : \u211d\u22650 a : \u211d\u22650\u221e b : \u211d ha : a \u2260 \u22a4 \u22a2 a < ENNReal.ofReal b \u2194 ENNReal.toReal a < b ** lift a to \u211d\u22650 using ha ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 a\u271d b\u271d c d : \u211d\u22650\u221e r p q : \u211d\u22650 b : \u211d a : \u211d\u22650 \u22a2 \u2191a < ENNReal.ofReal b \u2194 ENNReal.toReal \u2191a < b ** simpa [ENNReal.ofReal, ENNReal.toReal] using Real.lt_toNNReal_iff_coe_lt ** Qed", + "informal": "" + }, + { + "formal": "GaussianInt.normSq_le_normSq_of_re_le_of_im_le ** x y : \u2102 hre : |x.re| \u2264 |y.re| him : |x.im| \u2264 |y.im| \u22a2 \u2191normSq x \u2264 \u2191normSq y ** 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] ** x y : \u2102 hre : |x.re| \u2264 |y.re| him : |x.im| \u2264 |y.im| \u22a2 |x.re| * |x.re| + |x.im| * |x.im| \u2264 |y.re| * |y.re| + |y.im| * |y.im| ** exact\n add_le_add (mul_self_le_mul_self (abs_nonneg _) hre) (mul_self_le_mul_self (abs_nonneg _) him) ** Qed", + "informal": "" + }, + { + "formal": "isOpenMap_quotient_mk'_mul ** M : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u0393 : Type u_4 inst\u271d\u00b3 : Group \u0393 T : Type u_5 inst\u271d\u00b2 : TopologicalSpace T inst\u271d\u00b9 : MulAction \u0393 T inst\u271d : ContinuousConstSMul \u0393 T U : Set T hU : IsOpen U \u22a2 IsOpen (Quotient.mk' '' U) ** rw [isOpen_coinduced, MulAction.quotient_preimage_image_eq_union_mul U] ** M : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u0393 : Type u_4 inst\u271d\u00b3 : Group \u0393 T : Type u_5 inst\u271d\u00b2 : TopologicalSpace T inst\u271d\u00b9 : MulAction \u0393 T inst\u271d : ContinuousConstSMul \u0393 T U : Set T hU : IsOpen U \u22a2 IsOpen (\u22c3 g, (fun x x_1 => x \u2022 x_1) g '' U) ** exact isOpen_iUnion fun \u03b3 => isOpenMap_smul \u03b3 U hU ** Qed", + "informal": "" + }, + { + "formal": "AddSubmonoid.iSup_eq_mrange_dfinsupp_sumAddHom ** \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d : AddCommMonoid \u03b3 S : \u03b9 \u2192 AddSubmonoid \u03b3 \u22a2 iSup S = AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i)) ** apply le_antisymm ** case a \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d : AddCommMonoid \u03b3 S : \u03b9 \u2192 AddSubmonoid \u03b3 \u22a2 iSup S \u2264 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i)) ** apply iSup_le _ ** \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d : AddCommMonoid \u03b3 S : \u03b9 \u2192 AddSubmonoid \u03b3 \u22a2 \u2200 (i : \u03b9), S i \u2264 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i)) ** intro i y hy ** \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d : AddCommMonoid \u03b3 S : \u03b9 \u2192 AddSubmonoid \u03b3 i : \u03b9 y : \u03b3 hy : y \u2208 S i \u22a2 y \u2208 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i)) ** exact \u27e8DFinsupp.single i \u27e8y, hy\u27e9, DFinsupp.sumAddHom_single _ _ _\u27e9 ** case a \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d : AddCommMonoid \u03b3 S : \u03b9 \u2192 AddSubmonoid \u03b3 \u22a2 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i)) \u2264 iSup S ** rintro x \u27e8v, rfl\u27e9 ** case a.intro \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d : AddCommMonoid \u03b3 S : \u03b9 \u2192 AddSubmonoid \u03b3 v : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i } \u22a2 \u2191(sumAddHom fun i => AddSubmonoid.subtype (S i)) v \u2208 iSup S ** exact dfinsupp_sumAddHom_mem _ v _ fun i _ => (le_iSup S i : S i \u2264 _) (v i).prop ** Qed", + "informal": "" + }, + { + "formal": "Real.isBounded_iff_bddBelow_bddAbove ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Set \u211d bdd : IsBounded s \u22a2 BddBelow s \u2227 BddAbove s ** obtain \u27e8r, hr\u27e9 : \u2203 r : \u211d, s \u2286 Icc (-r) r := by\n simpa [Real.closedBall_eq_Icc] using bdd.subset_closedBall 0 ** case intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Set \u211d bdd : IsBounded s r : \u211d hr : s \u2286 Icc (-r) r \u22a2 BddBelow s \u2227 BddAbove s ** exact \u27e8bddBelow_Icc.mono hr, bddAbove_Icc.mono hr\u27e9 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Set \u211d bdd : IsBounded s \u22a2 \u2203 r, s \u2286 Icc (-r) r ** simpa [Real.closedBall_eq_Icc] using bdd.subset_closedBall 0 ** Qed", + "informal": "" + }, + { + "formal": "Tropical.add_eq_right ** R : Type u inst\u271d : LinearOrder R x y : Tropical R h : y \u2264 x \u22a2 untrop (x + y) = untrop y ** simpa using h ** Qed", + "informal": "" + }, + { + "formal": "AddChar.to_mulShift_inj_of_isPrimitive ** R : Type u inst\u271d\u00b9 : CommRing R R' : Type v inst\u271d : CommRing R' \u03c8 : AddChar R R' h\u03c8 : IsPrimitive \u03c8 \u22a2 Function.Injective (mulShift \u03c8) ** intro a b h ** R : Type u inst\u271d\u00b9 : CommRing R R' : Type v inst\u271d : CommRing R' \u03c8 : AddChar R R' h\u03c8 : IsPrimitive \u03c8 a b : R h : mulShift \u03c8 a = mulShift \u03c8 b \u22a2 a = b ** apply_fun fun x => x * mulShift \u03c8 (-b) at h ** R : Type u inst\u271d\u00b9 : CommRing R R' : Type v inst\u271d : CommRing R' \u03c8 : AddChar R R' h\u03c8 : IsPrimitive \u03c8 a b : R h : mulShift \u03c8 a * mulShift \u03c8 (-b) = mulShift \u03c8 b * mulShift \u03c8 (-b) \u22a2 a = b ** simp only [mulShift_mul, mulShift_zero, add_right_neg] at h ** R : Type u inst\u271d\u00b9 : CommRing R R' : Type v inst\u271d : CommRing R' \u03c8 : AddChar R R' h\u03c8 : IsPrimitive \u03c8 a b : R h : mulShift \u03c8 (a + -b) = 1 \u22a2 a = b ** have h\u2082 := h\u03c8 (a + -b) ** R : Type u inst\u271d\u00b9 : CommRing R R' : Type v inst\u271d : CommRing R' \u03c8 : AddChar R R' h\u03c8 : IsPrimitive \u03c8 a b : R h : mulShift \u03c8 (a + -b) = 1 h\u2082 : a + -b \u2260 0 \u2192 IsNontrivial (mulShift \u03c8 (a + -b)) \u22a2 a = b ** rw [h, isNontrivial_iff_ne_trivial, \u2190 sub_eq_add_neg, sub_ne_zero] at h\u2082 ** R : Type u inst\u271d\u00b9 : CommRing R R' : Type v inst\u271d : CommRing R' \u03c8 : AddChar R R' h\u03c8 : IsPrimitive \u03c8 a b : R h : mulShift \u03c8 (a + -b) = 1 h\u2082 : a \u2260 b \u2192 1 \u2260 1 \u22a2 a = b ** exact not_not.mp fun h => h\u2082 h rfl ** Qed", + "informal": "" + }, + { + "formal": "AffineEquiv.span_eq_top_iff ** k : Type u_1 V\u2081 : Type u_2 P\u2081 : Type u_3 V\u2082 : Type u_4 P\u2082 : Type u_5 V\u2083 : Type u_6 P\u2083 : Type u_7 inst\u271d\u2079 : Ring k inst\u271d\u2078 : AddCommGroup V\u2081 inst\u271d\u2077 : Module k V\u2081 inst\u271d\u2076 : AffineSpace V\u2081 P\u2081 inst\u271d\u2075 : AddCommGroup V\u2082 inst\u271d\u2074 : Module k V\u2082 inst\u271d\u00b3 : AffineSpace V\u2082 P\u2082 inst\u271d\u00b2 : AddCommGroup V\u2083 inst\u271d\u00b9 : Module k V\u2083 inst\u271d : AffineSpace V\u2083 P\u2083 f : P\u2081 \u2192\u1d43[k] P\u2082 s : Set P\u2081 e : P\u2081 \u2243\u1d43[k] P\u2082 \u22a2 affineSpan k s = \u22a4 \u2194 affineSpan k (\u2191e '' s) = \u22a4 ** refine' \u27e8(e : P\u2081 \u2192\u1d43[k] P\u2082).span_eq_top_of_surjective e.surjective, _\u27e9 ** k : Type u_1 V\u2081 : Type u_2 P\u2081 : Type u_3 V\u2082 : Type u_4 P\u2082 : Type u_5 V\u2083 : Type u_6 P\u2083 : Type u_7 inst\u271d\u2079 : Ring k inst\u271d\u2078 : AddCommGroup V\u2081 inst\u271d\u2077 : Module k V\u2081 inst\u271d\u2076 : AffineSpace V\u2081 P\u2081 inst\u271d\u2075 : AddCommGroup V\u2082 inst\u271d\u2074 : Module k V\u2082 inst\u271d\u00b3 : AffineSpace V\u2082 P\u2082 inst\u271d\u00b2 : AddCommGroup V\u2083 inst\u271d\u00b9 : Module k V\u2083 inst\u271d : AffineSpace V\u2083 P\u2083 f : P\u2081 \u2192\u1d43[k] P\u2082 s : Set P\u2081 e : P\u2081 \u2243\u1d43[k] P\u2082 \u22a2 affineSpan k (\u2191e '' s) = \u22a4 \u2192 affineSpan k s = \u22a4 ** intro h ** k : Type u_1 V\u2081 : Type u_2 P\u2081 : Type u_3 V\u2082 : Type u_4 P\u2082 : Type u_5 V\u2083 : Type u_6 P\u2083 : Type u_7 inst\u271d\u2079 : Ring k inst\u271d\u2078 : AddCommGroup V\u2081 inst\u271d\u2077 : Module k V\u2081 inst\u271d\u2076 : AffineSpace V\u2081 P\u2081 inst\u271d\u2075 : AddCommGroup V\u2082 inst\u271d\u2074 : Module k V\u2082 inst\u271d\u00b3 : AffineSpace V\u2082 P\u2082 inst\u271d\u00b2 : AddCommGroup V\u2083 inst\u271d\u00b9 : Module k V\u2083 inst\u271d : AffineSpace V\u2083 P\u2083 f : P\u2081 \u2192\u1d43[k] P\u2082 s : Set P\u2081 e : P\u2081 \u2243\u1d43[k] P\u2082 h : affineSpan k (\u2191e '' s) = \u22a4 \u22a2 affineSpan k s = \u22a4 ** have : s = e.symm '' (e '' s) := by rw [\u2190 image_comp]; simp ** k : Type u_1 V\u2081 : Type u_2 P\u2081 : Type u_3 V\u2082 : Type u_4 P\u2082 : Type u_5 V\u2083 : Type u_6 P\u2083 : Type u_7 inst\u271d\u2079 : Ring k inst\u271d\u2078 : AddCommGroup V\u2081 inst\u271d\u2077 : Module k V\u2081 inst\u271d\u2076 : AffineSpace V\u2081 P\u2081 inst\u271d\u2075 : AddCommGroup V\u2082 inst\u271d\u2074 : Module k V\u2082 inst\u271d\u00b3 : AffineSpace V\u2082 P\u2082 inst\u271d\u00b2 : AddCommGroup V\u2083 inst\u271d\u00b9 : Module k V\u2083 inst\u271d : AffineSpace V\u2083 P\u2083 f : P\u2081 \u2192\u1d43[k] P\u2082 s : Set P\u2081 e : P\u2081 \u2243\u1d43[k] P\u2082 h : affineSpan k (\u2191e '' s) = \u22a4 this : s = \u2191(symm e) '' (\u2191e '' s) \u22a2 affineSpan k s = \u22a4 ** rw [this] ** k : Type u_1 V\u2081 : Type u_2 P\u2081 : Type u_3 V\u2082 : Type u_4 P\u2082 : Type u_5 V\u2083 : Type u_6 P\u2083 : Type u_7 inst\u271d\u2079 : Ring k inst\u271d\u2078 : AddCommGroup V\u2081 inst\u271d\u2077 : Module k V\u2081 inst\u271d\u2076 : AffineSpace V\u2081 P\u2081 inst\u271d\u2075 : AddCommGroup V\u2082 inst\u271d\u2074 : Module k V\u2082 inst\u271d\u00b3 : AffineSpace V\u2082 P\u2082 inst\u271d\u00b2 : AddCommGroup V\u2083 inst\u271d\u00b9 : Module k V\u2083 inst\u271d : AffineSpace V\u2083 P\u2083 f : P\u2081 \u2192\u1d43[k] P\u2082 s : Set P\u2081 e : P\u2081 \u2243\u1d43[k] P\u2082 h : affineSpan k (\u2191e '' s) = \u22a4 this : s = \u2191(symm e) '' (\u2191e '' s) \u22a2 affineSpan k (\u2191(symm e) '' (\u2191e '' s)) = \u22a4 ** exact (e.symm : P\u2082 \u2192\u1d43[k] P\u2081).span_eq_top_of_surjective e.symm.surjective h ** k : Type u_1 V\u2081 : Type u_2 P\u2081 : Type u_3 V\u2082 : Type u_4 P\u2082 : Type u_5 V\u2083 : Type u_6 P\u2083 : Type u_7 inst\u271d\u2079 : Ring k inst\u271d\u2078 : AddCommGroup V\u2081 inst\u271d\u2077 : Module k V\u2081 inst\u271d\u2076 : AffineSpace V\u2081 P\u2081 inst\u271d\u2075 : AddCommGroup V\u2082 inst\u271d\u2074 : Module k V\u2082 inst\u271d\u00b3 : AffineSpace V\u2082 P\u2082 inst\u271d\u00b2 : AddCommGroup V\u2083 inst\u271d\u00b9 : Module k V\u2083 inst\u271d : AffineSpace V\u2083 P\u2083 f : P\u2081 \u2192\u1d43[k] P\u2082 s : Set P\u2081 e : P\u2081 \u2243\u1d43[k] P\u2082 h : affineSpan k (\u2191e '' s) = \u22a4 \u22a2 s = \u2191(symm e) '' (\u2191e '' s) ** rw [\u2190 image_comp] ** k : Type u_1 V\u2081 : Type u_2 P\u2081 : Type u_3 V\u2082 : Type u_4 P\u2082 : Type u_5 V\u2083 : Type u_6 P\u2083 : Type u_7 inst\u271d\u2079 : Ring k inst\u271d\u2078 : AddCommGroup V\u2081 inst\u271d\u2077 : Module k V\u2081 inst\u271d\u2076 : AffineSpace V\u2081 P\u2081 inst\u271d\u2075 : AddCommGroup V\u2082 inst\u271d\u2074 : Module k V\u2082 inst\u271d\u00b3 : AffineSpace V\u2082 P\u2082 inst\u271d\u00b2 : AddCommGroup V\u2083 inst\u271d\u00b9 : Module k V\u2083 inst\u271d : AffineSpace V\u2083 P\u2083 f : P\u2081 \u2192\u1d43[k] P\u2082 s : Set P\u2081 e : P\u2081 \u2243\u1d43[k] P\u2082 h : affineSpan k (\u2191e '' s) = \u22a4 \u22a2 s = \u2191(symm e) \u2218 \u2191e '' s ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Mem\u2112p.snorm_indicator_norm_ge_pos_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 M, 0 < M \u2227 snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8M, hM\u27e9 := hf.snorm_indicator_norm_ge_le \u03bc hmeas h\u03b5 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 M, 0 < M \u2227 snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine'\n \u27e8max M 1, lt_of_lt_of_le zero_lt_one (le_max_right _ _), le_trans (snorm_mono fun x => _) hM\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 x : \u03b1 \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 ** rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 x : \u03b1 \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 ** refine' Set.indicator_le_indicator_of_subset (fun x hx => _) (fun x => norm_nonneg (f x)) x ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 x\u271d x : \u03b1 hx : x \u2208 {x | max M 1 \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 x \u2208 {x | M \u2264 \u2191\u2016f x\u2016\u208a} ** rw [Set.mem_setOf_eq] at hx ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 x\u271d x : \u03b1 hx : max M 1 \u2264 \u2191\u2016f x\u2016\u208a \u22a2 x \u2208 {x | M \u2264 \u2191\u2016f x\u2016\u208a} ** exact (max_le_iff.1 hx).1 ** Qed", + "informal": "" + }, + { + "formal": "ONote.NFBelow.repr_lt ** o : ONote b : Ordinal.{0} h : NFBelow o b \u22a2 repr o < \u03c9 ^ b ** induction' h with _ e n a eb b h\u2081 h\u2082 h\u2083 _ IH ** case zero o : ONote b b\u271d : Ordinal.{0} \u22a2 repr 0 < \u03c9 ^ b\u271d ** exact opow_pos _ omega_pos ** case oadd' o : ONote b\u271d : Ordinal.{0} e : ONote n : \u2115+ a : ONote eb b : Ordinal.{0} h\u2081 : NFBelow e eb h\u2082 : NFBelow a (repr e) h\u2083 : repr e < b a_ih\u271d : repr e < \u03c9 ^ eb IH : repr a < \u03c9 ^ repr e \u22a2 repr (ONote.oadd e n a) < \u03c9 ^ b ** rw [repr] ** case oadd' o : ONote b\u271d : Ordinal.{0} e : ONote n : \u2115+ a : ONote eb b : Ordinal.{0} h\u2081 : NFBelow e eb h\u2082 : NFBelow a (repr e) h\u2083 : repr e < b a_ih\u271d : repr e < \u03c9 ^ eb IH : repr a < \u03c9 ^ repr e \u22a2 \u03c9 ^ repr e * \u2191\u2191n + repr a < \u03c9 ^ b ** apply ((add_lt_add_iff_left _).2 IH).trans_le ** case oadd' o : ONote b\u271d : Ordinal.{0} e : ONote n : \u2115+ a : ONote eb b : Ordinal.{0} h\u2081 : NFBelow e eb h\u2082 : NFBelow a (repr e) h\u2083 : repr e < b a_ih\u271d : repr e < \u03c9 ^ eb IH : repr a < \u03c9 ^ repr e \u22a2 \u03c9 ^ repr e * \u2191\u2191n + \u03c9 ^ repr e \u2264 \u03c9 ^ b ** rw [\u2190 mul_succ] ** case oadd' o : ONote b\u271d : Ordinal.{0} e : ONote n : \u2115+ a : ONote eb b : Ordinal.{0} h\u2081 : NFBelow e eb h\u2082 : NFBelow a (repr e) h\u2083 : repr e < b a_ih\u271d : repr e < \u03c9 ^ eb IH : repr a < \u03c9 ^ repr e \u22a2 \u03c9 ^ repr e * succ \u2191\u2191n \u2264 \u03c9 ^ b ** apply (mul_le_mul_left' (succ_le_of_lt (nat_lt_omega _)) _).trans ** case oadd' o : ONote b\u271d : Ordinal.{0} e : ONote n : \u2115+ a : ONote eb b : Ordinal.{0} h\u2081 : NFBelow e eb h\u2082 : NFBelow a (repr e) h\u2083 : repr e < b a_ih\u271d : repr e < \u03c9 ^ eb IH : repr a < \u03c9 ^ repr e \u22a2 \u03c9 ^ repr e * \u03c9 \u2264 \u03c9 ^ b ** rw [\u2190 opow_succ] ** case oadd' o : ONote b\u271d : Ordinal.{0} e : ONote n : \u2115+ a : ONote eb b : Ordinal.{0} h\u2081 : NFBelow e eb h\u2082 : NFBelow a (repr e) h\u2083 : repr e < b a_ih\u271d : repr e < \u03c9 ^ eb IH : repr a < \u03c9 ^ repr e \u22a2 \u03c9 ^ succ (repr e) \u2264 \u03c9 ^ b ** exact opow_le_opow_right omega_pos (succ_le_of_lt h\u2083) ** Qed", + "informal": "" + }, + { + "formal": "Set.sigmaToiUnion_surjective ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 t : \u03b1 \u2192 Set \u03b2 b : \u03b2 hb : b \u2208 \u22c3 i, t i \u22a2 \u2203 a, b \u2208 t a ** simpa using hb ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.vars_monomial_single ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r\u271d : R e\u271d : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S f : R \u2192+* S i : \u03c3 e : \u2115 r : R he : e \u2260 0 hr : r \u2260 0 \u22a2 vars (\u2191(monomial fun\u2080 | i => e) r) = {i} ** rw [vars_monomial hr, Finsupp.support_single_ne_zero _ he] ** Qed", + "informal": "" + }, + { + "formal": "jacobiSym.one_right ** a : \u2124 \u22a2 J(a | 1) = 1 ** simp only [jacobiSym, factors_one, List.prod_nil, List.pmap] ** Qed", + "informal": "" + }, + { + "formal": "HNNExtension.induction_on ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 \u22a2 motive x ** 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 _ } ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } \u22a2 motive x ** 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]) ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } f : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } := lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t } (_ : \u2200 (a : { x // x \u2208 A }), { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }) \u22a2 motive x ** have hf : S.subtype.comp f = MonoidHom.id _ :=\n hom_ext (by ext; simp) (by simp) ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } f : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } := lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t } (_ : \u2200 (a : { x // x \u2208 A }), { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }) hf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6) \u22a2 motive x ** show motive (MonoidHom.id _ x) ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } f : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } := lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t } (_ : \u2200 (a : { x // x \u2208 A }), { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }) hf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6) \u22a2 motive (\u2191(MonoidHom.id (HNNExtension G A B \u03c6)) x) ** rw [\u2190 hf] ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } f : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } := lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t } (_ : \u2200 (a : { x // x \u2208 A }), { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }) hf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6) \u22a2 motive (\u2191(MonoidHom.comp (Subgroup.subtype S) f) x) ** exact (f x).2 ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 \u22a2 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier ** simpa using of 1 ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } \u22a2 \u2200 (a : { x // x \u2208 A }), { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t } ** intro a ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } a : { x // x \u2208 A } \u22a2 { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t } ** ext ** case a G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } a : { x // x \u2208 A } \u22a2 \u2191({ val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a) = \u2191(\u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }) ** simp [equiv_eq_conj, mul_assoc] ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } f : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } := lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t } (_ : \u2200 (a : { x // x \u2208 A }), { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }) \u22a2 MonoidHom.comp (MonoidHom.comp (Subgroup.subtype S) f) HNNExtension.of = MonoidHom.comp (MonoidHom.id (HNNExtension G A B \u03c6)) HNNExtension.of ** ext ** case h G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } f : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } := lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t } (_ : \u2200 (a : { x // x \u2208 A }), { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }) x\u271d : G \u22a2 \u2191(MonoidHom.comp (MonoidHom.comp (Subgroup.subtype S) f) HNNExtension.of) x\u271d = \u2191(MonoidHom.comp (MonoidHom.id (HNNExtension G A B \u03c6)) HNNExtension.of) x\u271d ** simp ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M motive : HNNExtension G A B \u03c6 \u2192 Prop x : HNNExtension G A B \u03c6 of : \u2200 (g : G), motive (\u2191HNNExtension.of g) t : motive HNNExtension.t mul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y) inv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9 S : Subgroup (HNNExtension G A B \u03c6) := { toSubmonoid := { toSubsemigroup := { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }, one_mem' := (_ : 1 \u2208 { carrier := setOf motive, mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) }, inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) } f : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } := lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t } (_ : \u2200 (a : { x // x \u2208 A }), { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a = \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }) \u22a2 \u2191(MonoidHom.comp (Subgroup.subtype S) f) HNNExtension.t = \u2191(MonoidHom.id (HNNExtension G A B \u03c6)) HNNExtension.t ** simp ** Qed", + "informal": "" + }, + { + "formal": "Nat.card_of_isEmpty ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : IsEmpty \u03b1 \u22a2 Nat.card \u03b1 = 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Int.floor_sub_int ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedRing \u03b1 inst\u271d : FloorRing \u03b1 z\u271d : \u2124 a\u271d a : \u03b1 z : \u2124 \u22a2 \u230aa - \u2191z\u230b = \u230aa + \u2191(-z)\u230b ** rw [Int.cast_neg, sub_eq_add_neg] ** Qed", + "informal": "" + }, + { + "formal": "Finset.mem_convexHull ** R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s\u271d : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E s : Finset E x : E \u22a2 x \u2208 \u2191(convexHull R) \u2191s \u2194 \u2203 w x_1 x_2, centerMass s w id = x ** rw [Finset.convexHull_eq, Set.mem_setOf_eq] ** Qed", + "informal": "" + }, + { + "formal": "NNReal.coe_eq_one ** r : \u211d\u22650 \u22a2 \u2191r = 1 \u2194 r = 1 ** rw [\u2190 NNReal.coe_one, NNReal.coe_eq] ** Qed", + "informal": "" + }, + { + "formal": "mul_finprod_cond_ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) \u22a2 f a * \u220f\u1da0 (i : \u03b1) (_ : i \u2260 a), f i = \u220f\u1da0 (i : \u03b1), f i ** rw [finprod_eq_prod _ hf] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) \u22a2 f a * \u220f\u1da0 (i : \u03b1) (_ : i \u2260 a), f i = \u220f i in Finite.toFinset hf, f i ** 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 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) h : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}) \u22a2 f a * \u220f\u1da0 (i : \u03b1) (_ : i \u2260 a), f i = \u220f i in Finite.toFinset hf, f i ** rw [finprod_cond_eq_prod_of_cond_iff f (fun hx => h _ hx), Finset.sdiff_singleton_eq_erase] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) h : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}) \u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i ** by_cases ha : a \u2208 mulSupport f ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) \u22a2 \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}) ** intro x hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) x : \u03b1 hx : f x \u2260 1 \u22a2 x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a} ** rw [Finset.mem_sdiff, Finset.mem_singleton, Finite.mem_toFinset, mem_mulSupport] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) x : \u03b1 hx : f x \u2260 1 \u22a2 x \u2260 a \u2194 f x \u2260 1 \u2227 \u00acx = a ** exact \u27e8fun h => And.intro hx h, fun h => h.2\u27e9 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) h : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}) ha : a \u2208 mulSupport f \u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i ** apply Finset.mul_prod_erase _ _ ((Finite.mem_toFinset _).mpr ha) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) h : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}) ha : \u00aca \u2208 mulSupport f \u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i ** rw [mem_mulSupport, not_not] at ha ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) h : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}) ha : f a = 1 \u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i ** rw [ha, one_mul] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 G : Type u_4 M : Type u_5 N : Type u_6 inst\u271d\u00b9 : CommMonoid M inst\u271d : CommMonoid N f g : \u03b1 \u2192 M a\u271d b : \u03b1 s t : Set \u03b1 a : \u03b1 hf : Set.Finite (mulSupport f) h : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}) ha : f a = 1 \u22a2 \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i ** apply Finset.prod_erase _ ha ** Qed", + "informal": "" + }, + { + "formal": "Set.Ioi_add_bij ** M : Type u_1 inst\u271d\u00b9 : OrderedCancelAddCommMonoid M inst\u271d : ExistsAddOfLE M a b c d : M \u22a2 BijOn (fun x => x + d) (Ioi a) (Ioi (a + d)) ** refine'\n \u27e8fun x h => add_lt_add_right (mem_Ioi.mp h) _, fun _ _ _ _ h => add_right_cancel h, fun _ h =>\n _\u27e9 ** M : Type u_1 inst\u271d\u00b9 : OrderedCancelAddCommMonoid M inst\u271d : ExistsAddOfLE M a b c d x\u271d : M h : x\u271d \u2208 Ioi (a + d) \u22a2 x\u271d \u2208 (fun x => x + d) '' Ioi a ** obtain \u27e8c, rfl\u27e9 := exists_add_of_le (mem_Ioi.mp h).le ** case intro M : Type u_1 inst\u271d\u00b9 : OrderedCancelAddCommMonoid M inst\u271d : ExistsAddOfLE M a b c\u271d d c : M h : a + d + c \u2208 Ioi (a + d) \u22a2 a + d + c \u2208 (fun x => x + d) '' Ioi a ** rw [mem_Ioi, add_right_comm, add_lt_add_iff_right] at h ** case intro M : Type u_1 inst\u271d\u00b9 : OrderedCancelAddCommMonoid M inst\u271d : ExistsAddOfLE M a b c\u271d d c : M h : a < a + c \u22a2 a + d + c \u2208 (fun x => x + d) '' Ioi a ** exact \u27e8a + c, h, by rw [add_right_comm]\u27e9 ** M : Type u_1 inst\u271d\u00b9 : OrderedCancelAddCommMonoid M inst\u271d : ExistsAddOfLE M a b c\u271d d c : M h : a < a + c \u22a2 (fun x => x + d) (a + c) = a + d + c ** rw [add_right_comm] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.tensorLeftHomEquiv_symm_naturality ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : MonoidalCategory C X X' Y Y' Z : C inst\u271d : ExactPairing Y Y' f : X \u27f6 X' g : X' \u27f6 Y \u2297 Z \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 ** dsimp [tensorLeftHomEquiv] ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : MonoidalCategory C X X' Y Y' Z : C inst\u271d : ExactPairing Y Y' f : X \u27f6 X' g : X' \u27f6 Y \u2297 Z \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 = (\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 ** simp only [id_tensor_comp, Category.assoc] ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.single_apply_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : Zero M a\u271d a' : \u03b1 b\u271d : M a x : \u03b1 b : M \u22a2 \u2191(single a b) x = 0 \u2194 x = a \u2192 b = 0 ** simp [single_eq_set_indicator] ** Qed", + "informal": "" + }, + { + "formal": "PowerSeries.coeff_zero_eq_constantCoeff_apply ** R : Type u_1 inst\u271d : Semiring R \u03c6 : R\u27e6X\u27e7 \u22a2 \u2191(coeff R 0) \u03c6 = \u2191(constantCoeff R) \u03c6 ** rw [coeff_zero_eq_constantCoeff] ** Qed", + "informal": "" + }, + { + "formal": "Submonoid.mul_leftInvEquiv_symm ** M : Type u_1 inst\u271d : CommMonoid M S : Submonoid M hS : S \u2264 IsUnit.submonoid M x : { x // x \u2208 S } \u22a2 \u2191x * \u2191(\u2191(MulEquiv.symm (leftInvEquiv S hS)) x) = 1 ** convert S.leftInvEquiv_mul hS ((S.leftInvEquiv hS).symm x) ** case h.e'_2.h.e'_5.h.e'_3 M : Type u_1 inst\u271d : CommMonoid M S : Submonoid M hS : S \u2264 IsUnit.submonoid M x : { x // x \u2208 S } \u22a2 x = \u2191(leftInvEquiv S hS) (\u2191(MulEquiv.symm (leftInvEquiv S hS)) x) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Affine.Simplex.mongePoint_eq_of_range_eq ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P n : \u2115 s\u2081 s\u2082 : Simplex \u211d P n h : Set.range s\u2081.points = Set.range s\u2082.points \u22a2 mongePoint s\u2081 = mongePoint s\u2082 ** simp_rw [mongePoint_eq_smul_vsub_vadd_circumcenter, centroid_eq_of_range_eq h,\n circumcenter_eq_of_range_eq h] ** Qed", + "informal": "" + }, + { + "formal": "LaurentPolynomial.commute_T ** R : Type u_1 inst\u271d : Semiring R n : \u2124 f : R[T;T\u207b\u00b9] m : \u2124 a : R \u22a2 T n * (\u2191C a * T m) = \u2191C a * T m * T n ** rw [T, T, \u2190 single_eq_C, single_mul_single, single_mul_single, single_mul_single] ** R : Type u_1 inst\u271d : Semiring R n : \u2124 f : R[T;T\u207b\u00b9] m : \u2124 a : R \u22a2 AddMonoidAlgebra.single (n + (0 + m)) (1 * (a * 1)) = AddMonoidAlgebra.single (0 + m + n) (a * 1 * 1) ** simp [add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Submonoid.pow_smul_mem_closure_smul ** \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x : N hx : x \u2208 closure s \u22a2 \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s) ** refine' @closure_induction N _ s (fun x : N => \u2203 n : \u2115, r ^ n \u2022 x \u2208 closure (r \u2022 s)) _ hx _ _ _ ** case refine'_1 \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x : N hx : x \u2208 closure s \u22a2 \u2200 (x : N), x \u2208 s \u2192 (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) x ** intro x hx ** case refine'_1 \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x\u271d : N hx\u271d : x\u271d \u2208 closure s x : N hx : x \u2208 s \u22a2 \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s) ** exact \u27e81, subset_closure \u27e8_, hx, by rw [pow_one]\u27e9\u27e9 ** \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x\u271d : N hx\u271d : x\u271d \u2208 closure s x : N hx : x \u2208 s \u22a2 (fun x => r \u2022 x) x = r ^ 1 \u2022 x ** rw [pow_one] ** case refine'_2 \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x : N hx : x \u2208 closure s \u22a2 (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) 1 ** exact \u27e80, by simpa using one_mem _\u27e9 ** \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x : N hx : x \u2208 closure s \u22a2 r ^ 0 \u2022 1 \u2208 closure (r \u2022 s) ** simpa using one_mem _ ** case refine'_3 \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x : N hx : x \u2208 closure s \u22a2 \u2200 (x y : N), (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) x \u2192 (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) ** rintro x y \u27e8nx, hx\u27e9 \u27e8ny, hy\u27e9 ** case refine'_3.intro.intro \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x\u271d : N hx\u271d : x\u271d \u2208 closure s x y : N nx : \u2115 hx : r ^ nx \u2022 x \u2208 closure (r \u2022 s) ny : \u2115 hy : r ^ ny \u2022 y \u2208 closure (r \u2022 s) \u22a2 \u2203 n, r ^ n \u2022 (x * y) \u2208 closure (r \u2022 s) ** use ny + nx ** case h \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x\u271d : N hx\u271d : x\u271d \u2208 closure s x y : N nx : \u2115 hx : r ^ nx \u2022 x \u2208 closure (r \u2022 s) ny : \u2115 hy : r ^ ny \u2022 y \u2208 closure (r \u2022 s) \u22a2 r ^ (ny + nx) \u2022 (x * y) \u2208 closure (r \u2022 s) ** rw [pow_add, mul_smul, \u2190 smul_mul_assoc, mul_comm, \u2190 smul_mul_assoc] ** case h \u03b1 : Type u_1 G : Type u_2 M : Type u_3 R : Type u_4 A : Type u_5 inst\u271d\u2074 : Monoid M inst\u271d\u00b3 : AddMonoid A s\u271d t u : Set M N : Type u_6 inst\u271d\u00b2 : CommMonoid N inst\u271d\u00b9 : MulAction M N inst\u271d : IsScalarTower M N N r : M s : Set N x\u271d : N hx\u271d : x\u271d \u2208 closure s x y : N nx : \u2115 hx : r ^ nx \u2022 x \u2208 closure (r \u2022 s) ny : \u2115 hy : r ^ ny \u2022 y \u2208 closure (r \u2022 s) \u22a2 r ^ ny \u2022 y * r ^ nx \u2022 x \u2208 closure (r \u2022 s) ** exact mul_mem hy hx ** Qed", + "informal": "" + }, + { + "formal": "Part.union_get_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Union \u03b1 a b : Part \u03b1 hab : (a \u222a b).Dom \u22a2 get (a \u222a b) hab = get a (_ : a.Dom) \u222a get b (_ : b.Dom) ** simp [union_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Union \u03b1 a b : Part \u03b1 hab : (a \u222a b).Dom \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) = get a (_ : a.Dom) \u222a get b (_ : b.Dom) ** aesop ** Qed", + "informal": "" + }, + { + "formal": "Primrec.list_tail ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 l : List \u03b1 \u22a2 (List.casesOn (id l) [] fun b l_1 => (l, b, l_1).2.2) = List.tail l ** cases l <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "Set.range_dcomp ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i \u22a2 (range fun g i => f i (g i)) = pi univ fun i => range (f i) ** refine Subset.antisymm ?_ fun x hx => ?_ ** case refine_1 \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i \u22a2 (range fun g i => f i (g i)) \u2286 pi univ fun i => range (f i) ** rintro _ \u27e8x, rfl\u27e9 i - ** case refine_1.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i\u271d : \u03b9 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i x : (i : \u03b9) \u2192 \u03b1 i i : \u03b9 \u22a2 (fun g i => f i (g i)) x i \u2208 (fun i => range (f i)) i ** exact \u27e8x i, rfl\u27e9 ** case refine_2 \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i x : (i : \u03b9) \u2192 \u03b2 i hx : x \u2208 pi univ fun i => range (f i) \u22a2 x \u2208 range fun g i => f i (g i) ** choose y hy using hx ** case refine_2 \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i x : (i : \u03b9) \u2192 \u03b2 i y : (i : \u03b9) \u2192 i \u2208 univ \u2192 \u03b1 i hy : \u2200 (i : \u03b9) (a : i \u2208 univ), f i (y i a) = x i \u22a2 x \u2208 range fun g i => f i (g i) ** exact \u27e8fun i => y i trivial, funext fun i => hy i trivial\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "IntermediateField.coe_nat_mem ** K : Type u_1 L : Type u_2 L' : Type u_3 inst\u271d\u2074 : Field K inst\u271d\u00b3 : Field L inst\u271d\u00b2 : Field L' inst\u271d\u00b9 : Algebra K L inst\u271d : Algebra K L' S : IntermediateField K L n : \u2115 \u22a2 \u2191n \u2208 S ** simpa using coe_int_mem S n ** Qed", + "informal": "" + }, + { + "formal": "integral_pow ** a b : \u211d n : \u2115 \u22a2 \u222b (x : \u211d) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (\u2191n + 1) ** simpa only [\u2190 Int.ofNat_succ, zpow_ofNat] using integral_zpow (Or.inl (Int.coe_nat_nonneg n)) ** Qed", + "informal": "" + }, + { + "formal": "Matrix.det_eq_sum_mul_adjugate_row ** m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n \u22a2 det A = \u2211 j : n, A i j * adjugate A j i ** haveI : Nonempty n := \u27e8i\u27e9 ** m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n this : Nonempty n \u22a2 det A = \u2211 j : n, A i j * adjugate A j i ** obtain \u27e8n', hn'\u27e9 := Nat.exists_eq_succ_of_ne_zero (Fintype.card_ne_zero : Fintype.card n \u2260 0) ** case intro m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n this : Nonempty n n' : \u2115 hn' : Fintype.card n = Nat.succ n' \u22a2 det A = \u2211 j : n, A i j * adjugate A j i ** obtain \u27e8e\u27e9 := Fintype.truncEquivFinOfCardEq hn' ** case intro.mk m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n this : Nonempty n n' : \u2115 hn' : Fintype.card n = Nat.succ n' x\u271d : Trunc (n \u2243 Fin (Nat.succ n')) e : n \u2243 Fin (Nat.succ n') \u22a2 det A = \u2211 j : n, A i j * adjugate A j i ** let A' := reindex e e A ** case intro.mk m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n this : Nonempty n n' : \u2115 hn' : Fintype.card n = Nat.succ n' x\u271d : Trunc (n \u2243 Fin (Nat.succ n')) e : n \u2243 Fin (Nat.succ n') A' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A \u22a2 det A = \u2211 j : n, A i j * adjugate A j i ** 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 ** case intro.mk m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n this : Nonempty n n' : \u2115 hn' : Fintype.card n = Nat.succ n' x\u271d : Trunc (n \u2243 Fin (Nat.succ n')) e : n \u2243 Fin (Nat.succ n') A' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A \u22a2 det A' = \u2211 j : Fin (Nat.succ n'), A' (\u2191e i) j * adjugate A' j (\u2191e i) ** rw [det_succ_row A' (e i)] ** case intro.mk m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n this : Nonempty n n' : \u2115 hn' : Fintype.card n = Nat.succ n' x\u271d : Trunc (n \u2243 Fin (Nat.succ n')) e : n \u2243 Fin (Nat.succ n') A' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A \u22a2 \u2211 j : Fin (Nat.succ n'), (-1) ^ (\u2191(\u2191e i) + \u2191j) * A' (\u2191e i) j * det (submatrix A' (Fin.succAbove (\u2191e i)) (Fin.succAbove j)) = \u2211 j : Fin (Nat.succ n'), A' (\u2191e i) j * adjugate A' j (\u2191e i) ** simp_rw [mul_assoc, mul_left_comm _ (A' _ _), \u2190 adjugate_fin_succ_eq_det_submatrix] ** m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n this\u271d : Nonempty n n' : \u2115 hn' : Fintype.card n = Nat.succ n' x\u271d : Trunc (n \u2243 Fin (Nat.succ n')) e : n \u2243 Fin (Nat.succ n') A' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A this : det A' = \u2211 j : Fin (Nat.succ n'), A' (\u2191e i) j * adjugate A' j (\u2191e i) \u22a2 det A = \u2211 j : n, A i j * adjugate A j i ** simp_rw [det_reindex_self, adjugate_reindex, reindex_apply, submatrix_apply, \u2190 e.sum_comp,\n Equiv.symm_apply_apply] at this ** m : Type u n : Type v \u03b1 : Type w inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : Fintype m inst\u271d : CommRing \u03b1 A : Matrix n n \u03b1 i : n this\u271d : Nonempty n n' : \u2115 hn' : Fintype.card n = Nat.succ n' x\u271d : Trunc (n \u2243 Fin (Nat.succ n')) e : n \u2243 Fin (Nat.succ n') A' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A this : det A = \u2211 x : n, A i x * adjugate A x i \u22a2 det A = \u2211 j : n, A i j * adjugate A j i ** exact this ** Qed", + "informal": "" + }, + { + "formal": "dist_le_pi_dist ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : PseudoMetricSpace \u03b1 \u03c0 : \u03b2 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b2 inst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b) f g : (b : \u03b2) \u2192 \u03c0 b b : \u03b2 \u22a2 dist (f b) (g b) \u2264 dist f g ** simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b] ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.nat_card_dvd_of_surjective ** G\u271d : Type u_1 inst\u271d\u00b2 : Group G\u271d H\u271d K L : Subgroup G\u271d G : Type u_2 H : Type u_3 inst\u271d\u00b9 : Group G inst\u271d : Group H f : G \u2192* H hf : Function.Surjective \u2191f \u22a2 Nat.card H \u2223 Nat.card G ** rw [\u2190 Nat.card_congr (QuotientGroup.quotientKerEquivOfSurjective f hf).toEquiv] ** G\u271d : Type u_1 inst\u271d\u00b2 : Group G\u271d H\u271d K L : Subgroup G\u271d G : Type u_2 H : Type u_3 inst\u271d\u00b9 : Group G inst\u271d : Group H f : G \u2192* H hf : Function.Surjective \u2191f \u22a2 Nat.card (G \u29f8 MonoidHom.ker f) \u2223 Nat.card G ** exact Dvd.intro_left (Nat.card f.ker) f.ker.card_mul_index ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearEquiv.uniqueDiffOn_preimage_iff ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F f : E \u2192 F s : Set E f' : E \u2192L[\ud835\udd5c] F e : F \u2243L[\ud835\udd5c] E \u22a2 UniqueDiffOn \ud835\udd5c (\u2191e \u207b\u00b9' s) \u2194 UniqueDiffOn \ud835\udd5c s ** rw [\u2190 e.image_symm_eq_preimage, e.symm.uniqueDiffOn_image_iff] ** Qed", + "informal": "" + }, + { + "formal": "IsLocalization.map_smul ** R : Type u_1 inst\u271d\u2077 : CommSemiring R M : Submonoid R S : Type u_2 inst\u271d\u2076 : CommSemiring S inst\u271d\u2075 : Algebra R S P : Type u_3 inst\u271d\u2074 : CommSemiring P inst\u271d\u00b3 : IsLocalization M S g : R \u2192+* P hg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y) T : Submonoid P Q : Type u_4 inst\u271d\u00b2 : CommSemiring Q hy : M \u2264 Submonoid.comap g T inst\u271d\u00b9 : Algebra P Q inst\u271d : IsLocalization T Q x : S z : R \u22a2 \u2191(map Q g hy) (z \u2022 x) = \u2191g z \u2022 \u2191(map Q g hy) x ** rw [Algebra.smul_def, Algebra.smul_def, RingHom.map_mul, map_eq] ** Qed", + "informal": "" + }, + { + "formal": "Int.card_fintype_Icc ** a b : \u2124 \u22a2 Fintype.card \u2191(Set.Icc a b) = toNat (b + 1 - a) ** rw [\u2190 card_Icc, Fintype.card_ofFinset] ** Qed", + "informal": "" + }, + { + "formal": "PowerSeries.rescale_mk ** R : Type u_1 inst\u271d : CommSemiring R f : \u2115 \u2192 R a : R \u22a2 \u2191(rescale a) (mk f) = mk fun n => a ^ n * f n ** ext ** case h R : Type u_1 inst\u271d : CommSemiring R f : \u2115 \u2192 R a : R n\u271d : \u2115 \u22a2 \u2191(coeff R n\u271d) (\u2191(rescale a) (mk f)) = \u2191(coeff R n\u271d) (mk fun n => a ^ n * f n) ** rw [coeff_rescale, coeff_mk, coeff_mk] ** Qed", + "informal": "" + }, + { + "formal": "UpperHalfPlane.SL_neg_smul ** g\u271d : SL(2, \u2124) z\u271d : \u210d \u0393 : Subgroup SL(2, \u2124) g : SL(2, \u2124) z : \u210d \u22a2 -g \u2022 z = g \u2022 z ** simp only [coe_GLPos_neg, sl_moeb, coe_int_neg, neg_smul, coe'] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousMultilinearMap.alternatization_apply_toAlternatingMap ** R : Type u_1 M : Type u_2 N : Type u_3 \u03b9 : Type u_4 inst\u271d\u2079 : Semiring R inst\u271d\u2078 : AddCommMonoid M inst\u271d\u2077 : Module R M inst\u271d\u2076 : TopologicalSpace M inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N inst\u271d\u00b3 : TopologicalSpace N inst\u271d\u00b2 : TopologicalAddGroup N inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 f g : ContinuousMultilinearMap R (fun x => M) N \u22a2 ContinuousAlternatingMap.toAlternatingMap (\u2191alternatization f) = \u2191MultilinearMap.alternatization f.toMultilinearMap ** ext v ** Qed", + "informal": "" + }, + { + "formal": "YoungDiagram.mk_mem_col_iff ** \u03bc : YoungDiagram i j : \u2115 \u22a2 (i, j) \u2208 col \u03bc j \u2194 (i, j) \u2208 \u03bc ** simp [col] ** Qed", + "informal": "" + }, + { + "formal": "TopCat.Presheaf.Pushforward.id_inv_app' ** C : Type u inst\u271d : Category.{v, u} C X : TopCat \u2131 : Presheaf C X U : Set \u2191X p : IsOpen U \u22a2 (id \u2131).inv.app (op { carrier := U, is_open' := p }) = \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p })) ** dsimp [id] ** C : Type u inst\u271d : Category.{v, u} C X : TopCat \u2131 : Presheaf C X U : Set \u2191X p : IsOpen U \u22a2 ((Functor.leftUnitor \u2131).inv \u226b whiskerRight (NatTrans.op (Opens.mapId X).hom) \u2131).app (op { carrier := U, is_open' := p }) = \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p })) ** simp [CategoryStruct.comp] ** Qed", + "informal": "" + }, + { + "formal": "ClosedEmbedding.polishSpace ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : ClosedEmbedding f \u22a2 PolishSpace \u03b1 ** letI := upgradePolishSpace \u03b2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : ClosedEmbedding f this : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2 \u22a2 PolishSpace \u03b1 ** letI : MetricSpace \u03b1 := hf.toEmbedding.comapMetricSpace f ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : ClosedEmbedding f this\u271d : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2 this : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f) \u22a2 PolishSpace \u03b1 ** haveI : SecondCountableTopology \u03b1 := hf.toEmbedding.secondCountableTopology ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : ClosedEmbedding f this\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2 this\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f) this : SecondCountableTopology \u03b1 \u22a2 PolishSpace \u03b1 ** have : CompleteSpace \u03b1 := by\n rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing]\n exact hf.closed_range.isComplete ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : ClosedEmbedding f this\u271d\u00b2 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2 this\u271d\u00b9 : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f) this\u271d : SecondCountableTopology \u03b1 this : CompleteSpace \u03b1 \u22a2 PolishSpace \u03b1 ** infer_instance ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : ClosedEmbedding f this\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2 this\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f) this : SecondCountableTopology \u03b1 \u22a2 CompleteSpace \u03b1 ** rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : ClosedEmbedding f this\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2 this\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f) this : SecondCountableTopology \u03b1 \u22a2 IsComplete (range f) ** exact hf.closed_range.isComplete ** Qed", + "informal": "" + }, + { + "formal": "Fin.repeat_one ** m n : \u2115 \u03b1\u271d : Fin (n + 1) \u2192 Type u x : \u03b1\u271d 0 q : (i : Fin (n + 1)) \u2192 \u03b1\u271d i p : (i : Fin n) \u2192 \u03b1\u271d (succ i) i : Fin n y : \u03b1\u271d (succ i) z : \u03b1\u271d 0 \u03b1 : Type u_1 a : Fin n \u2192 \u03b1 \u22a2 repeat 1 a = a \u2218 cast (_ : 1 * n = n) ** generalize_proofs h ** m n : \u2115 \u03b1\u271d : Fin (n + 1) \u2192 Type u x : \u03b1\u271d 0 q : (i : Fin (n + 1)) \u2192 \u03b1\u271d i p : (i : Fin n) \u2192 \u03b1\u271d (succ i) i : Fin n y : \u03b1\u271d (succ i) z : \u03b1\u271d 0 \u03b1 : Type u_1 a : Fin n \u2192 \u03b1 h : 1 * n = n \u22a2 repeat 1 a = a \u2218 cast h ** apply funext ** case h m n : \u2115 \u03b1\u271d : Fin (n + 1) \u2192 Type u x : \u03b1\u271d 0 q : (i : Fin (n + 1)) \u2192 \u03b1\u271d i p : (i : Fin n) \u2192 \u03b1\u271d (succ i) i : Fin n y : \u03b1\u271d (succ i) z : \u03b1\u271d 0 \u03b1 : Type u_1 a : Fin n \u2192 \u03b1 h : 1 * n = n \u22a2 \u2200 (x : Fin (1 * n)), repeat 1 a x = (a \u2218 cast h) x ** rw [(Fin.rightInverse_cast h.symm).surjective.forall] ** case h m n : \u2115 \u03b1\u271d : Fin (n + 1) \u2192 Type u x : \u03b1\u271d 0 q : (i : Fin (n + 1)) \u2192 \u03b1\u271d i p : (i : Fin n) \u2192 \u03b1\u271d (succ i) i : Fin n y : \u03b1\u271d (succ i) z : \u03b1\u271d 0 \u03b1 : Type u_1 a : Fin n \u2192 \u03b1 h : 1 * n = n \u22a2 \u2200 (x : Fin n), repeat 1 a (cast (_ : n = 1 * n) x) = (a \u2218 cast h) (cast (_ : n = 1 * n) x) ** intro i ** case h m n : \u2115 \u03b1\u271d : Fin (n + 1) \u2192 Type u x : \u03b1\u271d 0 q : (i : Fin (n + 1)) \u2192 \u03b1\u271d i p : (i : Fin n) \u2192 \u03b1\u271d (succ i) i\u271d : Fin n y : \u03b1\u271d (succ i\u271d) z : \u03b1\u271d 0 \u03b1 : Type u_1 a : Fin n \u2192 \u03b1 h : 1 * n = n i : Fin n \u22a2 repeat 1 a (cast (_ : n = 1 * n) i) = (a \u2218 cast h) (cast (_ : n = 1 * n) i) ** simp [modNat, Nat.mod_eq_of_lt i.is_lt] ** Qed", + "informal": "" + }, + { + "formal": "List.IsPrefix.filter_map ** \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 h : l\u2081 <+: l\u2082 f : \u03b1 \u2192 Option \u03b2 \u22a2 filterMap f l\u2081 <+: filterMap f l\u2082 ** induction' l\u2081 with hd\u2081 tl\u2081 hl generalizing l\u2082 ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082\u271d l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 h\u271d : l\u2081 <+: l\u2082\u271d f : \u03b1 \u2192 Option \u03b2 l\u2082 : List \u03b1 h : [] <+: l\u2082 \u22a2 filterMap f [] <+: filterMap f l\u2082 ** simp only [nil_prefix, filterMap_nil] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082\u271d l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 h\u271d : l\u2081 <+: l\u2082\u271d f : \u03b1 \u2192 Option \u03b2 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082 l\u2082 : List \u03b1 h : hd\u2081 :: tl\u2081 <+: l\u2082 \u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f l\u2082 ** cases' l\u2082 with hd\u2082 tl\u2082 ** case cons.nil \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 h\u271d : l\u2081 <+: l\u2082 f : \u03b1 \u2192 Option \u03b2 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082 h : hd\u2081 :: tl\u2081 <+: [] \u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f [] ** simpa only using eq_nil_of_prefix_nil h ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 h\u271d : l\u2081 <+: l\u2082 f : \u03b1 \u2192 Option \u03b2 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h : hd\u2081 :: tl\u2081 <+: hd\u2082 :: tl\u2082 \u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f (hd\u2082 :: tl\u2082) ** rw [cons_prefix_iff] at h ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 h\u271d : l\u2081 <+: l\u2082 f : \u03b1 \u2192 Option \u03b2 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h : hd\u2081 = hd\u2082 \u2227 tl\u2081 <+: tl\u2082 \u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f (hd\u2082 :: tl\u2082) ** rw [\u2190 @singleton_append _ hd\u2081 _, \u2190 @singleton_append _ hd\u2082 _, filterMap_append,\n filterMap_append, h.left, prefix_append_right_inj] ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 h\u271d : l\u2081 <+: l\u2082 f : \u03b1 \u2192 Option \u03b2 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h : hd\u2081 = hd\u2082 \u2227 tl\u2081 <+: tl\u2082 \u22a2 filterMap f tl\u2081 <+: filterMap f tl\u2082 ** exact hl h.right ** Qed", + "informal": "" + }, + { + "formal": "norm_exp_mul_sq_le ** z : \u2102 hz : 0 < z.im n : \u2124 \u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n ** let y := rexp (-\u03c0 * z.im) ** z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) \u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n ** have h : y < 1 := exp_lt_one_iff.mpr (mul_neg_of_neg_of_pos (neg_lt_zero.mpr pi_pos) hz) ** z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n ** refine' (le_of_eq _).trans (_ : y ^ n ^ 2 \u2264 _) ** case refine'_1 z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 = y ^ n ^ 2 ** rw [Complex.norm_eq_abs, Complex.abs_exp] ** case refine'_1 z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2 ** 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 ** case refine'_1 z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 this : (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = -\u03c0 * z.im * \u2191n ^ 2 \u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2 ** obtain \u27e8m, hm\u27e9 := Int.eq_ofNat_of_zero_le (sq_nonneg n) ** case refine'_1.intro z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 this : (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = -\u03c0 * z.im * \u2191n ^ 2 m : \u2115 hm : n ^ 2 = \u2191m \u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2 ** rw [this, exp_mul, \u2190 Int.cast_pow, rpow_int_cast, hm, zpow_ofNat] ** z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = -\u03c0 * z.im * \u2191n ^ 2 ** 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] ** z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 \u03c0 * \u2191n ^ 2 * -z.im = -\u03c0 * z.im * \u2191n ^ 2 ** ring ** z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 \u2191\u03c0 * I * \u2191n ^ 2 * z = \u2191(\u03c0 * \u2191n ^ 2) * (z * I) ** push_cast ** z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 \u2191\u03c0 * I * \u2191n ^ 2 * z = \u2191\u03c0 * \u2191n ^ 2 * (z * I) ** ring ** case refine'_2 z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 y ^ n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n ** have : n ^ 2 = \u2191(n.natAbs ^ 2) := by rw [Nat.cast_pow, Int.natAbs_sq] ** case refine'_2 z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 this : n ^ 2 = \u2191(Int.natAbs n ^ 2) \u22a2 y ^ n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n ** rw [this, zpow_ofNat] ** case refine'_2 z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 this : n ^ 2 = \u2191(Int.natAbs n ^ 2) \u22a2 y ^ Int.natAbs n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n ** exact pow_le_pow_of_le_one (exp_pos _).le h.le ((sq n.natAbs).symm \u25b8 n.natAbs.le_mul_self) ** z : \u2102 hz : 0 < z.im n : \u2124 y : \u211d := rexp (-\u03c0 * z.im) h : y < 1 \u22a2 n ^ 2 = \u2191(Int.natAbs n ^ 2) ** rw [Nat.cast_pow, Int.natAbs_sq] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.rnDeriv_lt_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, rnDeriv \u03bc \u03bd x < \u22a4 ** 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 _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 rnDeriv \u03bc \u03bd x < \u22a4 ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bc n : \u2115 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 rnDeriv \u03bc \u03bd x < \u22a4 ** rw [\u2190 ae_restrict_iff' (measurable_spanningSets _ _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bc n : \u2115 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bd (spanningSets \u03bc n), rnDeriv \u03bc \u03bd x < \u22a4 ** apply ae_lt_top (measurable_rnDeriv _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bc n : \u2115 \u22a2 \u222b\u207b (x : \u03b1) in spanningSets \u03bc n, rnDeriv \u03bc \u03bd x \u2202\u03bd \u2260 \u22a4 ** refine' (lintegral_rnDeriv_lt_top_of_measure_ne_top _ _).ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc (spanningSets \u03bc n) \u2260 \u22a4 ** exact (measure_spanningSets_lt_top _ _).ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bc this : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 rnDeriv \u03bc \u03bd x < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, rnDeriv \u03bc \u03bd x < \u22a4 ** filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _) ** Qed", + "informal": "" + }, + { + "formal": "Basis.det_unitsSMul ** R : Type u_1 inst\u271d\u2078 : CommRing R M : Type u_2 inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M M' : Type u_3 inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : Module R M' \u03b9 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b9 inst\u271d\u00b2 : Fintype \u03b9 e\u271d : Basis \u03b9 R M A : Type u_5 inst\u271d\u00b9 : CommRing A inst\u271d : Module A M e : Basis \u03b9 R M w : \u03b9 \u2192 R\u02e3 \u22a2 det (unitsSMul e w) = \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 det e ** ext f ** case H R : Type u_1 inst\u271d\u2078 : CommRing R M : Type u_2 inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M M' : Type u_3 inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : Module R M' \u03b9 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b9 inst\u271d\u00b2 : Fintype \u03b9 e\u271d : Basis \u03b9 R M A : Type u_5 inst\u271d\u00b9 : CommRing A inst\u271d : Module A M e : Basis \u03b9 R M w : \u03b9 \u2192 R\u02e3 f : \u03b9 \u2192 M \u22a2 \u2191(det (unitsSMul e w)) f = \u2191(\u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 det e) f ** 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 ** case H R : Type u_1 inst\u271d\u2078 : CommRing R M : Type u_2 inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M M' : Type u_3 inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : Module R M' \u03b9 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b9 inst\u271d\u00b2 : Fintype \u03b9 e\u271d : Basis \u03b9 R M A : Type u_5 inst\u271d\u00b9 : CommRing A inst\u271d : Module A M e : Basis \u03b9 R M w : \u03b9 \u2192 R\u02e3 f : \u03b9 \u2192 M \u22a2 (Matrix.det fun i j => \u2191(\u2191(unitsSMul e w).repr (f j)) i) = \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 Matrix.det fun i j => \u2191(\u2191e.repr (f j)) i ** simp only [e.repr_unitsSMul] ** case H R : Type u_1 inst\u271d\u2078 : CommRing R M : Type u_2 inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M M' : Type u_3 inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : Module R M' \u03b9 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b9 inst\u271d\u00b2 : Fintype \u03b9 e\u271d : Basis \u03b9 R M A : Type u_5 inst\u271d\u00b9 : CommRing A inst\u271d : Module A M e : Basis \u03b9 R M w : \u03b9 \u2192 R\u02e3 f : \u03b9 \u2192 M \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 ** convert Matrix.det_mul_column (fun i => (\u2191(w i)\u207b\u00b9 : R)) fun i j => e.repr (f j) i ** case h.e'_3.h.e'_1 R : Type u_1 inst\u271d\u2078 : CommRing R M : Type u_2 inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M M' : Type u_3 inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : Module R M' \u03b9 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b9 inst\u271d\u00b2 : Fintype \u03b9 e\u271d : Basis \u03b9 R M A : Type u_5 inst\u271d\u00b9 : CommRing A inst\u271d : Module A M e : Basis \u03b9 R M w : \u03b9 \u2192 R\u02e3 f : \u03b9 \u2192 M \u22a2 \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 = \u220f i : \u03b9, \u2191(w i)\u207b\u00b9 ** simp only [\u2190 Finset.prod_inv_distrib] ** case h.e'_3.h.e'_1 R : Type u_1 inst\u271d\u2078 : CommRing R M : Type u_2 inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M M' : Type u_3 inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : Module R M' \u03b9 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b9 inst\u271d\u00b2 : Fintype \u03b9 e\u271d : Basis \u03b9 R M A : Type u_5 inst\u271d\u00b9 : CommRing A inst\u271d : Module A M e : Basis \u03b9 R M w : \u03b9 \u2192 R\u02e3 f : \u03b9 \u2192 M \u22a2 \u2191(\u220f x : \u03b9, (w x)\u207b\u00b9) = \u220f i : \u03b9, \u2191(w i)\u207b\u00b9 ** norm_cast ** Qed", + "informal": "" + }, + { + "formal": "List.bind_id ** \u03b1 : Type u_1 l : List (List \u03b1) \u22a2 List.bind l id = join l ** simp [List.bind] ** Qed", + "informal": "" + }, + { + "formal": "Metric.hasBasis_cobounded_compl_closedBall ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d : PseudoMetricSpace \u03b1 x : \u03b1 s t : Set \u03b1 r : \u211d c : \u03b1 x\u271d : Set \u03b1 \u22a2 (\u2203 r, x\u271d \u2286 closedBall c r) \u2194 \u2203 i, True \u2227 (closedBall c i)\u1d9c \u2286 x\u271d\u1d9c ** simp ** Qed", + "informal": "" + }, + { + "formal": "Complex.dist_self_conj ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E z : \u2102 \u22a2 dist z (\u2191(starRingEnd \u2102) z) = 2 * |z.im| ** rw [dist_comm, dist_conj_self] ** Qed", + "informal": "" + }, + { + "formal": "SetTheory.PGame.Domineering.moveRight_card ** b : Board m : \u2124 \u00d7 \u2124 h : m \u2208 right b \u22a2 Finset.card (moveRight b m) + 2 = Finset.card b ** dsimp [moveRight] ** b : Board m : \u2124 \u00d7 \u2124 h : m \u2208 right b \u22a2 Finset.card (Finset.erase (Finset.erase b m) (m.1 - 1, m.2)) + 2 = Finset.card b ** rw [Finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h)] ** b : Board m : \u2124 \u00d7 \u2124 h : m \u2208 right b \u22a2 Finset.card (Finset.erase b m) - 1 + 2 = Finset.card b ** rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)] ** b : Board m : \u2124 \u00d7 \u2124 h : m \u2208 right b \u22a2 Finset.card b - 1 - 1 + 2 = Finset.card b ** exact tsub_add_cancel_of_le (card_of_mem_right h) ** Qed", + "informal": "" + }, + { + "formal": "Set.univ_pi_subset_univ_pi_iff ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 \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 ** simp [pi_subset_pi_iff] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.eraseLead_card_support ** R : Type u_1 inst\u271d : Semiring R f : R[X] c : \u2115 fc : card (support f) = c \u22a2 card (support (eraseLead f)) = c - 1 ** by_cases f0 : f = 0 ** case pos R : Type u_1 inst\u271d : Semiring R f : R[X] c : \u2115 fc : card (support f) = c f0 : f = 0 \u22a2 card (support (eraseLead f)) = c - 1 ** rw [\u2190 fc, f0, eraseLead_zero, support_zero, card_empty] ** case neg R : Type u_1 inst\u271d : Semiring R f : R[X] c : \u2115 fc : card (support f) = c f0 : \u00acf = 0 \u22a2 card (support (eraseLead f)) = c - 1 ** rw [eraseLead_support, card_erase_of_mem (natDegree_mem_support_of_nonzero f0), fc] ** Qed", + "informal": "" + }, + { + "formal": "mem_const_vadd_affineSegment ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2076 : OrderedRing R inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' x y z : P v : V \u22a2 v +\u1d65 z \u2208 affineSegment R (v +\u1d65 x) (v +\u1d65 y) \u2194 z \u2208 affineSegment R x y ** rw [\u2190 affineSegment_const_vadd_image, (AddAction.injective v).mem_set_image] ** Qed", + "informal": "" + }, + { + "formal": "cmpLE_swap ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b1 : Type u_3 inst\u271d\u00b2 : LE \u03b1 inst\u271d\u00b9 : IsTotal \u03b1 fun x x_1 => x \u2264 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2264 x_1 x y : \u03b1 \u22a2 Ordering.swap (cmpLE x y) = cmpLE y x ** by_cases xy:x \u2264 y <;> by_cases yx:y \u2264 x <;> simp [cmpLE, *, Ordering.swap] ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b1 : Type u_3 inst\u271d\u00b2 : LE \u03b1 inst\u271d\u00b9 : IsTotal \u03b1 fun x x_1 => x \u2264 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2264 x_1 x y : \u03b1 xy : \u00acx \u2264 y yx : \u00acy \u2264 x \u22a2 False ** cases not_or_of_not xy yx (total_of _ _ _) ** Qed", + "informal": "" + }, + { + "formal": "List.get_of_append ** \u03b1 : Type u_1 l\u2081 : List \u03b1 a : \u03b1 l\u2082 : List \u03b1 n : Nat l : List \u03b1 eq : l = l\u2081 ++ a :: l\u2082 h : length l\u2081 = n \u22a2 some (get l { val := n, isLt := (_ : n < length l) }) = some a ** rw [\u2190 get?_eq_get, eq, get?_append_right (h \u25b8 Nat.le_refl _), h, Nat.sub_self] ** \u03b1 : Type u_1 l\u2081 : List \u03b1 a : \u03b1 l\u2082 : List \u03b1 n : Nat l : List \u03b1 eq : l = l\u2081 ++ a :: l\u2082 h : length l\u2081 = n \u22a2 get? (a :: l\u2082) 0 = some a ** rfl ** Qed", + "informal": "" + }, + { + "formal": "isPreconnected_iff_subset_of_disjoint_closed ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u v : Set \u03b1 \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 ** constructor <;> intro h ** case mp \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u v : Set \u03b1 h : IsPreconnected s \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 ** intro u v hu hv hs huv ** case mp \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d : Set \u03b1 h : IsPreconnected s u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v huv : s \u2229 (u \u2229 v) = \u2205 \u22a2 s \u2286 u \u2228 s \u2286 v ** rw [isPreconnected_closed_iff] at h ** case mp \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d : Set \u03b1 h : \u2200 (t t' : Set \u03b1), 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')) u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v huv : s \u2229 (u \u2229 v) = \u2205 \u22a2 s \u2286 u \u2228 s \u2286 v ** specialize h u v hu hv hs ** case mp \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v huv : s \u2229 (u \u2229 v) = \u2205 h : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u22a2 s \u2286 u \u2228 s \u2286 v ** contrapose! huv ** case mp \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v h : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) huv : \u00acs \u2286 u \u2227 \u00acs \u2286 v \u22a2 s \u2229 (u \u2229 v) \u2260 \u2205 ** rw [\u2190 nonempty_iff_ne_empty] ** case mp \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v h : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) huv : \u00acs \u2286 u \u2227 \u00acs \u2286 v \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** simp [not_subset] at huv ** case mp \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v h : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) huv : (\u2203 a, a \u2208 s \u2227 \u00aca \u2208 u) \u2227 \u2203 a, a \u2208 s \u2227 \u00aca \u2208 v \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** rcases huv with \u27e8\u27e8x, hxs, hxu\u27e9, \u27e8y, hys, hyv\u27e9\u27e9 ** case mp.intro.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v h : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) x : \u03b1 hxs : x \u2208 s hxu : \u00acx \u2208 u y : \u03b1 hys : y \u2208 s hyv : \u00acy \u2208 v \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** have hxv : x \u2208 v := or_iff_not_imp_left.mp (hs hxs) hxu ** case mp.intro.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v h : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) x : \u03b1 hxs : x \u2208 s hxu : \u00acx \u2208 u y : \u03b1 hys : y \u2208 s hyv : \u00acy \u2208 v hxv : x \u2208 v \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** have hyu : y \u2208 u := or_iff_not_imp_right.mp (hs hys) hyv ** case mp.intro.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v h : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) x : \u03b1 hxs : x \u2208 s hxu : \u00acx \u2208 u y : \u03b1 hys : y \u2208 s hyv : \u00acy \u2208 v hxv : x \u2208 v hyu : y \u2208 u \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** exact h \u27e8y, hys, hyu\u27e9 \u27e8x, hxs, hxv\u27e9 ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u v : Set \u03b1 h : \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 \u22a2 IsPreconnected s ** rw [isPreconnected_closed_iff] ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u v : Set \u03b1 h : \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 \u22a2 \u2200 (t t' : Set \u03b1), 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')) ** intro u v hu hv hs hsu hsv ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d : Set \u03b1 h : \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 u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) hsv : Set.Nonempty (s \u2229 v) \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** rw [nonempty_iff_ne_empty] ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d : Set \u03b1 h : \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 u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) hsv : Set.Nonempty (s \u2229 v) \u22a2 s \u2229 (u \u2229 v) \u2260 \u2205 ** intro H ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d : Set \u03b1 h : \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 u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) hsv : Set.Nonempty (s \u2229 v) H : s \u2229 (u \u2229 v) = \u2205 \u22a2 False ** specialize h u v hu hv hs H ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) hsv : Set.Nonempty (s \u2229 v) H : s \u2229 (u \u2229 v) = \u2205 h : s \u2286 u \u2228 s \u2286 v \u22a2 False ** contrapose H ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) hsv : Set.Nonempty (s \u2229 v) h : s \u2286 u \u2228 s \u2286 v H : \u00acFalse \u22a2 \u00acs \u2229 (u \u2229 v) = \u2205 ** apply Nonempty.ne_empty ** case mpr.a \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) hsv : Set.Nonempty (s \u2229 v) h : s \u2286 u \u2228 s \u2286 v H : \u00acFalse \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** cases' h with h h ** case mpr.a.inl \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) hsv : Set.Nonempty (s \u2229 v) H : \u00acFalse h : s \u2286 u \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** rcases hsv with \u27e8x, hxs, hxv\u27e9 ** case mpr.a.inl.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) H : \u00acFalse h : s \u2286 u x : \u03b1 hxs : x \u2208 s hxv : x \u2208 v \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** exact \u27e8x, hxs, \u27e8h hxs, hxv\u27e9\u27e9 ** case mpr.a.inr \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsu : Set.Nonempty (s \u2229 u) hsv : Set.Nonempty (s \u2229 v) H : \u00acFalse h : s \u2286 v \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** rcases hsu with \u27e8x, hxs, hxu\u27e9 ** case mpr.a.inr.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d : TopologicalSpace \u03b1 s t u\u271d v\u271d u v : Set \u03b1 hu : IsClosed u hv : IsClosed v hs : s \u2286 u \u222a v hsv : Set.Nonempty (s \u2229 v) H : \u00acFalse h : s \u2286 v x : \u03b1 hxs : x \u2208 s hxu : x \u2208 u \u22a2 Set.Nonempty (s \u2229 (u \u2229 v)) ** exact \u27e8x, hxs, \u27e8hxu, h hxs\u27e9\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "xor_comm ** a b : Prop \u22a2 Xor' a b = Xor' b a ** simp [Xor', and_comm, or_comm] ** Qed", + "informal": "" + }, + { + "formal": "AddLECancellable.tsub_lt_self_iff ** \u03b1 : Type u_1 inst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1 inst\u271d\u00b9 : Sub \u03b1 inst\u271d : OrderedSub \u03b1 a b c d : \u03b1 ha : AddLECancellable a \u22a2 a - b < a \u2194 0 < a \u2227 0 < b ** 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 ** \u03b1 : Type u_1 inst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1 inst\u271d\u00b9 : Sub \u03b1 inst\u271d : OrderedSub \u03b1 a b c d : \u03b1 ha : AddLECancellable a h : a - b < a \u22a2 0 \u2260 b ** rintro rfl ** \u03b1 : Type u_1 inst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1 inst\u271d\u00b9 : Sub \u03b1 inst\u271d : OrderedSub \u03b1 a c d : \u03b1 ha : AddLECancellable a h : a - 0 < a \u22a2 False ** rw [tsub_zero] at h ** \u03b1 : Type u_1 inst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1 inst\u271d\u00b9 : Sub \u03b1 inst\u271d : OrderedSub \u03b1 a c d : \u03b1 ha : AddLECancellable a h : a < a \u22a2 False ** exact h.false ** Qed", + "informal": "" + }, + { + "formal": "Std.TransCmp.gt_trans ** cmp\u271d : ?m.2419 \u2192 ?m.2419 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp\u271d x\u271d : Sort ?u.2417 cmp : x\u271d \u2192 x\u271d \u2192 Ordering inst\u271d : TransCmp cmp x y z : x\u271d h\u2081 : cmp x y = Ordering.gt h\u2082 : cmp y z = Ordering.gt \u22a2 cmp x z = Ordering.gt ** rw [cmp_eq_gt] at h\u2081 h\u2082 \u22a2 ** cmp\u271d : ?m.2419 \u2192 ?m.2419 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp\u271d x\u271d : Sort ?u.2417 cmp : x\u271d \u2192 x\u271d \u2192 Ordering inst\u271d : TransCmp cmp x y z : x\u271d h\u2081 : cmp y x = Ordering.lt h\u2082 : cmp z y = Ordering.lt \u22a2 cmp z x = Ordering.lt ** exact lt_trans h\u2082 h\u2081 ** Qed", + "informal": "" + }, + { + "formal": "Basis.finTwoProd_zero ** \u03b9 : Type u_1 \u03b9' : Type u_2 R\u271d : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 v : \u03b9 \u2192 M inst\u271d\u2079 : Ring R\u271d inst\u271d\u2078 : CommRing R\u2082 inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : AddCommGroup M' inst\u271d\u2075 : AddCommGroup M'' inst\u271d\u2074 : Module R\u271d M inst\u271d\u00b3 : Module R\u2082 M inst\u271d\u00b2 : Module R\u271d M' inst\u271d\u00b9 : Module R\u271d M'' c d : R\u271d x y : M b : Basis \u03b9 R\u271d M R : Type u_10 inst\u271d : Semiring R \u22a2 \u2191(Basis.finTwoProd R) 0 = (1, 0) ** simp [Basis.finTwoProd, LinearEquiv.finTwoArrow] ** Qed", + "informal": "" + }, + { + "formal": "integrableOn_Ioc_iff_integrableOn_Ioo ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : PartialOrder \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E \u03bc : Measure \u03b1 a b : \u03b1 inst\u271d : NoAtoms \u03bc \u22a2 \u2191\u2191\u03bc {b} \u2260 \u22a4 ** rw [measure_singleton] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : PartialOrder \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E \u03bc : Measure \u03b1 a b : \u03b1 inst\u271d : NoAtoms \u03bc \u22a2 0 \u2260 \u22a4 ** exact ENNReal.zero_ne_top ** Qed", + "informal": "" + }, + { + "formal": "Set.image_iInter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 f : \u03b1 \u2192 \u03b2 hf : Bijective f s : \u03b9 \u2192 Set \u03b1 \u22a2 f '' \u22c2 i, s i = \u22c2 i, f '' s i ** cases isEmpty_or_nonempty \u03b9 ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 f : \u03b1 \u2192 \u03b2 hf : Bijective f s : \u03b9 \u2192 Set \u03b1 h\u271d : IsEmpty \u03b9 \u22a2 f '' \u22c2 i, s i = \u22c2 i, f '' s i ** simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective] ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 f : \u03b1 \u2192 \u03b2 hf : Bijective f s : \u03b9 \u2192 Set \u03b1 h\u271d : Nonempty \u03b9 \u22a2 f '' \u22c2 i, s i = \u22c2 i, f '' s i ** exact (hf.injective.injOn _).image_iInter_eq ** Qed", + "informal": "" + }, + { + "formal": "Nat.le_mod_add_mod_of_dvd_add_of_not_dvd ** m n a\u271d b\u271d c\u271d d a b c : \u2115 h : c \u2223 a + b ha : \u00acc \u2223 a hc : \u00acc \u2264 a % c + b % c \u22a2 False ** have : (a + b) % c = a % c + b % c := add_mod_of_add_mod_lt (lt_of_not_ge hc) ** m n a\u271d b\u271d c\u271d d a b c : \u2115 h : c \u2223 a + b ha : \u00acc \u2223 a hc : \u00acc \u2264 a % c + b % c this : (a + b) % c = a % c + b % c \u22a2 False ** simp_all [dvd_iff_mod_eq_zero] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.rank_self_mul_conjTranspose ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 R : Type u_5 m_fin : Fintype m inst\u271d\u2075 : Fintype n inst\u271d\u2074 : Fintype o inst\u271d\u00b3 : Fintype m inst\u271d\u00b2 : Field R inst\u271d\u00b9 : PartialOrder R inst\u271d : StarOrderedRing R A : Matrix m n R \u22a2 rank (A * A\u1d34) = rank A ** simpa only [rank_conjTranspose, conjTranspose_conjTranspose] using\n rank_conjTranspose_mul_self A\u1d34 ** Qed", + "informal": "" + }, + { + "formal": "MulAction.card_eq_sum_card_group_div_card_stabilizer' ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 inst\u271d\u2075 : Group \u03b1 inst\u271d\u2074 : MulAction \u03b1 \u03b2 x : \u03b2 inst\u271d\u00b3 : Fintype \u03b1 inst\u271d\u00b2 : Fintype \u03b2 inst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2)) inst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b } \u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2 h\u03c6 : LeftInverse Quotient.mk'' \u03c6 \u22a2 Fintype.card \u03b2 = \u2211 \u03c9 : Quotient (orbitRel \u03b1 \u03b2), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } ** 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)] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 inst\u271d\u2075 : Group \u03b1 inst\u271d\u2074 : MulAction \u03b1 \u03b2 x : \u03b2 inst\u271d\u00b3 : Fintype \u03b1 inst\u271d\u00b2 : Fintype \u03b2 inst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2)) inst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b } \u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2 h\u03c6 : LeftInverse Quotient.mk'' \u03c6 \u22a2 Fintype.card \u03b2 = \u2211 \u03c9 : Quotient (orbitRel \u03b1 \u03b2), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } ** 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) ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 inst\u271d\u2075 : Group \u03b1 inst\u271d\u2074 : MulAction \u03b1 \u03b2 x : \u03b2 inst\u271d\u00b3 : Fintype \u03b1 inst\u271d\u00b2 : Fintype \u03b2 inst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2)) inst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b } \u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2 h\u03c6 : LeftInverse Quotient.mk'' \u03c6 this : \u2200 (\u03c9 : Quotient (orbitRel \u03b1 \u03b2)), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } = Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9)) \u22a2 Fintype.card \u03b2 = \u2211 \u03c9 : Quotient (orbitRel \u03b1 \u03b2), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } ** simp_rw [this, \u2190 Fintype.card_sigma,\n Fintype.card_congr (selfEquivSigmaOrbitsQuotientStabilizer' \u03b1 \u03b2 h\u03c6)] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 inst\u271d\u2075 : Group \u03b1 inst\u271d\u2074 : MulAction \u03b1 \u03b2 x : \u03b2 inst\u271d\u00b3 : Fintype \u03b1 inst\u271d\u00b2 : Fintype \u03b2 inst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2)) inst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b } \u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2 h\u03c6 : LeftInverse Quotient.mk'' \u03c6 \u22a2 \u2200 (\u03c9 : Quotient (orbitRel \u03b1 \u03b2)), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } = Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9)) ** intro \u03c9 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 inst\u271d\u2075 : Group \u03b1 inst\u271d\u2074 : MulAction \u03b1 \u03b2 x : \u03b2 inst\u271d\u00b3 : Fintype \u03b1 inst\u271d\u00b2 : Fintype \u03b2 inst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2)) inst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b } \u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2 h\u03c6 : LeftInverse Quotient.mk'' \u03c6 \u03c9 : Quotient (orbitRel \u03b1 \u03b2) \u22a2 Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } = Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9)) ** rw [Fintype.card_congr (@Subgroup.groupEquivQuotientProdSubgroup \u03b1 _ (stabilizer \u03b1 <| \u03c6 \u03c9)),\n Fintype.card_prod, Nat.mul_div_cancel] ** case H \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 inst\u271d\u2075 : Group \u03b1 inst\u271d\u2074 : MulAction \u03b1 \u03b2 x : \u03b2 inst\u271d\u00b3 : Fintype \u03b1 inst\u271d\u00b2 : Fintype \u03b2 inst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2)) inst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b } \u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2 h\u03c6 : LeftInverse Quotient.mk'' \u03c6 \u03c9 : Quotient (orbitRel \u03b1 \u03b2) \u22a2 0 < Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } ** exact Fintype.card_pos_iff.mpr (by infer_instance) ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2076 inst\u271d\u2075 : Group \u03b1 inst\u271d\u2074 : MulAction \u03b1 \u03b2 x : \u03b2 inst\u271d\u00b3 : Fintype \u03b1 inst\u271d\u00b2 : Fintype \u03b2 inst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2)) inst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b } \u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2 h\u03c6 : LeftInverse Quotient.mk'' \u03c6 \u03c9 : Quotient (orbitRel \u03b1 \u03b2) \u22a2 Nonempty { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } ** infer_instance ** Qed", + "informal": "" + }, + { + "formal": "WithZeroTopology.hasBasis_nhds_zero ** \u03b1 : Type u_1 \u0393\u2080 : Type u_2 inst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080 \u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080 l : Filter \u03b1 f : \u03b1 \u2192 \u0393\u2080 \u22a2 HasBasis (\ud835\udcdd 0) (fun \u03b3 => \u03b3 \u2260 0) Iio ** rw [nhds_zero] ** \u03b1 : Type u_1 \u0393\u2080 : Type u_2 inst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080 \u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080 l : Filter \u03b1 f : \u03b1 \u2192 \u0393\u2080 \u22a2 HasBasis (\u2a05 \u03b3, \u2a05 (_ : \u03b3 \u2260 0), \ud835\udcdf (Iio \u03b3)) (fun \u03b3 => \u03b3 \u2260 0) Iio ** refine' hasBasis_biInf_principal _ \u27e81, one_ne_zero\u27e9 ** \u03b1 : Type u_1 \u0393\u2080 : Type u_2 inst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080 \u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080 l : Filter \u03b1 f : \u03b1 \u2192 \u0393\u2080 \u22a2 DirectedOn ((fun \u03b3 => Iio \u03b3) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) fun \u03b3 => \u03b3 = 0 \u2192 False ** exact directedOn_iff_directed.2 (directed_of_inf fun a b hab => Iio_subset_Iio hab) ** Qed", + "informal": "" + }, + { + "formal": "Int.fdiv_self ** a : Int H : a \u2260 0 \u22a2 fdiv a a = 1 ** have := Int.mul_fdiv_cancel 1 H ** a : Int H : a \u2260 0 this : fdiv (1 * a) a = 1 \u22a2 fdiv a a = 1 ** rwa [Int.one_mul] at this ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.disjoint_supported_supported_iff ** \u03b1 : Type u_1 M : Type u_2 N : Type u_3 P : Type u_4 R : Type u_5 S : Type u_6 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : Semiring S inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : Module R M inst\u271d\u2074 : AddCommMonoid N inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : AddCommMonoid P inst\u271d\u00b9 : Module R P inst\u271d : Nontrivial M s t : Set \u03b1 \u22a2 Disjoint (supported M R s) (supported M R t) \u2194 Disjoint s t ** refine' \u27e8fun h => Set.disjoint_left.mpr fun x hx1 hx2 => _, disjoint_supported_supported\u27e9 ** \u03b1 : Type u_1 M : Type u_2 N : Type u_3 P : Type u_4 R : Type u_5 S : Type u_6 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : Semiring S inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : Module R M inst\u271d\u2074 : AddCommMonoid N inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : AddCommMonoid P inst\u271d\u00b9 : Module R P inst\u271d : Nontrivial M s t : Set \u03b1 h : Disjoint (supported M R s) (supported M R t) x : \u03b1 hx1 : x \u2208 s hx2 : x \u2208 t \u22a2 False ** rcases exists_ne (0 : M) with \u27e8y, hy\u27e9 ** case intro \u03b1 : Type u_1 M : Type u_2 N : Type u_3 P : Type u_4 R : Type u_5 S : Type u_6 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : Semiring S inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : Module R M inst\u271d\u2074 : AddCommMonoid N inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : AddCommMonoid P inst\u271d\u00b9 : Module R P inst\u271d : Nontrivial M s t : Set \u03b1 h : Disjoint (supported M R s) (supported M R t) x : \u03b1 hx1 : x \u2208 s hx2 : x \u2208 t y : M hy : y \u2260 0 \u22a2 False ** have := h.le_bot \u27e8single_mem_supported R y hx1, single_mem_supported R y hx2\u27e9 ** case intro \u03b1 : Type u_1 M : Type u_2 N : Type u_3 P : Type u_4 R : Type u_5 S : Type u_6 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : Semiring S inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : Module R M inst\u271d\u2074 : AddCommMonoid N inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : AddCommMonoid P inst\u271d\u00b9 : Module R P inst\u271d : Nontrivial M s t : Set \u03b1 h : Disjoint (supported M R s) (supported M R t) x : \u03b1 hx1 : x \u2208 s hx2 : x \u2208 t y : M hy : y \u2260 0 this : (fun\u2080 | x => y) \u2208 \u22a5 \u22a2 False ** rw [mem_bot, single_eq_zero] at this ** case intro \u03b1 : Type u_1 M : Type u_2 N : Type u_3 P : Type u_4 R : Type u_5 S : Type u_6 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : Semiring S inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : Module R M inst\u271d\u2074 : AddCommMonoid N inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : AddCommMonoid P inst\u271d\u00b9 : Module R P inst\u271d : Nontrivial M s t : Set \u03b1 h : Disjoint (supported M R s) (supported M R t) x : \u03b1 hx1 : x \u2208 s hx2 : x \u2208 t y : M hy : y \u2260 0 this : y = 0 \u22a2 False ** exact hy this ** Qed", + "informal": "" + }, + { + "formal": "EReal.bot_mul_of_neg ** x : EReal h : x < 0 \u22a2 \u22a5 * x = \u22a4 ** rw [EReal.mul_comm] ** x : EReal h : x < 0 \u22a2 x * \u22a5 = \u22a4 ** exact mul_bot_of_neg h ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.condexpL2_comp_continuousLinearMap ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 T : E' \u2192L[\u211d] E'' f : { x // x \u2208 Lp E' 2 } \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)) ** 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) _ _ _ ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 T : E' \u2192L[\u211d] E'' f : { x // x \u2208 Lp E' 2 } \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) x \u2202\u03bc ** intro s hs h\u03bcs ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 T : E' \u2192L[\u211d] E'' f : { x // x \u2208 Lp E' 2 } s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) x \u2202\u03bc ** 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)] ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 T : E' \u2192L[\u211d] E'' f : { x // x \u2208 Lp E' 2 } \u22a2 AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f))) \u03bc ** exact lpMeas.aeStronglyMeasurable' _ ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 T : E' \u2192L[\u211d] E'' f : { x // x \u2208 Lp E' 2 } \u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc ** have h_coe := T.coeFn_compLp (condexpL2 E' \ud835\udd5c hm f : \u03b1 \u2192\u2082[\u03bc] E') ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 T : E' \u2192L[\u211d] E'' f : { x // x \u2208 Lp E' 2 } h_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) \u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc ** rw [\u2190 EventuallyEq] at h_coe ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 T : E' \u2192L[\u211d] E'' f : { x // x \u2208 Lp E' 2 } h_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) \u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc ** refine' AEStronglyMeasurable'.congr _ h_coe.symm ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 T : E' \u2192L[\u211d] E'' f : { x // x \u2208 Lp E' 2 } h_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) \u22a2 AEStronglyMeasurable' m (fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)) \u03bc ** exact (lpMeas.aeStronglyMeasurable' (condexpL2 E' \ud835\udd5c hm f)).continuous_comp T.continuous ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.IsZero.of_iso ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u' inst\u271d : Category.{v', u'} D X Y : C hY : IsZero Y e : X \u2245 Y \u22a2 IsZero X ** refine' \u27e8fun Z => \u27e8\u27e8\u27e8e.hom \u226b hY.to_ Z\u27e9, fun f => _\u27e9\u27e9,\n fun Z => \u27e8\u27e8\u27e8hY.from_ Z \u226b e.inv\u27e9, fun f => _\u27e9\u27e9\u27e9 ** case refine'_1 C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u' inst\u271d : Category.{v', u'} D X Y : C hY : IsZero Y e : X \u2245 Y Z : C f : X \u27f6 Z \u22a2 f = default ** rw [\u2190 cancel_epi e.inv] ** case refine'_1 C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u' inst\u271d : Category.{v', u'} D X Y : C hY : IsZero Y e : X \u2245 Y Z : C f : X \u27f6 Z \u22a2 e.inv \u226b f = e.inv \u226b default ** apply hY.eq_of_src ** case refine'_2 C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u' inst\u271d : Category.{v', u'} D X Y : C hY : IsZero Y e : X \u2245 Y Z : C f : Z \u27f6 X \u22a2 f = default ** rw [\u2190 cancel_mono e.hom] ** case refine'_2 C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u' inst\u271d : Category.{v', u'} D X Y : C hY : IsZero Y e : X \u2245 Y Z : C f : Z \u27f6 X \u22a2 f \u226b e.hom = default \u226b e.hom ** apply hY.eq_of_tgt ** Qed", + "informal": "" + }, + { + "formal": "Matrix.isUnit_det_transpose ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 A B : Matrix n n \u03b1 h : IsUnit (det A) \u22a2 IsUnit (det A\u1d40) ** rw [det_transpose] ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 A B : Matrix n n \u03b1 h : IsUnit (det A) \u22a2 IsUnit (det A) ** exact h ** Qed", + "informal": "" + }, + { + "formal": "Matrix.isHermitian_inv ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : Type u_3 n : Type u_4 A\u271d : Matrix n n \u03b1 inst\u271d\u2074 : CommRing \u03b1 inst\u271d\u00b3 : StarRing \u03b1 inst\u271d\u00b2 : Fintype m inst\u271d\u00b9 : DecidableEq m A : Matrix m m \u03b1 inst\u271d : Invertible A h : IsHermitian A\u207b\u00b9 \u22a2 IsHermitian A ** rw [\u2190 inv_inv_of_invertible A] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : Type u_3 n : Type u_4 A\u271d : Matrix n n \u03b1 inst\u271d\u2074 : CommRing \u03b1 inst\u271d\u00b3 : StarRing \u03b1 inst\u271d\u00b2 : Fintype m inst\u271d\u00b9 : DecidableEq m A : Matrix m m \u03b1 inst\u271d : Invertible A h : IsHermitian A\u207b\u00b9 \u22a2 IsHermitian A\u207b\u00b9\u207b\u00b9 ** exact IsHermitian.inv h ** Qed", + "informal": "" + }, + { + "formal": "QuaternionAlgebra.mul_star_eq_coe ** S : Type u_1 T : Type u_2 R : Type u_3 inst\u271d : CommRing R c\u2081 c\u2082 r x y z : R a b c : \u210d[R,c\u2081,c\u2082] \u22a2 a * star a = \u2191(a * star a).re ** rw [\u2190 star_comm_self'] ** S : Type u_1 T : Type u_2 R : Type u_3 inst\u271d : CommRing R c\u2081 c\u2082 r x y z : R a b c : \u210d[R,c\u2081,c\u2082] \u22a2 star a * a = \u2191(star a * a).re ** exact a.star_mul_eq_coe ** Qed", + "informal": "" + }, + { + "formal": "Finset.image\u2082_union_right ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2078 : DecidableEq \u03b1' inst\u271d\u2077 : DecidableEq \u03b2' inst\u271d\u2076 : DecidableEq \u03b3 inst\u271d\u2075 : DecidableEq \u03b3' inst\u271d\u2074 : DecidableEq \u03b4 inst\u271d\u00b3 : DecidableEq \u03b4' inst\u271d\u00b2 : DecidableEq \u03b5 inst\u271d\u00b9 : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 inst\u271d : DecidableEq \u03b2 \u22a2 \u2191(image\u2082 f s (t \u222a t')) = \u2191(image\u2082 f s t \u222a image\u2082 f s t') ** push_cast ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2078 : DecidableEq \u03b1' inst\u271d\u2077 : DecidableEq \u03b2' inst\u271d\u2076 : DecidableEq \u03b3 inst\u271d\u2075 : DecidableEq \u03b3' inst\u271d\u2074 : DecidableEq \u03b4 inst\u271d\u00b3 : DecidableEq \u03b4' inst\u271d\u00b2 : DecidableEq \u03b5 inst\u271d\u00b9 : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 inst\u271d : DecidableEq \u03b2 \u22a2 image2 f (\u2191s) (\u2191t \u222a \u2191t') = image2 f \u2191s \u2191t \u222a image2 f \u2191s \u2191t' ** exact image2_union_right ** Qed", + "informal": "" + }, + { + "formal": "Nat.Primrec'.head ** n : \u2115 v : Vector \u2115 (Nat.succ n) \u22a2 Vector.get v 0 = Vector.head v ** simp [get_zero] ** Qed", + "informal": "" + }, + { + "formal": "norm_mk_zero ** M : Type u_1 N : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup M inst\u271d : SeminormedAddCommGroup N S : AddSubgroup M \u22a2 \u20160\u2016 = 0 ** erw [quotient_norm_eq_zero_iff] ** M : Type u_1 N : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup M inst\u271d : SeminormedAddCommGroup N S : AddSubgroup M \u22a2 0 \u2208 closure \u2191S ** exact subset_closure S.zero_mem ** Qed", + "informal": "" + }, + { + "formal": "bernoulli'_two ** A : Type u_1 inst\u271d\u00b9 : CommRing A inst\u271d : Algebra \u211a A \u22a2 bernoulli' 2 = 1 / 6 ** rw [bernoulli'_def] ** A : Type u_1 inst\u271d\u00b9 : CommRing A inst\u271d : Algebra \u211a A \u22a2 1 - \u2211 k in range 2, \u2191(Nat.choose 2 k) / (\u21912 - \u2191k + 1) * bernoulli' k = 1 / 6 ** norm_num [sum_range_succ, sum_range_succ, sum_range_zero] ** Qed", + "informal": "" + }, + { + "formal": "Stream'.even_cons_cons ** \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w a\u2081 a\u2082 : \u03b1 s : Stream' \u03b1 \u22a2 even (a\u2081 :: a\u2082 :: s) = a\u2081 :: even s ** unfold even ** \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w a\u2081 a\u2082 : \u03b1 s : Stream' \u03b1 \u22a2 corec (fun s => head s) (fun s => tail (tail s)) (a\u2081 :: a\u2082 :: s) = a\u2081 :: corec (fun s => head s) (fun s => tail (tail s)) s ** rw [corec_eq] ** \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w a\u2081 a\u2082 : \u03b1 s : Stream' \u03b1 \u22a2 head (a\u2081 :: a\u2082 :: s) :: corec (fun s => head s) (fun s => tail (tail s)) (tail (tail (a\u2081 :: a\u2082 :: s))) = a\u2081 :: corec (fun s => head s) (fun s => tail (tail s)) s ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Multiset.prod_toEnumFinset ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 m\u271d : Multiset \u03b1 \u03b2 : Type u_2 inst\u271d : CommMonoid \u03b2 m : Multiset \u03b1 f : \u03b1 \u2192 \u2115 \u2192 \u03b2 \u22a2 \u220f x in toEnumFinset m, f x.1 x.2 = \u220f x : ToType m, f x.fst \u2191x.snd ** rw [Fintype.prod_equiv m.coeEquiv (fun x \u21a6 f x x.2) fun x \u21a6 f x.1.1 x.1.2] ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 m\u271d : Multiset \u03b1 \u03b2 : Type u_2 inst\u271d : CommMonoid \u03b2 m : Multiset \u03b1 f : \u03b1 \u2192 \u2115 \u2192 \u03b2 \u22a2 \u220f x in toEnumFinset m, f x.1 x.2 = \u220f x : { x // x \u2208 toEnumFinset m }, f (\u2191x).1 (\u2191x).2 ** rw [\u2190 m.toEnumFinset.prod_coe_sort fun x \u21a6 f x.1 x.2] ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 m\u271d : Multiset \u03b1 \u03b2 : Type u_2 inst\u271d : CommMonoid \u03b2 m : Multiset \u03b1 f : \u03b1 \u2192 \u2115 \u2192 \u03b2 \u22a2 \u2200 (x : ToType m), f x.fst \u2191x.snd = f (\u2191(\u2191(coeEquiv m) x)).1 (\u2191(\u2191(coeEquiv m) x)).2 ** intro x ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 m\u271d : Multiset \u03b1 \u03b2 : Type u_2 inst\u271d : CommMonoid \u03b2 m : Multiset \u03b1 f : \u03b1 \u2192 \u2115 \u2192 \u03b2 x : ToType m \u22a2 f x.fst \u2191x.snd = f (\u2191(\u2191(coeEquiv m) x)).1 (\u2191(\u2191(coeEquiv m) x)).2 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MultilinearMap.map_sum_finset_aux ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d : \u2115 M : Fin (Nat.succ n\u271d) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n : \u2115 h : \u2211 i : \u03b9, 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) ** letI := fun i => Classical.decEq (\u03b1 i) ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d : \u2115 M : Fin (Nat.succ n\u271d) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n : \u2115 h : \u2211 i : \u03b9, Finset.card (A i) = n this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** induction' n using Nat.strong_induction_on with n IH generalizing A ** case h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, 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) ** by_cases Ai_empty : \u2203 i, A i = \u2205 ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u00ac\u2203 i, A i = \u2205 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** push_neg at Ai_empty ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** by_cases Ai_singleton : \u2200 i, (A i).card \u2264 1 ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u00ac\u2200 (i : \u03b9), Finset.card (A i) \u2264 1 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** push_neg at Ai_singleton ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2203 i, 1 < Finset.card (A i) \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** obtain \u27e8i\u2080, hi\u2080\u27e9 : \u2203 i, 1 < (A i).card := Ai_singleton ** case neg.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** 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 ** case neg.intro.intro.intro.intro.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** let B := Function.update A i\u2080 (A i\u2080 \\ {j\u2082}) ** case neg.intro.intro.intro.intro.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** let C := Function.update A i\u2080 {j\u2082} ** case neg.intro.intro.intro.intro.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** 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 ** case neg.intro.intro.intro.intro.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** 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 ** case neg.intro.intro.intro.intro.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** 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 ** case neg.intro.intro.intro.intro.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) pi_BC : piFinset A = piFinset B \u222a piFinset C \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** rw [A_eq_BC] ** case neg.intro.intro.intro.intro.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) pi_BC : piFinset A = piFinset B \u222a piFinset C \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)) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** simp only [MultilinearMap.map_add, Beq, Ceq, Brec, Crec, pi_BC] ** case neg.intro.intro.intro.intro.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) pi_BC : piFinset A = piFinset B \u222a piFinset C \u22a2 ((\u2211 x in piFinset (update A i\u2080 (A i\u2080 \\ {j\u2082})), \u2191f fun i => g i (x i)) + \u2211 x in piFinset (update A i\u2080 {j\u2082}), \u2191f fun i => g i (x i)) = \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) ** rw [\u2190 Finset.sum_union D] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2203 i, A i = \u2205 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** rcases Ai_empty with \u27e8i, hi\u27e9 ** case pos.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n i : \u03b9 hi : A i = \u2205 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** have : \u2211 j in A i, g i j = 0 := by rw [hi, Finset.sum_empty] ** case pos.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n i : \u03b9 hi : A i = \u2205 this : \u2211 j in A i, g i j = 0 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** rw [f.map_coord_zero i this] ** case pos.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n i : \u03b9 hi : A i = \u2205 this : \u2211 j in A i, g i j = 0 \u22a2 0 = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** 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 ** case pos.intro R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n i : \u03b9 hi : A i = \u2205 this\u271d : \u2211 j in A i, g i j = 0 this : piFinset A = \u2205 \u22a2 0 = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** rw [this, Finset.sum_empty] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n i : \u03b9 hi : A i = \u2205 \u22a2 \u2211 j in A i, g i j = 0 ** rw [hi, Finset.sum_empty] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n i : \u03b9 hi : A i = \u2205 this : \u2211 j in A i, g i j = 0 \u22a2 piFinset A = \u2205 ** refine Finset.eq_empty_of_forall_not_mem fun r hr => ?_ ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n i : \u03b9 hi : A i = \u2205 this : \u2211 j in A i, g i j = 0 r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A \u22a2 False ** have : r i \u2208 A i := mem_piFinset.mp hr i ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n i : \u03b9 hi : A i = \u2205 this\u271d : \u2211 j in A i, g i j = 0 r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A this : r i \u2208 A i \u22a2 False ** simp [hi] at this ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** 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)) ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** 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] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 this : \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 \u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i) ** simp only [Finset.sum_congr rfl this, Ai_card, card_piFinset, prod_const_one, one_nsmul,\n Finset.sum_const] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 \u22a2 \u2200 (i : \u03b9), Finset.card (A i) = 1 ** intro i ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 i : \u03b9 \u22a2 Finset.card (A i) = 1 ** have pos : Finset.card (A i) \u2260 0 := by simp [Finset.card_eq_zero, Ai_empty i] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 i : \u03b9 pos : Finset.card (A i) \u2260 0 \u22a2 Finset.card (A i) = 1 ** have : Finset.card (A i) \u2264 1 := Ai_singleton i ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 i : \u03b9 pos : Finset.card (A i) \u2260 0 this : Finset.card (A i) \u2264 1 \u22a2 Finset.card (A i) = 1 ** exact le_antisymm this (Nat.succ_le_of_lt (_root_.pos_iff_ne_zero.mpr pos)) ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 i : \u03b9 \u22a2 Finset.card (A i) \u2260 0 ** simp [Finset.card_eq_zero, Ai_empty i] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 \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 ** intro r hr ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 r : (i : \u03b9) \u2192 \u03b1 i hr : r \u2208 piFinset A \u22a2 (\u2191f fun i => g i (r i)) = \u2191f fun i => \u2211 j in A i, g i j ** congr with i ** case h.e_6.h.h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 r : (i : \u03b9) \u2192 \u03b1 i hr : r \u2208 piFinset A i : \u03b9 \u22a2 g i (r i) = \u2211 j in A i, g i j ** 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 ** case h.e_6.h.h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 r : (i : \u03b9) \u2192 \u03b1 i hr : r \u2208 piFinset A i : \u03b9 this : \u2200 (j : \u03b1 i), j \u2208 A i \u2192 g i j = g i (r i) \u22a2 g i (r i) = \u2211 j in A i, g i j ** simp only [Finset.sum_congr rfl this, Finset.mem_univ, Finset.sum_const, Ai_card i, one_nsmul] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 r : (i : \u03b9) \u2192 \u03b1 i hr : r \u2208 piFinset A i : \u03b9 \u22a2 \u2200 (j : \u03b1 i), j \u2208 A i \u2192 g i j = g i (r i) ** intro j hj ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 r : (i : \u03b9) \u2192 \u03b1 i hr : r \u2208 piFinset A i : \u03b9 j : \u03b1 i hj : j \u2208 A i \u22a2 g i j = g i (r i) ** congr ** case e_a R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 r : (i : \u03b9) \u2192 \u03b1 i hr : r \u2208 piFinset A i : \u03b9 j : \u03b1 i hj : j \u2208 A i \u22a2 j = r i ** apply Finset.card_le_one_iff.1 (Ai_singleton i) hj ** case e_a R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 Ai_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1 Ai_card : \u2200 (i : \u03b9), Finset.card (A i) = 1 r : (i : \u03b9) \u2192 \u03b1 i hr : r \u2208 piFinset A i : \u03b9 j : \u03b1 i hj : j \u2208 A i \u22a2 r i \u2208 A i ** exact mem_piFinset.mp hr i ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} \u22a2 \u2200 (i : \u03b9), B i \u2286 A i ** intro i ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} i : \u03b9 \u22a2 B i \u2286 A i ** by_cases hi : i = i\u2080 ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} i : \u03b9 hi : i = i\u2080 \u22a2 B i \u2286 A i ** rw [hi] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} i : \u03b9 hi : i = i\u2080 \u22a2 B i\u2080 \u2286 A i\u2080 ** simp only [sdiff_subset, update_same] ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} i : \u03b9 hi : \u00aci = i\u2080 \u22a2 B i \u2286 A i ** simp only [hi, update_noteq, Ne.def, not_false_iff, Finset.Subset.refl] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i \u22a2 \u2200 (i : \u03b9), C i \u2286 A i ** intro i ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i i : \u03b9 \u22a2 C i \u2286 A i ** by_cases hi : i = i\u2080 ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i i : \u03b9 hi : i = i\u2080 \u22a2 C i \u2286 A i ** rw [hi] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i i : \u03b9 hi : i = i\u2080 \u22a2 C i\u2080 \u2286 A i\u2080 ** simp only [hj\u2082, Finset.singleton_subset_iff, update_same] ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i i : \u03b9 hi : \u00aci = i\u2080 \u22a2 C i \u2286 A i ** simp only [hi, update_noteq, Ne.def, not_false_iff, Finset.Subset.refl] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i \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) ** ext i ** case h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 \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 ** by_cases hi : i = i\u2080 ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 \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 ** rw [hi, update_same] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 \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 ** 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] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 this : A i\u2080 = B i\u2080 \u222a C i\u2080 \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 ** rw [this] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 this : A i\u2080 = B i\u2080 \u222a C i\u2080 \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 ** refine Finset.sum_union <| Finset.disjoint_right.2 fun j hj => ?_ ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 this : A i\u2080 = B i\u2080 \u222a C i\u2080 j : \u03b1 i\u2080 hj : j \u2208 C i\u2080 \u22a2 \u00acj \u2208 B i\u2080 ** have : j = j\u2082 := by\n simpa using hj ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 this\u271d : A i\u2080 = B i\u2080 \u222a C i\u2080 j : \u03b1 i\u2080 hj : j \u2208 C i\u2080 this : j = j\u2082 \u22a2 \u00acj \u2208 B i\u2080 ** rw [this] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 this\u271d : A i\u2080 = B i\u2080 \u222a C i\u2080 j : \u03b1 i\u2080 hj : j \u2208 C i\u2080 this : j = j\u2082 \u22a2 \u00acj\u2082 \u2208 B i\u2080 ** simp only [mem_sdiff, eq_self_iff_true, not_true, not_false_iff, Finset.mem_singleton,\n update_same, and_false_iff] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 \u22a2 A i\u2080 = B i\u2080 \u222a C i\u2080 ** simp only [Function.update_same, Finset.sdiff_union_self_eq_union] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 \u22a2 A i\u2080 = A i\u2080 \u222a {j\u2082} ** symm ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 \u22a2 A i\u2080 \u222a {j\u2082} = A i\u2080 ** simp only [hj\u2082, Finset.singleton_subset_iff, Finset.union_eq_left] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : i = i\u2080 this : A i\u2080 = B i\u2080 \u222a C i\u2080 j : \u03b1 i\u2080 hj : j \u2208 C i\u2080 \u22a2 j = j\u2082 ** simpa using hj ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i i : \u03b9 hi : \u00aci = i\u2080 \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 ** simp [hi] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) \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 ** ext i ** case h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) i : \u03b9 \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 ** by_cases hi : i = i\u2080 ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) i : \u03b9 hi : i = i\u2080 \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 ** rw [hi] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) i : \u03b9 hi : i = i\u2080 \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 ** simp only [update_same] ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) i : \u03b9 hi : \u00aci = i\u2080 \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 ** simp only [hi, update_noteq, Ne.def, not_false_iff] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 \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 ** ext i ** case h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 i : \u03b9 \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 ** by_cases hi : i = i\u2080 ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 i : \u03b9 hi : i = i\u2080 \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 ** rw [hi] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 i : \u03b9 hi : i = i\u2080 \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 ** simp only [update_same] ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 i : \u03b9 hi : \u00aci = i\u2080 \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 ** simp only [hi, update_noteq, Ne.def, not_false_iff] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 \u22a2 (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) ** 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)) ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 this : \u2211 i : \u03b9, Finset.card (B i) < \u2211 i : \u03b9, Finset.card (A i) \u22a2 (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) ** rw [h] at this ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 this : \u2211 i : \u03b9, Finset.card (B 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) ** exact IH _ this B rfl ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 \u22a2 \u2211 i : \u03b9, Finset.card (B i) < \u2211 i : \u03b9, Finset.card (A i) ** refine'\n Finset.sum_lt_sum (fun i _ => Finset.card_le_of_subset (B_subset_A i))\n \u27e8i\u2080, Finset.mem_univ _, _\u27e9 ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 \u22a2 Finset.card (B i\u2080) < Finset.card (A i\u2080) ** have : {j\u2082} \u2286 A i\u2080 := by simp [hj\u2082] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 this : {j\u2082} \u2286 A i\u2080 \u22a2 Finset.card (B i\u2080) < Finset.card (A i\u2080) ** simp only [Finset.card_sdiff this, Function.update_same, Finset.card_singleton] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 this : {j\u2082} \u2286 A i\u2080 \u22a2 Finset.card (A i\u2080) - 1 < Finset.card (A i\u2080) ** exact Nat.pred_lt (ne_of_gt (lt_trans Nat.zero_lt_one hi\u2080)) ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 \u22a2 {j\u2082} \u2286 A i\u2080 ** simp [hj\u2082] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) \u22a2 (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) ** 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 ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) this : \u2211 i : \u03b9, Finset.card (C i) < \u2211 i : \u03b9, Finset.card (A i) \u22a2 (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) ** rw [h] at this ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) this : \u2211 i : \u03b9, Finset.card (C 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) ** exact IH _ this C rfl ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) \u22a2 Finset.card (C i\u2080) < Finset.card (A i\u2080) ** simp [hi\u2080] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) \u22a2 Disjoint (B i\u2080) (C i\u2080) ** simp ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) \u22a2 piFinset A = piFinset B \u222a piFinset C ** apply Finset.Subset.antisymm ** case H\u2081 R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) \u22a2 piFinset A \u2286 piFinset B \u222a piFinset C ** intro r hr ** case H\u2081 R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A \u22a2 r \u2208 piFinset B \u222a piFinset C ** by_cases hri\u2080 : r i\u2080 = j\u2082 ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : r i\u2080 = j\u2082 \u22a2 r \u2208 piFinset B \u222a piFinset C ** apply Finset.mem_union_right ** case pos.h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : r i\u2080 = j\u2082 \u22a2 r \u2208 piFinset C ** refine mem_piFinset.2 fun i => ?_ ** case pos.h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : r i\u2080 = j\u2082 i : \u03b9 \u22a2 r i \u2208 C i ** by_cases hi : i = i\u2080 ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : r i\u2080 = j\u2082 i : \u03b9 hi : i = i\u2080 \u22a2 r i \u2208 C i ** have : r i\u2080 \u2208 C i\u2080 := by simp [hri\u2080] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : r i\u2080 = j\u2082 i : \u03b9 hi : i = i\u2080 this : r i\u2080 \u2208 C i\u2080 \u22a2 r i \u2208 C i ** rwa [hi] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : r i\u2080 = j\u2082 i : \u03b9 hi : i = i\u2080 \u22a2 r i\u2080 \u2208 C i\u2080 ** simp [hri\u2080] ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : r i\u2080 = j\u2082 i : \u03b9 hi : \u00aci = i\u2080 \u22a2 r i \u2208 C i ** simp [hi, mem_piFinset.1 hr i] ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : \u00acr i\u2080 = j\u2082 \u22a2 r \u2208 piFinset B \u222a piFinset C ** apply Finset.mem_union_left ** case neg.h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : \u00acr i\u2080 = j\u2082 \u22a2 r \u2208 piFinset B ** refine mem_piFinset.2 fun i => ?_ ** case neg.h R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : \u00acr i\u2080 = j\u2082 i : \u03b9 \u22a2 r i \u2208 B i ** by_cases hi : i = i\u2080 ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : \u00acr i\u2080 = j\u2082 i : \u03b9 hi : i = i\u2080 \u22a2 r i \u2208 B i ** have : r i\u2080 \u2208 B i\u2080 := by simp [hri\u2080, mem_piFinset.1 hr i\u2080] ** case pos R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : \u00acr i\u2080 = j\u2082 i : \u03b9 hi : i = i\u2080 this : r i\u2080 \u2208 B i\u2080 \u22a2 r i \u2208 B i ** rwa [hi] ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : \u00acr i\u2080 = j\u2082 i : \u03b9 hi : i = i\u2080 \u22a2 r i\u2080 \u2208 B i\u2080 ** simp [hri\u2080, mem_piFinset.1 hr i\u2080] ** case neg R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) r : (a : \u03b9) \u2192 \u03b1 a hr : r \u2208 piFinset A hri\u2080 : \u00acr i\u2080 = j\u2082 i : \u03b9 hi : \u00aci = i\u2080 \u22a2 r i \u2208 B i ** simp [hi, mem_piFinset.1 hr i] ** case H\u2082 R : Type uR S : Type uS \u03b9 : Type u\u03b9 n\u271d\u00b9 : \u2115 M : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : AddCommMonoid M\u2083 inst\u271d\u2077 : AddCommMonoid M' inst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i) inst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u2074 : Module R M\u2082 inst\u271d\u00b3 : Module R M\u2083 inst\u271d\u00b2 : Module R M' f f' : MultilinearMap R M\u2081 M\u2082 \u03b1 : \u03b9 \u2192 Type u_1 g : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i A\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Fintype \u03b9 n\u271d : \u2115 h\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d this : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i) n : \u2115 IH : \u2200 (m : \u2115), m < n \u2192 \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)), \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) A : (i : \u03b9) \u2192 Finset (\u03b1 i) h : \u2211 i : \u03b9, Finset.card (A i) = n Ai_empty : \u2200 (i : \u03b9), A i \u2260 \u2205 i\u2080 : \u03b9 hi\u2080 : 1 < Finset.card (A i\u2080) j\u2081 j\u2082 : \u03b1 i\u2080 left\u271d : j\u2081 \u2208 A i\u2080 hj\u2082 : j\u2082 \u2208 A i\u2080 right\u271d : j\u2081 \u2260 j\u2082 B : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082}) C : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082} B_subset_A : \u2200 (i : \u03b9), B i \u2286 A i C_subset_A : \u2200 (i : \u03b9), C i \u2286 A i A_eq_BC : (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) Beq : 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 Ceq : 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 Brec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i) Crec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i) D : Disjoint (piFinset B) (piFinset C) \u22a2 piFinset B \u222a piFinset C \u2286 piFinset A ** exact\n Finset.union_subset (piFinset_subset _ _ fun i => B_subset_A i)\n (piFinset_subset _ _ fun i => C_subset_A i) ** Qed", + "informal": "" + }, + { + "formal": "Surreal.nsmul_pow_two_powHalf' ** n k : \u2115 \u22a2 2 ^ n \u2022 powHalf (n + k) = powHalf k ** induction' k with k hk ** case zero n : \u2115 \u22a2 2 ^ n \u2022 powHalf (n + Nat.zero) = powHalf Nat.zero ** simp only [add_zero, Surreal.nsmul_pow_two_powHalf, Nat.zero_eq, eq_self_iff_true,\n Surreal.powHalf_zero] ** case succ n k : \u2115 hk : 2 ^ n \u2022 powHalf (n + k) = powHalf k \u22a2 2 ^ n \u2022 powHalf (n + Nat.succ k) = powHalf (Nat.succ k) ** rw [\u2190 double_powHalf_succ_eq_powHalf (n + k), \u2190 double_powHalf_succ_eq_powHalf k,\n smul_algebra_smul_comm] at hk ** case succ n k : \u2115 hk : 2 \u2022 2 ^ n \u2022 powHalf (Nat.succ (n + k)) = 2 \u2022 powHalf (Nat.succ k) \u22a2 2 ^ n \u2022 powHalf (n + Nat.succ k) = powHalf (Nat.succ k) ** rwa [\u2190 zsmul_eq_zsmul_iff' two_ne_zero] ** Qed", + "informal": "" + }, + { + "formal": "eq_iff_eq_of_cmp_eq_cmp ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 x y : \u03b1 \u03b2 : Type u_3 inst\u271d : LinearOrder \u03b2 x' y' : \u03b2 h : cmp x y = cmp x' y' \u22a2 x = y \u2194 x' = y' ** 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)] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.IsPushout.of_map ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D F : C \u2964 D W X Y Z : C f : W \u27f6 X g : W \u27f6 Y h : X \u27f6 Z i : Y \u27f6 Z inst\u271d : ReflectsColimit (span f g) F e : f \u226b h = g \u226b i H : IsPushout (F.map f) (F.map g) (F.map h) (F.map i) \u22a2 IsPushout f g h i ** refine' \u27e8\u27e8e\u27e9, \u27e8isColimitOfReflects F <| _\u27e9\u27e9 ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D F : C \u2964 D W X Y Z : C f : W \u27f6 X g : W \u27f6 Y h : X \u27f6 Z i : Y \u27f6 Z inst\u271d : ReflectsColimit (span f g) F e : f \u226b h = g \u226b i H : IsPushout (F.map f) (F.map g) (F.map h) (F.map i) \u22a2 IsColimit (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i))) ** refine'\n (IsColimit.equivOfNatIsoOfIso (spanCompIso F f g) _ _ (WalkingSpan.ext _ _ _)).symm\n H.isColimit ** case refine'_1 C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D F : C \u2964 D W X Y Z : C f : W \u27f6 X g : W \u27f6 Y h : X \u27f6 Z i : Y \u27f6 Z inst\u271d : ReflectsColimit (span f g) F e : f \u226b h = g \u226b i H : IsPushout (F.map f) (F.map g) (F.map h) (F.map i) \u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).pt \u2245 (cocone H).pt case refine'_2 C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D F : C \u2964 D W X Y Z : C f : W \u27f6 X g : W \u27f6 Y h : X \u27f6 Z i : Y \u27f6 Z inst\u271d : ReflectsColimit (span f g) F e : f \u226b h = g \u226b i H : IsPushout (F.map f) (F.map g) (F.map h) (F.map i) \u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).\u03b9.app WalkingCospan.left \u226b ?refine'_1.hom = (cocone H).\u03b9.app WalkingCospan.left case refine'_3 C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D F : C \u2964 D W X Y Z : C f : W \u27f6 X g : W \u27f6 Y h : X \u27f6 Z i : Y \u27f6 Z inst\u271d : ReflectsColimit (span f g) F e : f \u226b h = g \u226b i H : IsPushout (F.map f) (F.map g) (F.map h) (F.map i) \u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).\u03b9.app WalkingCospan.right \u226b ?refine'_1.hom = (cocone H).\u03b9.app WalkingCospan.right ** exacts [Iso.refl _, (Category.comp_id _).trans (Category.id_comp _),\n (Category.comp_id _).trans (Category.id_comp _)] ** Qed", + "informal": "" + }, + { + "formal": "Ideal.isJacobson_of_isIntegral ** R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R \u22a2 IsJacobson S ** rw [isJacobson_iff_prime_eq] ** R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R \u22a2 \u2200 (P : Ideal S), IsPrime P \u2192 jacobson P = P ** intro P hP ** R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P \u22a2 jacobson P = P ** by_cases hP_top : comap (algebraMap R S) P = \u22a4 ** case pos R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : comap (algebraMap R S) P = \u22a4 \u22a2 jacobson P = P ** simp [comap_eq_top_iff.1 hP_top] ** case neg R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 \u22a2 jacobson P = P ** haveI : Nontrivial (R \u29f8 comap (algebraMap R S) P) := Quotient.nontrivial hP_top ** case neg R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 this : Nontrivial (R \u29f8 comap (algebraMap R S) P) \u22a2 jacobson P = P ** rw [jacobson_eq_iff_jacobson_quotient_eq_bot] ** case neg R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 this : Nontrivial (R \u29f8 comap (algebraMap R S) P) \u22a2 jacobson \u22a5 = \u22a5 ** refine' eq_bot_of_comap_eq_bot (isIntegral_quotient_of_isIntegral hRS) _ ** case neg R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 this : Nontrivial (R \u29f8 comap (algebraMap R S) P) \u22a2 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) (jacobson \u22a5) = \u22a5 ** 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] ** case neg R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 this : Nontrivial (R \u29f8 comap (algebraMap R S) P) \u22a2 sInf (comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J}) \u2264 jacobson \u22a5 ** refine' sInf_le_sInf fun J hJ => _ ** case neg R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 this : Nontrivial (R \u29f8 comap (algebraMap R S) P) J : Ideal (R \u29f8 comap (algebraMap R S) P) hJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J} \u22a2 J \u2208 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J} ** simp only [true_and_iff, Set.mem_image, bot_le, Set.mem_setOf_eq] ** case neg R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 this : Nontrivial (R \u29f8 comap (algebraMap R S) P) J : Ideal (R \u29f8 comap (algebraMap R S) P) hJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J} \u22a2 \u2203 x, IsMaximal x \u2227 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) x = J ** have : J.IsMaximal := by simpa using hJ ** case neg R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 this\u271d : Nontrivial (R \u29f8 comap (algebraMap R S) P) J : Ideal (R \u29f8 comap (algebraMap R S) P) hJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J} this : IsMaximal J \u22a2 \u2203 x, IsMaximal x \u2227 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) x = J ** exact exists_ideal_over_maximal_of_isIntegral (isIntegral_quotient_of_isIntegral hRS) J\n (comap_bot_le_of_injective _ algebraMap_quotient_injective) ** R : Type u_1 S : Type u_2 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S I : Ideal R inst\u271d : Algebra R S hRS : Algebra.IsIntegral R S hR : IsJacobson R P : Ideal S hP : IsPrime P hP_top : \u00accomap (algebraMap R S) P = \u22a4 this : Nontrivial (R \u29f8 comap (algebraMap R S) P) J : Ideal (R \u29f8 comap (algebraMap R S) P) hJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J} \u22a2 IsMaximal J ** simpa using hJ ** Qed", + "informal": "" + }, + { + "formal": "ExponentExists.isTorsion ** G : Type u_1 H : Type u_2 inst\u271d\u00b9 : Group G N : Subgroup G inst\u271d : Group H h : ExponentExists G g : G \u22a2 IsOfFinOrder g ** obtain \u27e8n, npos, hn\u27e9 := h ** case intro.intro G : Type u_1 H : Type u_2 inst\u271d\u00b9 : Group G N : Subgroup G inst\u271d : Group H g : G n : \u2115 npos : 0 < n hn : \u2200 (g : G), g ^ n = 1 \u22a2 IsOfFinOrder g ** exact (isOfFinOrder_iff_pow_eq_one g).mpr \u27e8n, npos, hn g\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "SetTheory.PGame.zero_lf_le ** x : PGame \u22a2 0 \u29cf x \u2194 \u2203 i, 0 \u2264 moveLeft x i ** rw [lf_iff_exists_le] ** x : PGame \u22a2 ((\u2203 i, 0 \u2264 moveLeft x i) \u2228 \u2203 j, moveRight 0 j \u2264 x) \u2194 \u2203 i, 0 \u2264 moveLeft x i ** simp ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.mem_vars_bind\u2081 ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S inst\u271d : CommSemiring T f\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R j : \u03c4 h : j \u2208 vars (\u2191(bind\u2081 f) \u03c6) \u22a2 \u2203 i, i \u2208 vars \u03c6 \u2227 j \u2208 vars (f i) ** classical\nsimpa only [exists_prop, Finset.mem_biUnion, mem_support_iff, Ne.def] using vars_bind\u2081 f \u03c6 h ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S inst\u271d : CommSemiring T f\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R j : \u03c4 h : j \u2208 vars (\u2191(bind\u2081 f) \u03c6) \u22a2 \u2203 i, i \u2208 vars \u03c6 \u2227 j \u2208 vars (f i) ** simpa only [exists_prop, Finset.mem_biUnion, mem_support_iff, Ne.def] using vars_bind\u2081 f \u03c6 h ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.fp_iff_derivBFamily ** o : Ordinal.{u} f : (b : Ordinal.{u}) \u2192 b < o \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} \u22a2 (\u2200 (i : Ordinal.{u}) (hi : i < o), f i hi a = a) \u2194 \u2203 b, derivBFamily o f b = a ** rw [\u2190 le_iff_derivBFamily H] ** o : Ordinal.{u} f : (b : Ordinal.{u}) \u2192 b < o \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} \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 ** refine' \u27e8fun h i hi => le_of_eq (h i hi), fun h i hi => _\u27e9 ** o : Ordinal.{u} f : (b : Ordinal.{u}) \u2192 b < o \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} h : \u2200 (i : Ordinal.{u}) (hi : i < o), f i hi a \u2264 a i : Ordinal.{u} hi : i < o \u22a2 f i hi a = a ** rw [\u2190 (H i hi).le_iff_eq] ** o : Ordinal.{u} f : (b : Ordinal.{u}) \u2192 b < o \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} h : \u2200 (i : Ordinal.{u}) (hi : i < o), f i hi a \u2264 a i : Ordinal.{u} hi : i < o \u22a2 f i hi a \u2264 a ** exact h i hi ** Qed", + "informal": "" + }, + { + "formal": "le_nhds_mul ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : Mul M inst\u271d : ContinuousMul M a b : M \u22a2 \ud835\udcdd a * \ud835\udcdd b \u2264 \ud835\udcdd (a * b) ** rw [\u2190 map\u2082_mul, \u2190 map_uncurry_prod, \u2190 nhds_prod_eq] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : Mul M inst\u271d : ContinuousMul M a b : M \u22a2 map (Function.uncurry fun x x_1 => x * x_1) (\ud835\udcdd (a, b)) \u2264 \ud835\udcdd (a * b) ** exact continuous_mul.tendsto _ ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.condexp_mono ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm) ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** exact (condexp_ae_eq_condexpL1 hm _).trans_le\n ((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm) ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** simp_rw [condexp_of_not_le hm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 0 \u2264\u1d50[\u03bc] 0 ** rfl ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 \u2264\u1d50[\u03bc] 0 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "le_one_of_mul_le_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MulOneClass \u03b1 inst\u271d\u00b9 : LE \u03b1 inst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 a b : \u03b1 h : a * b \u2264 b \u22a2 a * ?m.22212 h \u2264 1 * ?m.22212 h ** simpa only [one_mul] ** Qed", + "informal": "" + }, + { + "formal": "leftLim_eq_of_tendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 h\u03b1 : TopologicalSpace \u03b1 h'\u03b1 : OrderTopology \u03b1 inst\u271d : T2Space \u03b2 f : \u03b1 \u2192 \u03b2 a : \u03b1 y : \u03b2 h : \ud835\udcdd[Iio a] a \u2260 \u22a5 h' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y) \u22a2 leftLim f a = y ** have h'' : \u2203 y, Tendsto f (\ud835\udcdd[<] a) (\ud835\udcdd y) := \u27e8y, h'\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 h\u03b1 : TopologicalSpace \u03b1 h'\u03b1 : OrderTopology \u03b1 inst\u271d : T2Space \u03b2 f : \u03b1 \u2192 \u03b2 a : \u03b1 y : \u03b2 h : \ud835\udcdd[Iio a] a \u2260 \u22a5 h' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y) h'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y) \u22a2 leftLim f a = y ** simp only [leftLim, h, h'', not_true, or_self_iff, if_false] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 h\u03b1 : TopologicalSpace \u03b1 h'\u03b1 : OrderTopology \u03b1 inst\u271d : T2Space \u03b2 f : \u03b1 \u2192 \u03b2 a : \u03b1 y : \u03b2 h : \ud835\udcdd[Iio a] a \u2260 \u22a5 h' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y) h'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y) \u22a2 limUnder (\ud835\udcdd[Iio a] a) f = y ** haveI := neBot_iff.2 h ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 h\u03b1 : TopologicalSpace \u03b1 h'\u03b1 : OrderTopology \u03b1 inst\u271d : T2Space \u03b2 f : \u03b1 \u2192 \u03b2 a : \u03b1 y : \u03b2 h : \ud835\udcdd[Iio a] a \u2260 \u22a5 h' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y) h'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y) this : NeBot (\ud835\udcdd[Iio a] a) \u22a2 limUnder (\ud835\udcdd[Iio a] a) f = y ** exact lim_eq h' ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Perm.support_swap ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 f g : Perm \u03b1 x y : \u03b1 h : x \u2260 y \u22a2 support (swap x y) = {x, y} ** ext z ** case a \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 f g : Perm \u03b1 x y : \u03b1 h : x \u2260 y z : \u03b1 \u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y} ** by_cases hx : z = x ** case pos \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 f g : Perm \u03b1 x y : \u03b1 h : x \u2260 y z : \u03b1 hx : z = x \u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y} case neg \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 f g : Perm \u03b1 x y : \u03b1 h : x \u2260 y z : \u03b1 hx : \u00acz = x \u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y} ** any_goals simpa [hx] using h.symm ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 f g : Perm \u03b1 x y : \u03b1 h : x \u2260 y z : \u03b1 hx : \u00acz = x \u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y} ** simpa [hx] using h.symm ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 f g : Perm \u03b1 x y : \u03b1 h : x \u2260 y z : \u03b1 hx : \u00acz = x hy : \u00acz = y \u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y} ** simp [swap_apply_of_ne_of_ne, hx, hy] <;>\nexact h ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.JordanDecomposition.toSignedMeasure_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j : JordanDecomposition \u03b1 r : \u211d\u22650 \u22a2 toSignedMeasure (r \u2022 j) = r \u2022 toSignedMeasure j ** ext1 i hi ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j : JordanDecomposition \u03b1 r : \u211d\u22650 i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(toSignedMeasure (r \u2022 j)) i = \u2191(r \u2022 toSignedMeasure j) i ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.nfp_le_iff ** f : Ordinal.{u} \u2192 Ordinal.{u} a b : Ordinal.{u} \u22a2 nfp f a \u2264 b \u2194 \u2200 (n : \u2115), f^[n] a \u2264 b ** rw [\u2190 sup_iterate_eq_nfp] ** f : Ordinal.{u} \u2192 Ordinal.{u} a b : Ordinal.{u} \u22a2 (fun a => sup fun n => f^[n] a) a \u2264 b \u2194 \u2200 (n : \u2115), f^[n] a \u2264 b ** exact sup_le_iff ** Qed", + "informal": "" + }, + { + "formal": "Complex.continuous_sinh ** \u22a2 Continuous sinh ** change Continuous fun z => (exp z - exp (-z)) / 2 ** \u22a2 Continuous fun z => (cexp z - cexp (-z)) / 2 ** continuity ** Qed", + "informal": "" + }, + { + "formal": "IsClosed.mul_right_of_isCompact ** \u03b1 : Type u \u03b2 : Type v G : Type w H : Type x inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : Group \u03b1 inst\u271d : TopologicalGroup \u03b1 s t : Set \u03b1 ht : IsClosed t hs : IsCompact s \u22a2 IsClosed (t * s) ** rw [\u2190 image_op_smul] ** \u03b1 : Type u \u03b2 : Type v G : Type w H : Type x inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : Group \u03b1 inst\u271d : TopologicalGroup \u03b1 s t : Set \u03b1 ht : IsClosed t hs : IsCompact s \u22a2 IsClosed (op '' s \u2022 t) ** exact IsClosed.smul_left_of_isCompact ht (hs.image continuous_op) ** Qed", + "informal": "" + }, + { + "formal": "Complex.cos_neg ** x y : \u2102 \u22a2 cos (-x) = cos x ** simp [cos, sub_eq_add_neg, exp_neg, add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Ideal.pow_sup_eq_top ** R : Type u \u03b9 : Type u_1 inst\u271d : CommSemiring R I J K L : Ideal R n : \u2115 h : I \u2294 J = \u22a4 \u22a2 I ^ n \u2294 J = \u22a4 ** rw [\u2190 Finset.card_range n, \u2190 Finset.prod_const] ** R : Type u \u03b9 : Type u_1 inst\u271d : CommSemiring R I J K L : Ideal R n : \u2115 h : I \u2294 J = \u22a4 \u22a2 (\u220f _x in Finset.range n, I) \u2294 J = \u22a4 ** exact prod_sup_eq_top fun _ _ => h ** Qed", + "informal": "" + }, + { + "formal": "Matrix.rank_mul_le_left ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 R : Type u_5 m_fin : Fintype m inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : Fintype o inst\u271d\u00b9 : CommRing R inst\u271d : StrongRankCondition R A : Matrix m n R B : Matrix n o R \u22a2 rank (A * B) \u2264 rank A ** rw [rank, rank, mulVecLin_mul] ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 R : Type u_5 m_fin : Fintype m inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : Fintype o inst\u271d\u00b9 : CommRing R inst\u271d : StrongRankCondition R A : Matrix m n R B : Matrix n o R \u22a2 finrank R { x // x \u2208 LinearMap.range (LinearMap.comp (mulVecLin A) (mulVecLin B)) } \u2264 finrank R { x // x \u2208 LinearMap.range (mulVecLin A) } ** exact Cardinal.toNat_le_of_le_of_lt_aleph0 (rank_lt_aleph0 _ _) (LinearMap.rank_comp_le_left _ _) ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Perm.sumCongr_swap_refl ** \u03b1\u271d : Sort ?u.118581 inst\u271d\u00b2 : DecidableEq \u03b1\u271d \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 i j : \u03b1 \u22a2 sumCongr (swap i j) (Equiv.refl \u03b2) = swap (Sum.inl i) (Sum.inl j) ** ext x ** case H \u03b1\u271d : Sort ?u.118581 inst\u271d\u00b2 : DecidableEq \u03b1\u271d \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 i j : \u03b1 x : \u03b1 \u2295 \u03b2 \u22a2 \u2191(sumCongr (swap i j) (Equiv.refl \u03b2)) x = \u2191(swap (Sum.inl i) (Sum.inl j)) x ** cases x ** case H.inl \u03b1\u271d : Sort ?u.118581 inst\u271d\u00b2 : DecidableEq \u03b1\u271d \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 i j val\u271d : \u03b1 \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) ** simp only [Equiv.sumCongr_apply, Sum.map, coe_refl, comp.right_id, Sum.elim_inl, comp_apply,\n swap_apply_def, Sum.inl.injEq] ** case H.inl \u03b1\u271d : Sort ?u.118581 inst\u271d\u00b2 : DecidableEq \u03b1\u271d \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 i j val\u271d : \u03b1 \u22a2 Sum.inl (if val\u271d = i then j else if val\u271d = j then i else val\u271d) = if val\u271d = i then Sum.inl j else if val\u271d = j then Sum.inl i else Sum.inl val\u271d ** split_ifs <;> rfl ** case H.inr \u03b1\u271d : Sort ?u.118581 inst\u271d\u00b2 : DecidableEq \u03b1\u271d \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 i j : \u03b1 val\u271d : \u03b2 \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) ** simp [Sum.map, swap_apply_of_ne_of_ne] ** Qed", + "informal": "" + }, + { + "formal": "dvd_of_eq ** \u03b1 : Type u_1 inst\u271d : Monoid \u03b1 a b : \u03b1 h : a = b \u22a2 a \u2223 b ** rw [h] ** Qed", + "informal": "" + }, + { + "formal": "ZFSet.singleton_injective ** x y : ZFSet H : {x} = {y} \u22a2 x = y ** let this := congr_arg sUnion H ** x y : ZFSet H : {x} = {y} this : \u22c3\u2080 {x} = \u22c3\u2080 {y} := congr_arg sUnion H \u22a2 x = y ** rwa [sUnion_singleton, sUnion_singleton] at this ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Abelian.exact_epi_comp_iff ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z W : C h : W \u27f6 X inst\u271d : Epi h \u22a2 Exact (h \u226b f) g \u2194 Exact f g ** refine' \u27e8fun hfg => _, fun h => exact_epi_comp h\u27e9 ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z W : C h : W \u27f6 X inst\u271d : Epi h hfg : Exact (h \u226b f) g \u22a2 Exact f g ** 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 ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z W : C h : W \u27f6 X inst\u271d : Epi h hfg : Exact (h \u226b f) g hc : IsColimit (CokernelCofork.of\u03c0 (Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0))) (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0)) := isCokernelOfComp h (h \u226b f) (colimit.isColimit (parallelPair (h \u226b f) 0)) (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0) (_ : h \u226b f = h \u226b f) \u22a2 Exact f g ** refine' (exact_iff' _ _ (limit.isLimit _) hc).2 \u27e8_, ((exact_iff _ _).1 hfg).2\u27e9 ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z W : C h : W \u27f6 X inst\u271d : Epi h hfg : Exact (h \u226b f) g hc : IsColimit (CokernelCofork.of\u03c0 (Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0))) (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0)) := isCokernelOfComp h (h \u226b f) (colimit.isColimit (parallelPair (h \u226b f) 0)) (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0) (_ : h \u226b f = h \u226b f) \u22a2 f \u226b g = 0 ** exact zero_of_epi_comp h (by rw [\u2190 hfg.1, Category.assoc]) ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z W : C h : W \u27f6 X inst\u271d : Epi h hfg : Exact (h \u226b f) g \u22a2 f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0 ** rw [\u2190 cancel_epi h, \u2190 Category.assoc, CokernelCofork.condition, comp_zero] ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : Abelian C X Y Z : C f : X \u27f6 Y g : Y \u27f6 Z W : C h : W \u27f6 X inst\u271d : Epi h hfg : Exact (h \u226b f) g hc : IsColimit (CokernelCofork.of\u03c0 (Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0))) (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0)) := isCokernelOfComp h (h \u226b f) (colimit.isColimit (parallelPair (h \u226b f) 0)) (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0) (_ : h \u226b f = h \u226b f) \u22a2 h \u226b f \u226b g = 0 ** rw [\u2190 hfg.1, Category.assoc] ** Qed", + "informal": "" + }, + { + "formal": "Function.extend_div ** I : Type u \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f\u271d : I \u2192 Type v\u2081 g : I \u2192 Type v\u2082 h : I \u2192 Type v\u2083 x y : (i : I) \u2192 f\u271d i i : I inst\u271d : Div \u03b3 f : \u03b1 \u2192 \u03b2 g\u2081 g\u2082 : \u03b1 \u2192 \u03b3 e\u2081 e\u2082 : \u03b2 \u2192 \u03b3 \u22a2 extend f (g\u2081 / g\u2082) (e\u2081 / e\u2082) = extend f g\u2081 e\u2081 / extend f g\u2082 e\u2082 ** classical\nfunext x\nsimp [Function.extend_def, apply_dite\u2082] ** I : Type u \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f\u271d : I \u2192 Type v\u2081 g : I \u2192 Type v\u2082 h : I \u2192 Type v\u2083 x y : (i : I) \u2192 f\u271d i i : I inst\u271d : Div \u03b3 f : \u03b1 \u2192 \u03b2 g\u2081 g\u2082 : \u03b1 \u2192 \u03b3 e\u2081 e\u2082 : \u03b2 \u2192 \u03b3 \u22a2 extend f (g\u2081 / g\u2082) (e\u2081 / e\u2082) = extend f g\u2081 e\u2081 / extend f g\u2082 e\u2082 ** funext x ** case h I : Type u \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f\u271d : I \u2192 Type v\u2081 g : I \u2192 Type v\u2082 h : I \u2192 Type v\u2083 x\u271d y : (i : I) \u2192 f\u271d i i : I inst\u271d : Div \u03b3 f : \u03b1 \u2192 \u03b2 g\u2081 g\u2082 : \u03b1 \u2192 \u03b3 e\u2081 e\u2082 : \u03b2 \u2192 \u03b3 x : \u03b2 \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 ** simp [Function.extend_def, apply_dite\u2082] ** Qed", + "informal": "" + }, + { + "formal": "Int.four_dvd_add_or_sub_of_odd ** m n a b : \u2124 ha : Odd a hb : Odd b \u22a2 4 \u2223 a + b \u2228 4 \u2223 a - b ** obtain \u27e8m, rfl\u27e9 := ha ** case intro m\u271d n b : \u2124 hb : Odd b m : \u2124 \u22a2 4 \u2223 2 * m + 1 + b \u2228 4 \u2223 2 * m + 1 - b ** obtain \u27e8n, rfl\u27e9 := hb ** case intro.intro m\u271d n\u271d m n : \u2124 \u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1) ** obtain h | h := Int.even_or_odd (m + n) ** case intro.intro.inl m\u271d n\u271d m n : \u2124 h : Even (m + n) \u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1) ** right ** case intro.intro.inl.h m\u271d n\u271d m n : \u2124 h : Even (m + n) \u22a2 4 \u2223 2 * m + 1 - (2 * n + 1) ** rw [Int.even_add, \u2190 Int.even_sub] at h ** case intro.intro.inl.h m\u271d n\u271d m n : \u2124 h : Even (m - n) \u22a2 4 \u2223 2 * m + 1 - (2 * n + 1) ** obtain \u27e8k, hk\u27e9 := h ** case intro.intro.inl.h.intro m\u271d n\u271d m n k : \u2124 hk : m - n = k + k \u22a2 4 \u2223 2 * m + 1 - (2 * n + 1) ** convert dvd_mul_right 4 k using 1 ** case h.e'_4 m\u271d n\u271d m n k : \u2124 hk : m - n = k + k \u22a2 2 * m + 1 - (2 * n + 1) = 4 * k ** rw [eq_add_of_sub_eq hk, mul_add, add_assoc, add_sub_cancel, \u2190 two_mul, \u2190 mul_assoc] ** case h.e'_4 m\u271d n\u271d m n k : \u2124 hk : m - n = k + k \u22a2 2 * 2 * k = 4 * k ** rfl ** case intro.intro.inr m\u271d n\u271d m n : \u2124 h : Odd (m + n) \u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1) ** left ** case intro.intro.inr.h m\u271d n\u271d m n : \u2124 h : Odd (m + n) \u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) ** obtain \u27e8k, hk\u27e9 := h ** case intro.intro.inr.h.intro m\u271d n\u271d m n k : \u2124 hk : m + n = 2 * k + 1 \u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) ** convert dvd_mul_right 4 (k + 1) using 1 ** case h.e'_4 m\u271d n\u271d m n k : \u2124 hk : m + n = 2 * k + 1 \u22a2 2 * m + 1 + (2 * n + 1) = 4 * (k + 1) ** 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] ** case h.e'_4 m\u271d n\u271d m n k : \u2124 hk : m + n = 2 * k + 1 \u22a2 2 * 2 * k + (2 * 1 + (1 + 1)) = 4 * k + 4 * 1 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "IsGalois.of_separable_splitting_field_aux ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) = Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef } ** have h : IsIntegral K x :=\n isIntegral_of_isScalarTower (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x) ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x \u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) = Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef } ** have h1 : p \u2260 0 := fun hp => by\n rw [hp, Polynomial.aroots_zero] at hx\n exact Multiset.not_mem_zero x hx ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 \u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) = Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef } ** 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 ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p \u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) = Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef } ** 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 ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this \u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) = Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef } ** 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 ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) \u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) = Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef } ** rw [Fintype.card_congr key_equiv, Fintype.card_sigma, IntermediateField.adjoin.finrank h] ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) \u22a2 (Finset.sum Finset.univ fun a => Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)) = Fintype.card (K \u2192\u2090[F] E) * Polynomial.natDegree (minpoly K x) ** apply Finset.sum_const_nat ** case h\u2081 F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) \u22a2 \u2200 (x_1 : K \u2192\u2090[F] E), 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) ** intro f _ ** case h\u2081 F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) f : K \u2192\u2090[F] E a\u271d : f \u2208 Finset.univ \u22a2 Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) = Polynomial.natDegree (minpoly K x) ** rw [\u2190 @IntermediateField.card_algHom_adjoin_integral K _ E _ _ x E _ (RingHom.toAlgebra f) h] ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp\u271d : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x hp : p = 0 \u22a2 False ** rw [hp, Polynomial.aroots_zero] at hx ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp\u271d : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 0 inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x hp : p = 0 \u22a2 False ** exact Multiset.not_mem_zero x hx ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 \u22a2 minpoly K x \u2223 Polynomial.map (algebraMap F K) p ** apply minpoly.dvd ** case hp F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 \u22a2 \u2191(Polynomial.aeval x) (Polynomial.map (algebraMap F K) p) = 0 ** rw [Polynomial.aeval_def, Polynomial.eval\u2082_map, \u2190 Polynomial.eval_map, \u2190\n IsScalarTower.algebraMap_eq] ** case hp F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 \u22a2 Polynomial.eval x (Polynomial.map (algebraMap F E) p) = 0 ** exact (Polynomial.mem_roots (Polynomial.map_ne_zero h1)).mp hx ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p \u22a2 ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) ** change (K\u27eex\u27ef \u2192\u2090[F] E) \u2243 \u03a3 f : K \u2192\u2090[F] E, _ ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p \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) ** exact algHomEquivSigma ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this f : K \u2192\u2090[F] E \u22a2 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) ** have := Fintype.ofEquiv _ key_equiv ** F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this f : K \u2192\u2090[F] E this : Fintype ((f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)) \u22a2 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) ** apply Fintype.ofInjective (Sigma.mk f) fun _ _ H => eq_of_heq (Sigma.ext_iff.mp H).2 ** case h\u2081 F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) f : K \u2192\u2090[F] E a\u271d : f \u2208 Finset.univ \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) ** congr! ** case h\u2081.h_sep F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) f : K \u2192\u2090[F] E a\u271d : f \u2208 Finset.univ \u22a2 Polynomial.Separable (minpoly K x) ** exact Polynomial.Separable.of_dvd ((Polynomial.separable_map (algebraMap F K)).mpr hp) h2 ** case h\u2081.h_splits F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) f : K \u2192\u2090[F] E a\u271d : f \u2208 Finset.univ \u22a2 Polynomial.Splits (algebraMap K E) (minpoly K x) ** refine' Polynomial.splits_of_splits_of_dvd _ (Polynomial.map_ne_zero h1) _ h2 ** case h\u2081.h_splits F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) f : K \u2192\u2090[F] E a\u271d : f \u2208 Finset.univ \u22a2 Polynomial.Splits (algebraMap K E) (Polynomial.map (algebraMap F K) p) ** rw [Polynomial.splits_map_iff, \u2190 @IsScalarTower.algebraMap_eq _ _ _ _ _ _ _ (_) _ _] ** case h\u2081.h_splits F : Type u_1 inst\u271d\u2078 : Field F E : Type u_2 inst\u271d\u2077 : Field E inst\u271d\u2076 : Algebra F E p : F[X] hFE : FiniteDimensional F E sp : Polynomial.IsSplittingField F E p hp : Polynomial.Separable p K : Type u_3 inst\u271d\u2075 : Field K inst\u271d\u2074 : Algebra F K inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : IsScalarTower F K E x : E hx : x \u2208 Polynomial.aroots p E inst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E) inst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) h : IsIntegral K x h1 : p \u2260 0 h2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p key_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F 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) := let_fun this := algHomEquivSigma; this this : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) f : K \u2192\u2090[F] E a\u271d : f \u2208 Finset.univ \u22a2 Polynomial.Splits (algebraMap F E) p ** exact sp.splits ** Qed", + "informal": "" + }, + { + "formal": "Bundle.contMDiffAt_section ** \ud835\udd5c : Type u_1 B : Type u_2 B' : Type u_3 F : Type u_4 M : Type u_5 E : B \u2192 Type u_6 inst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2075 : NormedAddCommGroup F inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E) inst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x) EB : Type u_7 inst\u271d\u00b9\u00b9 : NormedAddCommGroup EB inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB HB : Type u_8 inst\u271d\u2079 : TopologicalSpace HB IB : ModelWithCorners \ud835\udd5c EB HB E' : B \u2192 Type u_9 inst\u271d\u2078 : (x : B) \u2192 Zero (E' x) EM : Type u_10 inst\u271d\u2077 : NormedAddCommGroup EM inst\u271d\u2076 : NormedSpace \ud835\udd5c EM HM : Type u_11 inst\u271d\u2075 : TopologicalSpace HM IM : ModelWithCorners \ud835\udd5c EM HM inst\u271d\u2074 : TopologicalSpace M inst\u271d\u00b3 : ChartedSpace HM M Is : SmoothManifoldWithCorners IM M n : \u2115\u221e inst\u271d\u00b2 : TopologicalSpace B inst\u271d\u00b9 : ChartedSpace HB B inst\u271d : FiberBundle F E s : (x : B) \u2192 E x x\u2080 : B \u22a2 ContMDiffAt IB (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n (fun x => TotalSpace.mk' F x (s x)) x\u2080 \u2194 ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) { proj := x, snd := s x }).2) x\u2080 ** simp_rw [contMDiffAt_totalSpace, and_iff_right_iff_imp] ** \ud835\udd5c : Type u_1 B : Type u_2 B' : Type u_3 F : Type u_4 M : Type u_5 E : B \u2192 Type u_6 inst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2075 : NormedAddCommGroup F inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E) inst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x) EB : Type u_7 inst\u271d\u00b9\u00b9 : NormedAddCommGroup EB inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB HB : Type u_8 inst\u271d\u2079 : TopologicalSpace HB IB : ModelWithCorners \ud835\udd5c EB HB E' : B \u2192 Type u_9 inst\u271d\u2078 : (x : B) \u2192 Zero (E' x) EM : Type u_10 inst\u271d\u2077 : NormedAddCommGroup EM inst\u271d\u2076 : NormedSpace \ud835\udd5c EM HM : Type u_11 inst\u271d\u2075 : TopologicalSpace HM IM : ModelWithCorners \ud835\udd5c EM HM inst\u271d\u2074 : TopologicalSpace M inst\u271d\u00b3 : ChartedSpace HM M Is : SmoothManifoldWithCorners IM M n : \u2115\u221e inst\u271d\u00b2 : TopologicalSpace B inst\u271d\u00b9 : ChartedSpace HB B inst\u271d : FiberBundle F E s : (x : B) \u2192 E x x\u2080 : B \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 ContMDiffAt IB IB n (fun x => x) x\u2080 ** intro ** \ud835\udd5c : Type u_1 B : Type u_2 B' : Type u_3 F : Type u_4 M : Type u_5 E : B \u2192 Type u_6 inst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2075 : NormedAddCommGroup F inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E) inst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x) EB : Type u_7 inst\u271d\u00b9\u00b9 : NormedAddCommGroup EB inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB HB : Type u_8 inst\u271d\u2079 : TopologicalSpace HB IB : ModelWithCorners \ud835\udd5c EB HB E' : B \u2192 Type u_9 inst\u271d\u2078 : (x : B) \u2192 Zero (E' x) EM : Type u_10 inst\u271d\u2077 : NormedAddCommGroup EM inst\u271d\u2076 : NormedSpace \ud835\udd5c EM HM : Type u_11 inst\u271d\u2075 : TopologicalSpace HM IM : ModelWithCorners \ud835\udd5c EM HM inst\u271d\u2074 : TopologicalSpace M inst\u271d\u00b3 : ChartedSpace HM M Is : SmoothManifoldWithCorners IM M n : \u2115\u221e inst\u271d\u00b2 : TopologicalSpace B inst\u271d\u00b9 : ChartedSpace HB B inst\u271d : FiberBundle F E s : (x : B) \u2192 E x x\u2080 : B a\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 \u22a2 ContMDiffAt IB IB n (fun x => x) x\u2080 ** exact contMDiffAt_id ** Qed", + "informal": "" + }, + { + "formal": "le_min ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 h\u2081 : c \u2264 a h\u2082 : c \u2264 b \u22a2 c \u2264 min a b ** 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 ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 h\u2081 : c \u2264 a h\u2082 : c \u2264 b h : a \u2264 b \u22a2 c \u2264 min a b ** simp [min_def, if_pos h] ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 h\u2081 : c \u2264 a h\u2082 : c \u2264 b h : a \u2264 b \u22a2 c \u2264 a ** exact h\u2081 ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 h\u2081 : c \u2264 a h\u2082 : c \u2264 b h : \u00aca \u2264 b \u22a2 c \u2264 min a b ** simp [min_def, if_neg h] ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 h\u2081 : c \u2264 a h\u2082 : c \u2264 b h : \u00aca \u2264 b \u22a2 c \u2264 b ** exact h\u2082 ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.rootSet_X_pow ** R : Type u S : Type v k : Type y A : Type z a b : R n\u271d : \u2115 inst\u271d\u00b3 : Field R p q : R[X] inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : IsDomain S inst\u271d : Algebra R S n : \u2115 hn : n \u2260 0 \u22a2 rootSet (X ^ n) S = {0} ** rw [\u2190 one_mul (X ^ n : R[X]), \u2190 C_1, rootSet_C_mul_X_pow hn] ** R : Type u S : Type v k : Type y A : Type z a b : R n\u271d : \u2115 inst\u271d\u00b3 : Field R p q : R[X] inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : IsDomain S inst\u271d : Algebra R S n : \u2115 hn : n \u2260 0 \u22a2 1 \u2260 0 ** exact one_ne_zero ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.GrothendieckTopology.plusMap_comp ** C : Type u inst\u271d\u00b3 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b2 : Category.{max v u, w} D inst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P) P\u271d : C\u1d52\u1d56 \u2964 D inst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D P Q R : C\u1d52\u1d56 \u2964 D \u03b7 : P \u27f6 Q \u03b3 : Q \u27f6 R \u22a2 plusMap J (\u03b7 \u226b \u03b3) = plusMap J \u03b7 \u226b plusMap J \u03b3 ** ext : 2 ** case w.h C : Type u inst\u271d\u00b3 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b2 : Category.{max v u, w} D inst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P) P\u271d : C\u1d52\u1d56 \u2964 D inst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D P Q R : C\u1d52\u1d56 \u2964 D \u03b7 : P \u27f6 Q \u03b3 : Q \u27f6 R x\u271d : C\u1d52\u1d56 \u22a2 (plusMap J (\u03b7 \u226b \u03b3)).app x\u271d = (plusMap J \u03b7 \u226b plusMap J \u03b3).app x\u271d ** refine' colimit.hom_ext (fun S => _) ** case w.h C : Type u inst\u271d\u00b3 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b2 : Category.{max v u, w} D inst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P) P\u271d : C\u1d52\u1d56 \u2964 D inst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D P Q R : C\u1d52\u1d56 \u2964 D \u03b7 : P \u27f6 Q \u03b3 : Q \u27f6 R x\u271d : C\u1d52\u1d56 S : (Cover J x\u271d.unop)\u1d52\u1d56 \u22a2 colimit.\u03b9 (diagram J P x\u271d.unop) S \u226b (plusMap J (\u03b7 \u226b \u03b3)).app x\u271d = colimit.\u03b9 (diagram J P x\u271d.unop) S \u226b (plusMap J \u03b7 \u226b plusMap J \u03b3).app x\u271d ** simp [plusMap, J.diagramNatTrans_comp] ** Qed", + "informal": "" + }, + { + "formal": "HasStrictDerivAt.cpow ** f g : \u2102 \u2192 \u2102 s : Set \u2102 f' g' x c : \u2102 hf : HasStrictDerivAt f f' x hg : HasStrictDerivAt g g' x h0 : 0 < (f x).re \u2228 (f x).im \u2260 0 \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 ** simpa using (hf.cpow hg h0).hasStrictDerivAt ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.lintegral_iSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f \u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** set c : \u211d\u22650 \u2192 \u211d\u22650\u221e := (\u2191) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some \u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** set F := fun a : \u03b1 => \u2a06 n, f n a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a \u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** have _ : Measurable F := measurable_iSup hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F \u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' le_antisymm _ (iSup_lintegral_le _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F \u22a2 lintegral \u03bc F \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** rw [lintegral_eq_nnreal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F \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 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' iSup_le fun s => iSup_le fun hsf => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' ENNReal.le_of_forall_lt_one_mul_le fun a ha => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x a : \u211d\u22650\u221e ha : a < 1 \u22a2 a * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** rcases ENNReal.lt_iff_exists_coe.1 ha with \u27e8r, rfl, _\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha : \u2191r < 1 \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** have ha : r < 1 := ENNReal.coe_lt_coe.1 ha ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** let rs := s.map fun a => r * a ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** 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 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** 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) ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** 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) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s \u22a2 const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs ** ext1 a ** case H \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s a : \u03b1 \u22a2 \u2191(const \u03b1 \u2191r * SimpleFunc.map c s) a = \u2191(SimpleFunc.map c rs) a ** exact ENNReal.coe_mul.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs \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} ** intro p ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e \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} ** rw [\u2190 inter_iUnion] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e \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} ** nth_rw 1 [\u2190 inter_univ (map c rs \u207b\u00b9' {p})] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e \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} ** refine' Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e x : \u03b1 hx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a} ** by_cases p_eq : p = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e x : \u03b1 hx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} p_eq : \u00acp = 0 \u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a} ** simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e x : \u03b1 p_eq : \u00acp = 0 hx : \u2191(r * \u2191s x) = p \u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a} ** subst hx ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** have : r * s x \u2260 0 := by rwa [Ne, \u2190 ENNReal.coe_eq_zero] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this : r * \u2191s x \u2260 0 \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** have : s x \u2260 0 := by\n refine' mt _ this\n intro h\n rw [h, mul_zero] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d : r * \u2191s x \u2260 0 this : \u2191s x \u2260 0 \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** 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) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d\u00b9 : r * \u2191s x \u2260 0 this\u271d : \u2191s x \u2260 0 this : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** rcases lt_iSup_iff.1 this with \u27e8i, hi\u27e9 ** case neg.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d\u00b9 : r * \u2191s x \u2260 0 this\u271d : \u2191s x \u2260 0 this : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x i : \u2115 hi : \u2191(SimpleFunc.map c rs) x < f i x \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** exact mem_iUnion.2 \u27e8i, le_of_lt hi\u27e9 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e x : \u03b1 hx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} p_eq : p = 0 \u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a} ** simp [p_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 \u22a2 r * \u2191s x \u2260 0 ** rwa [Ne, \u2190 ENNReal.coe_eq_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this : r * \u2191s x \u2260 0 \u22a2 \u2191s x \u2260 0 ** refine' mt _ this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this : r * \u2191s x \u2260 0 \u22a2 \u2191s x = 0 \u2192 r * \u2191s x = 0 ** intro h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this : r * \u2191s x \u2260 0 h : \u2191s x = 0 \u22a2 r * \u2191s x = 0 ** rw [h, mul_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d : r * \u2191s x \u2260 0 this : \u2191s x \u2260 0 \u22a2 \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x ** refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d : r * \u2191s x \u2260 0 this : \u2191s x \u2260 0 \u22a2 \u2191rs x < \u2191s x ** suffices r * s x < 1 * s x by simpa ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d : r * \u2191s x \u2260 0 this : \u2191s x \u2260 0 \u22a2 r * \u2191s x < 1 * \u2191s x ** exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d\u00b9 : r * \u2191s x \u2260 0 this\u271d : \u2191s x \u2260 0 this : r * \u2191s x < 1 * \u2191s x \u22a2 \u2191rs x < \u2191s x ** simpa ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} \u22a2 \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} ** intro r i j h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} r : \u211d\u22650\u221e i j : \u2115 h : i \u2264 j \u22a2 (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) i \u2264 (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) j ** refine' inter_subset_inter (Subset.refl _) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} r : \u211d\u22650\u221e i j : \u2115 h : i \u2264 j \u22a2 {a | r \u2264 f i a} \u2286 {a | r \u2264 f j a} ** intro x (hx : r \u2264 f i x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} r : \u211d\u22650\u221e i j : \u2115 h : i \u2264 j x : \u03b1 hx : r \u2264 f i x \u22a2 x \u2208 {a | r \u2264 f j a} ** exact le_trans hx (h_mono h x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map c s) \u03bc = \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r}) ** rw [\u2190 const_mul_lintegral, eq_rs, SimpleFunc.lintegral] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r}) = \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}) ** simp only [(eq _).symm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d\u00b9 : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} x : \u211d\u22650\u221e x\u271d : x \u2208 SimpleFunc.range (SimpleFunc.map c rs) \u22a2 x * \u2191\u2191\u03bc (\u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a}) = \u2a06 n, x * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a}) ** rw [measure_iUnion_eq_iSup (directed_of_sup <| mono x), ENNReal.mul_iSup] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \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}) = \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}) ** rw [ENNReal.finset_sum_iSup_nat] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \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}) ** intro p i j h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} p : \u211d\u22650\u221e i j : \u2115 h : i \u2264 j \u22a2 (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) i \u2264 (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) j ** exact mul_le_mul_left' (measure_mono <| mono p h) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \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 \u2a06 n, SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc ** refine' iSup_mono fun n => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 \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 SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc ** rw [restrict_lintegral _ (h_meas n)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 \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 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) ** refine' le_of_eq (Finset.sum_congr rfl fun r _ => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d\u00b9 : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 r : \u211d\u22650\u221e x\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs) \u22a2 r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) = r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) ** congr 2 with a ** case e_a.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d\u00b9 : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 r : \u211d\u22650\u221e x\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs) a : \u03b1 \u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u2194 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} ** refine' and_congr_right _ ** case e_a.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d\u00b9 : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 r : \u211d\u22650\u221e x\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs) a : \u03b1 \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}) ** simp (config := { contextual := true }) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2a06 n, SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' iSup_mono fun n => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 \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 ** rw [\u2190 SimpleFunc.lintegral_eq_lintegral] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 \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 ** refine' lintegral_mono fun a => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 a : \u03b1 \u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a ** simp only [map_apply] at h_meas ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \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} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(r * \u2191s a) \u2264 f n a} n : \u2115 a : \u03b1 \u22a2 indicator {a | \u2191(r * \u2191s a) \u2264 f n a} (fun x => \u2191(r * \u2191s x)) a \u2264 f n a ** exact indicator_apply_le id ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.OuterMeasure.toMeasure_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u22a2 inst\u271d \u2264 OuterMeasure.caratheodory \u22a4 ** rw [OuterMeasure.top_caratheodory] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u22a2 inst\u271d \u2264 \u22a4 ** exact le_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u22a4 s \u2264 \u2191\u2191(OuterMeasure.toMeasure \u22a4 (_ : inst\u271d \u2264 OuterMeasure.caratheodory \u22a4)) s ** cases' s.eq_empty_or_nonempty with h h <;>\n simp [h, toMeasure_apply \u22a4 _ hs, OuterMeasure.top_apply] ** Qed", + "informal": "" + }, + { + "formal": "Liouville.liouvilleWith ** x : \u211d hx : Liouville x p : \u211d \u22a2 LiouvilleWith p x ** suffices : LiouvilleWith \u2308p\u2309\u208a x ** x : \u211d hx : Liouville x p : \u211d this : LiouvilleWith (\u2191\u2308p\u2309\u208a) x \u22a2 LiouvilleWith p x case this x : \u211d hx : Liouville x p : \u211d \u22a2 LiouvilleWith (\u2191\u2308p\u2309\u208a) x ** exact this.mono (Nat.le_ceil p) ** case this x : \u211d hx : Liouville x p : \u211d \u22a2 LiouvilleWith (\u2191\u2308p\u2309\u208a) x ** refine \u27e81, ((eventually_gt_atTop 1).and_frequently (hx.frequently_exists_num \u2308p\u2309\u208a)).mono ?_\u27e9 ** case this x : \u211d hx : Liouville x p : \u211d \u22a2 \u2200 (x_1 : \u2115), (1 < x_1 \u2227 \u2203 a, x \u2260 \u2191a / \u2191x_1 \u2227 |x - \u2191a / \u2191x_1| < 1 / \u2191x_1 ^ \u2308p\u2309\u208a) \u2192 \u2203 m, x \u2260 \u2191m / \u2191x_1 \u2227 |x - \u2191m / \u2191x_1| < 1 / \u2191x_1 ^ \u2191\u2308p\u2309\u208a ** rintro b \u27e8_hb, a, hne, hlt\u27e9 ** case this.intro.intro.intro x : \u211d hx : Liouville x p : \u211d b : \u2115 _hb : 1 < b a : \u2124 hne : x \u2260 \u2191a / \u2191b hlt : |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2308p\u2309\u208a \u22a2 \u2203 m, x \u2260 \u2191m / \u2191b \u2227 |x - \u2191m / \u2191b| < 1 / \u2191b ^ \u2191\u2308p\u2309\u208a ** refine \u27e8a, hne, ?_\u27e9 ** case this.intro.intro.intro x : \u211d hx : Liouville x p : \u211d b : \u2115 _hb : 1 < b a : \u2124 hne : x \u2260 \u2191a / \u2191b hlt : |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2308p\u2309\u208a \u22a2 |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2191\u2308p\u2309\u208a ** rwa [rpow_nat_cast] ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.oangle_midpoint_rev_left ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P hd2 : Fact (finrank \u211d V = 2) inst\u271d : Module.Oriented \u211d V (Fin 2) p\u2081 p\u2082 p\u2083 : P \u22a2 \u2221 (midpoint \u211d p\u2082 p\u2081) p\u2082 p\u2083 = \u2221 p\u2081 p\u2082 p\u2083 ** rw [midpoint_comm, oangle_midpoint_left] ** Qed", + "informal": "" + }, + { + "formal": "Int.le_max_left ** a b : Int \u22a2 a \u2264 max a b ** rw [Int.max_def] ** a b : Int \u22a2 a \u2264 if a \u2264 b then b else a ** split <;> simp [*] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.noZeroSMulDivisors_iff_torsion_eq_bot ** R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R \u22a2 NoZeroSMulDivisors R M \u2194 torsion R M = \u22a5 ** constructor <;> intro h ** case mp R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h : NoZeroSMulDivisors R M \u22a2 torsion R M = \u22a5 ** haveI : NoZeroSMulDivisors R M := h ** case mp R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h this : NoZeroSMulDivisors R M \u22a2 torsion R M = \u22a5 ** rw [eq_bot_iff] ** case mp R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h this : NoZeroSMulDivisors R M \u22a2 torsion R M \u2264 \u22a5 ** rintro x \u27e8a, hax\u27e9 ** case mp.intro R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h this : NoZeroSMulDivisors R M x : M a : { x // x \u2208 R\u2070 } hax : a \u2022 x = 0 \u22a2 x \u2208 \u22a5 ** change (a : R) \u2022 x = 0 at hax ** case mp.intro R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h this : NoZeroSMulDivisors R M x : M a : { x // x \u2208 R\u2070 } hax : \u2191a \u2022 x = 0 \u22a2 x \u2208 \u22a5 ** cases' eq_zero_or_eq_zero_of_smul_eq_zero hax with h0 h0 ** case mp.intro.inl R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h this : NoZeroSMulDivisors R M x : M a : { x // x \u2208 R\u2070 } hax : \u2191a \u2022 x = 0 h0 : \u2191a = 0 \u22a2 x \u2208 \u22a5 ** exfalso ** case mp.intro.inl.h R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h this : NoZeroSMulDivisors R M x : M a : { x // x \u2208 R\u2070 } hax : \u2191a \u2022 x = 0 h0 : \u2191a = 0 \u22a2 False ** exact nonZeroDivisors.coe_ne_zero a h0 ** case mp.intro.inr R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h this : NoZeroSMulDivisors R M x : M a : { x // x \u2208 R\u2070 } hax : \u2191a \u2022 x = 0 h0 : x = 0 \u22a2 x \u2208 \u22a5 ** exact h0 ** R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h : torsion R M = \u22a5 a : R x : M hax : a \u2022 x = 0 \u22a2 a = 0 \u2228 x = 0 ** by_cases ha : a = 0 ** case pos R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h : torsion R M = \u22a5 a : R x : M hax : a \u2022 x = 0 ha : a = 0 \u22a2 a = 0 \u2228 x = 0 ** left ** case pos.h R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h : torsion R M = \u22a5 a : R x : M hax : a \u2022 x = 0 ha : a = 0 \u22a2 a = 0 ** exact ha ** case neg R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h : torsion R M = \u22a5 a : R x : M hax : a \u2022 x = 0 ha : \u00aca = 0 \u22a2 a = 0 \u2228 x = 0 ** right ** case neg.h R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h : torsion R M = \u22a5 a : R x : M hax : a \u2022 x = 0 ha : \u00aca = 0 \u22a2 x = 0 ** rw [\u2190 mem_bot R, \u2190 h] ** case neg.h R : Type u_1 M : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : NoZeroDivisors R inst\u271d : Nontrivial R h : torsion R M = \u22a5 a : R x : M hax : a \u2022 x = 0 ha : \u00aca = 0 \u22a2 x \u2208 torsion R M ** exact \u27e8\u27e8a, mem_nonZeroDivisors_of_ne_zero ha\u27e9, hax\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "List.Pairwise.forall ** \u03b1 : Type u_1 \u03b2 : Type u_2 R S T : \u03b1 \u2192 \u03b1 \u2192 Prop a : \u03b1 l : List \u03b1 hR : Symmetric R hl : Pairwise R l \u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 l \u2192 \u2200 \u2983b : \u03b1\u2984, b \u2208 l \u2192 a \u2260 b \u2192 R a b ** apply Pairwise.forall_of_forall ** case H \u03b1 : Type u_1 \u03b2 : Type u_2 R S T : \u03b1 \u2192 \u03b1 \u2192 Prop a : \u03b1 l : List \u03b1 hR : Symmetric R hl : Pairwise R l \u22a2 Symmetric fun x y => x \u2260 y \u2192 R x y ** exact fun a b h hne => hR (h hne.symm) ** case H\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 R S T : \u03b1 \u2192 \u03b1 \u2192 Prop a : \u03b1 l : List \u03b1 hR : Symmetric R hl : Pairwise R l \u22a2 \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2260 x \u2192 R x x ** exact fun _ _ hx => (hx rfl).elim ** case H\u2082 \u03b1 : Type u_1 \u03b2 : Type u_2 R S T : \u03b1 \u2192 \u03b1 \u2192 Prop a : \u03b1 l : List \u03b1 hR : Symmetric R hl : Pairwise R l \u22a2 Pairwise (fun x y => x \u2260 y \u2192 R x y) l ** exact hl.imp (@fun a b h _ => by exact h) ** \u03b1 : Type u_1 \u03b2 : Type u_2 R S T : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 l : List \u03b1 hR : Symmetric R hl : Pairwise R l a b : \u03b1 h : R a b x\u271d : a \u2260 b \u22a2 R a b ** exact h ** Qed", + "informal": "" + }, + { + "formal": "CauchyFilter.nonempty_cauchyFilter_iff ** \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u22a2 Nonempty (CauchyFilter \u03b1) \u2194 Nonempty \u03b1 ** constructor <;> rintro \u27e8c\u27e9 ** case mp.intro \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 c : CauchyFilter \u03b1 \u22a2 Nonempty \u03b1 ** have := eq_univ_iff_forall.1 denseEmbedding_pureCauchy.toDenseInducing.closure_range c ** case mp.intro \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 c : CauchyFilter \u03b1 this : c \u2208 closure (range pureCauchy) \u22a2 Nonempty \u03b1 ** obtain \u27e8_, \u27e8_, a, _\u27e9\u27e9 := mem_closure_iff.1 this _ isOpen_univ trivial ** case mp.intro.intro.intro.intro \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 c : CauchyFilter \u03b1 this : c \u2208 closure (range pureCauchy) w\u271d : CauchyFilter \u03b1 left\u271d : w\u271d \u2208 univ a : \u03b1 h\u271d : pureCauchy a = w\u271d \u22a2 Nonempty \u03b1 ** exact \u27e8a\u27e9 ** case mpr.intro \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 c : \u03b1 \u22a2 Nonempty (CauchyFilter \u03b1) ** exact \u27e8pureCauchy c\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Filter.tendsto_lift ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 f f\u2081 f\u2082 : Filter \u03b1 g g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2 m : \u03b3 \u2192 \u03b2 l : Filter \u03b3 \u22a2 Tendsto m l (Filter.lift f g) \u2194 \u2200 (s : Set \u03b1), s \u2208 f \u2192 Tendsto m l (g s) ** simp only [Filter.lift, tendsto_iInf] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.comap_map_eq_self ** R : Type u_1 R\u2082 : Type u_2 K : Type u_3 M : Type u_4 M\u2082 : Type u_5 V : Type u_6 S : Type u_7 inst\u271d\u2076 : Semiring R inst\u271d\u2075 : Semiring R\u2082 inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : AddCommGroup M\u2082 inst\u271d\u00b9 : Module R\u2082 M\u2082 \u03c4\u2081\u2082 : R \u2192+* R\u2082 inst\u271d : RingHomSurjective \u03c4\u2081\u2082 F : Type u_8 sc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082 f : F p : Submodule R M h : LinearMap.ker f \u2264 p \u22a2 comap f (map f p) = p ** rw [Submodule.comap_map_eq, sup_of_le_left h] ** Qed", + "informal": "" + }, + { + "formal": "toIocDiv_sub ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c : \u03b1 n : \u2124 a b : \u03b1 \u22a2 toIocDiv hp a (b - p) = toIocDiv hp a b - 1 ** simpa only [one_zsmul] using toIocDiv_sub_zsmul hp a b 1 ** Qed", + "informal": "" + }, + { + "formal": "Nat.coprime_mul_left_add_left ** m n k : \u2115 \u22a2 Coprime (n * k + m) n \u2194 Coprime m n ** rw [Coprime, Coprime, gcd_mul_left_add_left] ** Qed", + "informal": "" + }, + { + "formal": "List.not_of_mem_foldl_argAux ** \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o : Option \u03b1 a m : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r \u22a2 \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 l \u2192 m \u2208 foldl (argAux r) o l \u2192 \u00acr a m ** induction' l using List.reverseRecOn with tl a ih ** case H1 \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o : Option \u03b1 a\u271d m : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m \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 ** intro b m o hb ho ** case H1 \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m\u271d : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b m : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] ho : m \u2208 foldl (argAux r) o (tl ++ [a]) \u22a2 \u00acr b m ** rw [foldl_append, foldl_cons, foldl_nil, argAux] at ho ** case H1 \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m\u271d : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b m : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] ho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c \u22a2 \u00acr b m ** cases' hf : foldl (argAux r) o tl with c ** case H1.some \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m\u271d : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b m : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] ho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c c : \u03b1 hf : foldl (argAux r) o tl = some c \u22a2 \u00acr b m ** rw [hf, Option.mem_def] at ho ** case H1.some \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m\u271d : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b m : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] c : \u03b1 ho : (Option.casesOn (some c) (some a) fun c => if r a c then some a else some c) = some m hf : foldl (argAux r) o tl = some c \u22a2 \u00acr b m ** dsimp only at ho ** case H1.some \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m\u271d : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b m : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] c : \u03b1 ho : (if r a c then some a else some c) = some m hf : foldl (argAux r) o tl = some c \u22a2 \u00acr b m ** split_ifs at ho with hac <;> cases' mem_append.1 hb with h h <;>\n injection ho with ho <;> subst ho ** case H0 \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o : Option \u03b1 a m : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r \u22a2 \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 [] \u2192 m \u2208 foldl (argAux r) o [] \u2192 \u00acr a m ** simp ** case H1.none \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m\u271d : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b m : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] ho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c hf : foldl (argAux r) o tl = none \u22a2 \u00acr b m ** rw [hf] at ho ** case H1.none \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m\u271d : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b m : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] ho : m \u2208 Option.casesOn none (some a) fun c => if r a c then some a else some c hf : foldl (argAux r) o tl = none \u22a2 \u00acr b m ** rw [foldl_argAux_eq_none] at hf ** case H1.none \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m\u271d : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b m : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] ho : m \u2208 Option.casesOn none (some a) fun c => if r a c then some a else some c hf : tl = [] \u2227 o = none \u22a2 \u00acr b m ** simp_all [hf.1, hf.2, hr\u2080 _] ** case pos.inl \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] c : \u03b1 hf : foldl (argAux r) o tl = some c hac : r a c h : b \u2208 tl \u22a2 \u00acr b a ** exact fun hba => ih h hf (hr\u2081 hba hac) ** case pos.inr \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] c : \u03b1 hf : foldl (argAux r) o tl = some c hac : r a c h : b \u2208 [a] \u22a2 \u00acr b a ** simp_all [hr\u2080 _] ** case neg.inl \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] c : \u03b1 hf : foldl (argAux r) o tl = some c hac : \u00acr a c h : b \u2208 tl \u22a2 \u00acr b c ** exact ih h hf ** case neg.inr \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r l : List \u03b1 o\u271d : Option \u03b1 a\u271d m : \u03b1 hr\u2080 : Irreflexive r hr\u2081 : Transitive r tl : List \u03b1 a : \u03b1 ih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m b : \u03b1 o : Option \u03b1 hb : b \u2208 tl ++ [a] c : \u03b1 hf : foldl (argAux r) o tl = some c hac : \u00acr a c h : b \u2208 [a] \u22a2 \u00acr b c ** simp_all ** Qed", + "informal": "" + }, + { + "formal": "String.find_of_valid ** p : Char \u2192 Bool s : String \u22a2 find s p = { byteIdx := utf8Len (List.takeWhile (fun x => !p x) s.data) } ** simpa using findAux_of_valid p [] s.1 [] ** Qed", + "informal": "" + }, + { + "formal": "Filter.HasBasis.eventuallyConst_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 l : Filter \u03b1 f : \u03b1 \u2192 \u03b2 \u03b9 : Sort u_5 p : \u03b9 \u2192 Prop s : \u03b9 \u2192 Set \u03b1 h : HasBasis l p s \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 ** simp only [Set.Subsingleton, ball_image_iff] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.iterate_derivative_comp_one_sub_X ** R : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n : \u2115 inst\u271d : CommRing R p : R[X] k : \u2115 \u22a2 (\u2191derivative)^[k] (comp p (1 - X)) = (-1) ^ k * comp ((\u2191derivative)^[k] p) (1 - X) ** induction' k with k ih generalizing p ** case zero R : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n : \u2115 inst\u271d : CommRing R p\u271d p : R[X] \u22a2 (\u2191derivative)^[Nat.zero] (comp p (1 - X)) = (-1) ^ Nat.zero * comp ((\u2191derivative)^[Nat.zero] p) (1 - X) ** simp ** case succ R : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n : \u2115 inst\u271d : CommRing R p\u271d : R[X] k : \u2115 ih : \u2200 (p : R[X]), (\u2191derivative)^[k] (comp p (1 - X)) = (-1) ^ k * comp ((\u2191derivative)^[k] p) (1 - X) p : R[X] \u22a2 (\u2191derivative)^[Nat.succ k] (comp p (1 - X)) = (-1) ^ Nat.succ k * comp ((\u2191derivative)^[Nat.succ k] p) (1 - X) ** simp [ih (derivative p), iterate_derivative_neg, derivative_comp, pow_succ] ** Qed", + "informal": "" + }, + { + "formal": "Finset.map_inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f\u271d : \u03b1 \u21aa \u03b2 s : Finset \u03b1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 f : \u03b1 \u21aa \u03b2 s\u2081 s\u2082 : Finset \u03b1 \u22a2 \u2191(map f (s\u2081 \u2229 s\u2082)) = \u2191(map f s\u2081 \u2229 map f s\u2082) ** simp only [coe_map, coe_inter, Set.image_inter f.injective] ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.le_sub_of_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop a b c : Ordinal.{u_4} h : b \u2264 a \u22a2 c \u2264 a - b \u2194 b + c \u2264 a ** rw [\u2190 add_le_add_iff_left b, Ordinal.add_sub_cancel_of_le h] ** Qed", + "informal": "" + }, + { + "formal": "norm_inner_div_norm_mul_norm_eq_one_iff ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E \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 ** constructor ** case mp \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E \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 ** intro h ** case mp \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E h : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 \u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x ** have hx\u2080 : x \u2260 0 := fun h\u2080 => by simp [h\u2080] at h ** case mp \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E h : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 hx\u2080 : x \u2260 0 \u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x ** have hy\u2080 : y \u2260 0 := fun h\u2080 => by simp [h\u2080] at h ** case mp \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E h : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 hx\u2080 : x \u2260 0 hy\u2080 : y \u2260 0 \u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x ** refine' \u27e8hx\u2080, (norm_inner_eq_norm_iff hx\u2080 hy\u2080).1 <| eq_of_div_eq_one _\u27e9 ** case mp \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E h : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 hx\u2080 : x \u2260 0 hy\u2080 : y \u2260 0 \u22a2 \u2016inner x y\u2016 / (\u2016x\u2016 * \u2016y\u2016) = 1 ** simpa using h ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E h : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 h\u2080 : x = 0 \u22a2 False ** simp [h\u2080] at h ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E h : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 hx\u2080 : x \u2260 0 h\u2080 : y = 0 \u22a2 False ** simp [h\u2080] at h ** case mpr \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E \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 ** rintro \u27e8hx, \u27e8r, \u27e8hr, rfl\u27e9\u27e9\u27e9 ** case mpr.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x : E hx : x \u2260 0 r : \ud835\udd5c hr : r \u2260 0 \u22a2 \u2016inner x (r \u2022 x) / (\u2191\u2016x\u2016 * \u2191\u2016r \u2022 x\u2016)\u2016 = 1 ** simp only [norm_div, norm_mul, norm_ofReal, abs_norm] ** case mpr.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x : E hx : x \u2260 0 r : \ud835\udd5c hr : r \u2260 0 \u22a2 \u2016inner x (r \u2022 x)\u2016 / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1 ** exact norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr ** Qed", + "informal": "" + }, + { + "formal": "Ideal.spanNorm_mul_of_bot_or_top ** R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S \u22a2 spanNorm R (I * J) = spanNorm R I * spanNorm R J ** refine le_antisymm ?_ (spanNorm_mul_spanNorm_le R _ _) ** R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S \u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J ** cases' eq_bot_or_top (spanNorm R I) with hI hI ** case inr R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S hI : spanNorm R I = \u22a4 \u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J ** rw [hI, Ideal.top_mul] ** case inr R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S hI : spanNorm R I = \u22a4 \u22a2 spanNorm R (I * J) \u2264 spanNorm R J ** cases' eq_bot_or_top (spanNorm R J) with hJ hJ ** case inr.inr R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S hI : spanNorm R I = \u22a4 hJ : spanNorm R J = \u22a4 \u22a2 spanNorm R (I * J) \u2264 spanNorm R J ** rw [hJ] ** case inr.inr R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S hI : spanNorm R I = \u22a4 hJ : spanNorm R J = \u22a4 \u22a2 spanNorm R (I * J) \u2264 \u22a4 ** exact le_top ** case inl R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S hI : spanNorm R I = \u22a5 \u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J ** rw [hI, spanNorm_eq_bot_iff.mp hI, bot_mul, spanNorm_bot] ** case inl R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S hI : spanNorm R I = \u22a5 \u22a2 \u22a5 \u2264 \u22a5 * spanNorm R J ** exact bot_le ** case inr.inl R : Type u_1 inst\u271d\u2076 : CommRing R S : Type u_2 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : Algebra R S inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsDomain S inst\u271d\u00b9 : Module.Free R S inst\u271d : Module.Finite R S eq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4 I J : Ideal S hI : spanNorm R I = \u22a4 hJ : spanNorm R J = \u22a5 \u22a2 spanNorm R (I * J) \u2264 spanNorm R J ** rw [hJ, spanNorm_eq_bot_iff.mp hJ, mul_bot, spanNorm_bot] ** Qed", + "informal": "" + }, + { + "formal": "NNReal.not_summable_iff_tendsto_nat_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u2115 \u2192 \u211d\u22650 \u22a2 \u00acSummable f \u2194 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop ** constructor ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u2115 \u2192 \u211d\u22650 \u22a2 \u00acSummable f \u2192 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop ** intro h ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u2115 \u2192 \u211d\u22650 h : \u00acSummable f \u22a2 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop ** refine' ((tendsto_of_monotone _).resolve_right h).comp _ ** case mp.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u2115 \u2192 \u211d\u22650 h : \u00acSummable f \u22a2 Monotone fun s => \u2211 b in s, f b case mp.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u2115 \u2192 \u211d\u22650 h : \u00acSummable f \u22a2 Tendsto (fun n => Finset.range n) atTop atTop ** exacts [Finset.sum_mono_set _, tendsto_finset_range] ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u2115 \u2192 \u211d\u22650 \u22a2 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop \u2192 \u00acSummable f ** rintro hnat \u27e8r, hr\u27e9 ** case mpr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u2115 \u2192 \u211d\u22650 hnat : Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop r : \u211d\u22650 hr : HasSum f r \u22a2 False ** exact not_tendsto_nhds_of_tendsto_atTop hnat _ (hasSum_iff_tendsto_nat.1 hr) ** Qed", + "informal": "" + }, + { + "formal": "integral_comp_neg_Iic ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E \u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x ** have A : MeasurableEmbedding fun x : \u211d => -x :=\n (Homeomorph.neg \u211d).closedEmbedding.measurableEmbedding ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E A : MeasurableEmbedding fun x => -x \u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x ** have := MeasurableEmbedding.set_integral_map (\u03bc := volume) A f (Ici (-c)) ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E A : MeasurableEmbedding fun x => -x this : \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) \u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x ** rw [Measure.map_neg_eq_self (volume : Measure \u211d)] at this ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E A : MeasurableEmbedding fun x => -x this : \u222b (y : \u211d) in Ici (-c), f y = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x) \u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x ** simp_rw [\u2190 integral_Ici_eq_integral_Ioi, this, neg_preimage, preimage_neg_Ici, neg_neg] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.mem_orthogonalGroup_iff ** n : Type u inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n \u03b1 : Type v inst\u271d\u00b2 : CommRing \u03b1 inst\u271d\u00b9 : StarRing \u03b1 A\u271d : Matrix n n \u03b1 \u03b2 : Type v inst\u271d : CommRing \u03b2 A : Matrix n n \u03b2 \u22a2 A \u2208 orthogonalGroup n \u03b2 \u2194 A * star A = 1 ** refine' \u27e8And.right, fun hA => \u27e8_, hA\u27e9\u27e9 ** n : Type u inst\u271d\u2074 : DecidableEq n inst\u271d\u00b3 : Fintype n \u03b1 : Type v inst\u271d\u00b2 : CommRing \u03b1 inst\u271d\u00b9 : StarRing \u03b1 A\u271d : Matrix n n \u03b1 \u03b2 : Type v inst\u271d : CommRing \u03b2 A : Matrix n n \u03b2 hA : A * star A = 1 \u22a2 star A * A = 1 ** simpa only [mul_eq_one_comm] using hA ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.mk_eq_two_iff' ** \u03b1 \u03b2 : Type u x : \u03b1 \u22a2 #\u03b1 = 2 \u2194 \u2203! y, y \u2260 x ** rw [mk_eq_two_iff] ** \u03b1 \u03b2 : Type u x : \u03b1 \u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2194 \u2203! y, y \u2260 x ** constructor ** case mp \u03b1 \u03b2 : Type u x : \u03b1 \u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2192 \u2203! y, y \u2260 x ** rintro \u27e8a, b, hne, h\u27e9 ** case mp.intro.intro.intro \u03b1 \u03b2 : Type u x a b : \u03b1 hne : a \u2260 b h : {a, b} = univ \u22a2 \u2203! y, y \u2260 x ** simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h ** case mp.intro.intro.intro \u03b1 \u03b2 : Type u x a b : \u03b1 hne : a \u2260 b h : \u2200 (x : \u03b1), x = a \u2228 x = b \u22a2 \u2203! y, y \u2260 x ** rcases h x with (rfl | rfl) ** case mp.intro.intro.intro.inl \u03b1 \u03b2 : Type u x b : \u03b1 hne : x \u2260 b h : \u2200 (x_1 : \u03b1), x_1 = x \u2228 x_1 = b \u22a2 \u2203! y, y \u2260 x case mp.intro.intro.intro.inr \u03b1 \u03b2 : Type u x a : \u03b1 hne : a \u2260 x h : \u2200 (x_1 : \u03b1), x_1 = a \u2228 x_1 = x \u22a2 \u2203! y, y \u2260 x ** exacts [\u27e8b, hne.symm, fun z => (h z).resolve_left\u27e9, \u27e8a, hne, fun z => (h z).resolve_right\u27e9] ** case mpr \u03b1 \u03b2 : Type u x : \u03b1 \u22a2 (\u2203! y, y \u2260 x) \u2192 \u2203 x y, x \u2260 y \u2227 {x, y} = univ ** rintro \u27e8y, hne, hy\u27e9 ** case mpr.intro.intro \u03b1 \u03b2 : Type u x y : \u03b1 hne : y \u2260 x hy : \u2200 (y_1 : \u03b1), (fun y => y \u2260 x) y_1 \u2192 y_1 = y \u22a2 \u2203 x y, x \u2260 y \u2227 {x, y} = univ ** exact \u27e8x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "tendsto_inv_nhdsWithin_Iic_inv ** \u03b1 : Type u \u03b2 : Type v G : Type w H : Type x inst\u271d\u2076 : TopologicalSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : TopologicalGroup G inst\u271d\u00b3 : TopologicalSpace \u03b1 f : \u03b1 \u2192 G s : Set \u03b1 x : \u03b1 inst\u271d\u00b2 : TopologicalSpace H inst\u271d\u00b9 : OrderedCommGroup H inst\u271d : ContinuousInv H a : H \u22a2 Tendsto Inv.inv (\ud835\udcdd[Iic a\u207b\u00b9] a\u207b\u00b9) (\ud835\udcdd[Ici a] a) ** simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Iic _ _ _ _ a\u207b\u00b9 ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.hasTerminalChangeDiagram ** C : Type u\u2081 inst\u271d : Category.{v\u2081, u\u2081} C X : C F\u2081 : Discrete PEmpty.{w + 1} \u2964 C F\u2082 : Discrete PEmpty.{w' + 1} \u2964 C h : HasLimit F\u2081 \u22a2 (X : Discrete PEmpty.{w' + 1}) \u2192 ((Functor.const (Discrete PEmpty.{w' + 1})).obj (limit F\u2081)).obj X \u27f6 F\u2082.obj X ** aesop_cat ** C : Type u\u2081 inst\u271d : Category.{v\u2081, u\u2081} C X : C F\u2081 : Discrete PEmpty.{w + 1} \u2964 C F\u2082 : Discrete PEmpty.{w' + 1} \u2964 C h : HasLimit F\u2081 \u22a2 \u2200 \u2983X Y : Discrete PEmpty.{w' + 1}\u2984 (f : X \u27f6 Y), ((Functor.const (Discrete PEmpty.{w' + 1})).obj (limit F\u2081)).map f \u226b id (Discrete.casesOn Y fun as => (_ : False).elim) = id (Discrete.casesOn X fun as => (_ : False).elim) \u226b F\u2082.map f ** aesop_cat ** Qed", + "informal": "" + }, + { + "formal": "Multiset.inter_le_right ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 inst\u271d : DecidableEq \u03b1 s\u271d\u00b9 t\u271d u : Multiset \u03b1 a\u271d b : \u03b1 s\u271d : Multiset \u03b1 a : \u03b1 s : Multiset \u03b1 IH : \u2200 (t : Multiset \u03b1), s \u2229 t \u2264 t t : Multiset \u03b1 h : a \u2208 t \u22a2 (a ::\u2098 s) \u2229 t \u2264 t ** simpa [h] using cons_le_cons a (IH (t.erase a)) ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 inst\u271d : DecidableEq \u03b1 s\u271d\u00b9 t\u271d u : Multiset \u03b1 a\u271d b : \u03b1 s\u271d : Multiset \u03b1 a : \u03b1 s : Multiset \u03b1 IH : \u2200 (t : Multiset \u03b1), s \u2229 t \u2264 t t : Multiset \u03b1 h : \u00aca \u2208 t \u22a2 (a ::\u2098 s) \u2229 t \u2264 t ** simp [h, IH] ** Qed", + "informal": "" + }, + { + "formal": "Set.einfsep_ne_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : EDist \u03b1 x y : \u03b1 s t : Set \u03b1 \u22a2 einfsep s \u2260 \u22a4 \u2194 \u2203 x x_1 y x_2 _hxy, edist x y \u2260 \u22a4 ** simp_rw [\u2190 lt_top_iff_ne_top, einfsep_lt_top] ** Qed", + "informal": "" + }, + { + "formal": "Filter.coprod\u1d62_cocompact ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 s t : Set \u03b1 inst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) \u03b4 : Type u_3 \u03ba : \u03b4 \u2192 Type u_4 inst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d) \u22a2 (Filter.coprod\u1d62 fun d => cocompact (\u03ba d)) = cocompact ((d : \u03b4) \u2192 \u03ba d) ** refine' le_antisymm (iSup_le fun i => Filter.comap_cocompact_le (continuous_apply i)) _ ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 s t : Set \u03b1 inst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) \u03b4 : Type u_3 \u03ba : \u03b4 \u2192 Type u_4 inst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d) \u22a2 cocompact ((d : \u03b4) \u2192 \u03ba d) \u2264 Filter.coprod\u1d62 fun d => cocompact (\u03ba d) ** refine' compl_surjective.forall.2 fun s H => _ ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 s\u271d t : Set \u03b1 inst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) \u03b4 : Type u_3 \u03ba : \u03b4 \u2192 Type u_4 inst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d) s : Set ((i : \u03b4) \u2192 \u03ba i) H : s\u1d9c \u2208 Filter.coprod\u1d62 fun d => cocompact (\u03ba d) \u22a2 s\u1d9c \u2208 cocompact ((d : \u03b4) \u2192 \u03ba d) ** simp only [compl_mem_coprod\u1d62, Filter.mem_cocompact, compl_subset_compl, image_subset_iff] at H \u22a2 ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 s\u271d t : Set \u03b1 inst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) \u03b4 : Type u_3 \u03ba : \u03b4 \u2192 Type u_4 inst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d) s : Set ((i : \u03b4) \u2192 \u03ba i) H : \u2200 (i : \u03b4), \u2203 t, IsCompact t \u2227 s \u2286 Function.eval i \u207b\u00b9' t \u22a2 \u2203 t, IsCompact t \u2227 s \u2286 t ** choose K hKc htK using H ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 s\u271d t : Set \u03b1 inst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) \u03b4 : Type u_3 \u03ba : \u03b4 \u2192 Type u_4 inst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d) s : Set ((i : \u03b4) \u2192 \u03ba i) K : (i : \u03b4) \u2192 Set (\u03ba i) hKc : \u2200 (i : \u03b4), IsCompact (K i) htK : \u2200 (i : \u03b4), s \u2286 Function.eval i \u207b\u00b9' K i \u22a2 \u2203 t, IsCompact t \u2227 s \u2286 t ** exact \u27e8Set.pi univ K, isCompact_univ_pi hKc, fun f hf i _ => htK i hf\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MeasurableSpace.le_invariants_iterate ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 n : \u2115 \u22a2 invariants f \u2264 invariants f^[n] ** induction n with\n| zero => simp [invariants_le]\n| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _) ** case zero \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 \u22a2 invariants f \u2264 invariants f^[Nat.zero] ** simp [invariants_le] ** case succ \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 n : \u2115 ihn : invariants f \u2264 invariants f^[n] \u22a2 invariants f \u2264 invariants f^[Nat.succ n] ** exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _) ** Qed", + "informal": "" + }, + { + "formal": "integral_convolution ** \ud835\udd5c : Type u\ud835\udd5c G : Type uG E : Type uE E' : Type uE' E'' : Type uE'' F : Type uF F' : Type uF' F'' : Type uF'' P : Type uP inst\u271d\u00b2\u2079 : NormedAddCommGroup E inst\u271d\u00b2\u2078 : NormedAddCommGroup E' inst\u271d\u00b2\u2077 : NormedAddCommGroup E'' inst\u271d\u00b2\u2076 : NormedAddCommGroup F f f' : G \u2192 E g g' : G \u2192 E' x x' : G y y' : E inst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c inst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E' inst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E'' inst\u271d\u00b2\u00b9 : NormedSpace \u211d F inst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F n : \u2115\u221e inst\u271d\u00b9\u2079 : CompleteSpace F inst\u271d\u00b9\u2078 : MeasurableSpace G \u03bc \u03bd : Measure G L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F inst\u271d\u00b9\u2077 : NormedAddCommGroup F' inst\u271d\u00b9\u2076 : NormedSpace \u211d F' inst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9\u2074 : CompleteSpace F' inst\u271d\u00b9\u00b3 : NormedAddCommGroup F'' inst\u271d\u00b9\u00b2 : NormedSpace \u211d F'' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'' inst\u271d\u00b9\u2070 : CompleteSpace F'' k : G \u2192 E'' L\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F' L\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F' L\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'' inst\u271d\u2079 : AddGroup G inst\u271d\u2078 : SigmaFinite \u03bc inst\u271d\u2077 : SigmaFinite \u03bd inst\u271d\u2076 : IsAddRightInvariant \u03bc inst\u271d\u2075 : MeasurableAdd\u2082 G inst\u271d\u2074 : MeasurableNeg G inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9 : CompleteSpace E inst\u271d : CompleteSpace E' hf : Integrable f hg : Integrable g \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) ** refine' (integral_integral_swap (by apply hf.rst.imnvolution_integrand L hg)).trans _ ** \ud835\udd5c : Type u\ud835\udd5c G : Type uG E : Type uE E' : Type uE' E'' : Type uE'' F : Type uF F' : Type uF' F'' : Type uF'' P : Type uP inst\u271d\u00b2\u2079 : NormedAddCommGroup E inst\u271d\u00b2\u2078 : NormedAddCommGroup E' inst\u271d\u00b2\u2077 : NormedAddCommGroup E'' inst\u271d\u00b2\u2076 : NormedAddCommGroup F f f' : G \u2192 E g g' : G \u2192 E' x x' : G y y' : E inst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c inst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E' inst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E'' inst\u271d\u00b2\u00b9 : NormedSpace \u211d F inst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F n : \u2115\u221e inst\u271d\u00b9\u2079 : CompleteSpace F inst\u271d\u00b9\u2078 : MeasurableSpace G \u03bc \u03bd : Measure G L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F inst\u271d\u00b9\u2077 : NormedAddCommGroup F' inst\u271d\u00b9\u2076 : NormedSpace \u211d F' inst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9\u2074 : CompleteSpace F' inst\u271d\u00b9\u00b3 : NormedAddCommGroup F'' inst\u271d\u00b9\u00b2 : NormedSpace \u211d F'' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'' inst\u271d\u00b9\u2070 : CompleteSpace F'' k : G \u2192 E'' L\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F' L\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F' L\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'' inst\u271d\u2079 : AddGroup G inst\u271d\u2078 : SigmaFinite \u03bc inst\u271d\u2077 : SigmaFinite \u03bd inst\u271d\u2076 : IsAddRightInvariant \u03bc inst\u271d\u2075 : MeasurableAdd\u2082 G inst\u271d\u2074 : MeasurableNeg G inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9 : CompleteSpace E inst\u271d : CompleteSpace E' hf : Integrable f hg : Integrable g \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) ** simp_rw [integral_comp_comm _ (hg.comp_sub_right _), integral_sub_right_eq_self] ** \ud835\udd5c : Type u\ud835\udd5c G : Type uG E : Type uE E' : Type uE' E'' : Type uE'' F : Type uF F' : Type uF' F'' : Type uF'' P : Type uP inst\u271d\u00b2\u2079 : NormedAddCommGroup E inst\u271d\u00b2\u2078 : NormedAddCommGroup E' inst\u271d\u00b2\u2077 : NormedAddCommGroup E'' inst\u271d\u00b2\u2076 : NormedAddCommGroup F f f' : G \u2192 E g g' : G \u2192 E' x x' : G y y' : E inst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c inst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E' inst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E'' inst\u271d\u00b2\u00b9 : NormedSpace \u211d F inst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F n : \u2115\u221e inst\u271d\u00b9\u2079 : CompleteSpace F inst\u271d\u00b9\u2078 : MeasurableSpace G \u03bc \u03bd : Measure G L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F inst\u271d\u00b9\u2077 : NormedAddCommGroup F' inst\u271d\u00b9\u2076 : NormedSpace \u211d F' inst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9\u2074 : CompleteSpace F' inst\u271d\u00b9\u00b3 : NormedAddCommGroup F'' inst\u271d\u00b9\u00b2 : NormedSpace \u211d F'' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'' inst\u271d\u00b9\u2070 : CompleteSpace F'' k : G \u2192 E'' L\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F' L\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F' L\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'' inst\u271d\u2079 : AddGroup G inst\u271d\u2078 : SigmaFinite \u03bc inst\u271d\u2077 : SigmaFinite \u03bd inst\u271d\u2076 : IsAddRightInvariant \u03bc inst\u271d\u2075 : MeasurableAdd\u2082 G inst\u271d\u2074 : MeasurableNeg G inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9 : CompleteSpace E inst\u271d : CompleteSpace E' hf : Integrable f hg : Integrable g \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) ** exact (L.flip (\u222b x, g x \u2202\u03bc)).integral_comp_comm hf ** \ud835\udd5c : Type u\ud835\udd5c G : Type uG E : Type uE E' : Type uE' E'' : Type uE'' F : Type uF F' : Type uF' F'' : Type uF'' P : Type uP inst\u271d\u00b2\u2079 : NormedAddCommGroup E inst\u271d\u00b2\u2078 : NormedAddCommGroup E' inst\u271d\u00b2\u2077 : NormedAddCommGroup E'' inst\u271d\u00b2\u2076 : NormedAddCommGroup F f f' : G \u2192 E g g' : G \u2192 E' x x' : G y y' : E inst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c inst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E' inst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E'' inst\u271d\u00b2\u00b9 : NormedSpace \u211d F inst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F n : \u2115\u221e inst\u271d\u00b9\u2079 : CompleteSpace F inst\u271d\u00b9\u2078 : MeasurableSpace G \u03bc \u03bd : Measure G L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F inst\u271d\u00b9\u2077 : NormedAddCommGroup F' inst\u271d\u00b9\u2076 : NormedSpace \u211d F' inst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9\u2074 : CompleteSpace F' inst\u271d\u00b9\u00b3 : NormedAddCommGroup F'' inst\u271d\u00b9\u00b2 : NormedSpace \u211d F'' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F'' inst\u271d\u00b9\u2070 : CompleteSpace F'' k : G \u2192 E'' L\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F' L\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F' L\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'' inst\u271d\u2079 : AddGroup G inst\u271d\u2078 : SigmaFinite \u03bc inst\u271d\u2077 : SigmaFinite \u03bd inst\u271d\u2076 : IsAddRightInvariant \u03bc inst\u271d\u2075 : MeasurableAdd\u2082 G inst\u271d\u2074 : MeasurableNeg G inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9 : CompleteSpace E inst\u271d : CompleteSpace E' hf : Integrable f hg : Integrable g \u22a2 Integrable (uncurry fun x t => \u2191(\u2191L (f t)) (g (x - t))) ** apply hf.rst.imnvolution_integrand L hg ** Qed", + "informal": "" + }, + { + "formal": "sSup_compact_eq_top ** \u03b9 : Sort u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : CompleteLattice \u03b1 f : \u03b9 \u2192 \u03b1 inst\u271d\u00b9 : CompleteLattice \u03b1 inst\u271d : IsCompactlyGenerated \u03b1 a b : \u03b1 s : Set \u03b1 \u22a2 sSup {a | CompleteLattice.IsCompactElement a} = \u22a4 ** refine' Eq.trans (congr rfl (Set.ext fun x => _)) (sSup_compact_le_eq \u22a4) ** \u03b9 : Sort u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : CompleteLattice \u03b1 f : \u03b9 \u2192 \u03b1 inst\u271d\u00b9 : CompleteLattice \u03b1 inst\u271d : IsCompactlyGenerated \u03b1 a b : \u03b1 s : Set \u03b1 x : \u03b1 \u22a2 x \u2208 {a | CompleteLattice.IsCompactElement a} \u2194 x \u2208 {c | CompleteLattice.IsCompactElement c \u2227 c \u2264 \u22a4} ** exact (and_iff_left le_top).symm ** Qed", + "informal": "" + }, + { + "formal": "MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s \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 ** refine' \u27e8not_imp_not.1 fun h \u21a6 _, fun h \u21a6 (hfg.sum_smul_comp_perm_le_sum_smul h\u03c3).antisymm _\u27e9 ** case refine'_1 \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s h : \u00acMonovaryOn f (g \u2218 \u2191\u03c3) \u2191s \u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i ** rw [MonovaryOn] at h ** case refine'_1 \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s h : \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 \u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i ** push_neg at h ** case refine'_1 \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s h : 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 \u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i ** obtain \u27e8x, hx, y, hy, hgxy, hfxy\u27e9 := h ** case refine'_1.intro.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i ** set \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 ** case refine'_1.intro.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 \u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i ** 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 ** case refine'_1.intro.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 h\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s \u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i ** refine' ((hfg.sum_smul_comp_perm_le_sum_smul h\u03c4s).trans_lt' _).ne ** case refine'_1.intro.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 h\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s \u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i) ** obtain rfl | hxy := eq_or_ne x y ** case refine'_1.intro.intro.intro.intro.intro.inr \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 h\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s hxy : x \u2260 y \u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i) ** 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] ** case refine'_1.intro.intro.intro.intro.intro.inr \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 h\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s hxy : x \u2260 y \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)) < \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)) ** 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) ** case refine'_1.intro.intro.intro.intro.intro.inr \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 h\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s hxy : x \u2260 y z : \u03b9 hz : z \u2208 erase (erase s x) y \u22a2 f z \u2022 g (\u2191\u03c3 z) = f z \u2022 g (\u2191\u03c3 (\u2191(Equiv.swap x y) z)) ** simp_rw [mem_erase] at hz ** case refine'_1.intro.intro.intro.intro.intro.inr \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 h\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s hxy : x \u2260 y z : \u03b9 hz : z \u2260 y \u2227 z \u2260 x \u2227 z \u2208 s \u22a2 f z \u2022 g (\u2191\u03c3 z) = f z \u2022 g (\u2191\u03c3 (\u2191(Equiv.swap x y) z)) ** rw [swap_apply_of_ne_of_ne hz.2.1 hz.1] ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 \u22a2 {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s ** refine' (set_support_mul_subset \u03c3 <| swap x y).trans (Set.union_subset h\u03c3 fun z hz \u21a6 _) ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx : x \u2208 \u2191s y : \u03b9 hy : y \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y hfxy : f y < f x \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3 z : \u03b9 hz : z \u2208 {x_1 | \u2191(Equiv.swap x y) x_1 \u2260 x_1} \u22a2 z \u2208 \u2191s ** obtain \u27e8_, rfl | rfl\u27e9 := swap_apply_ne_self_iff.1 hz <;> assumption ** case refine'_1.intro.intro.intro.intro.intro.inl \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s x : \u03b9 hx hy : x \u2208 \u2191s hgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) x hfxy : f x < f x \u03c4 : Perm \u03b9 := (Equiv.swap x x).trans \u03c3 h\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s \u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i) ** cases lt_irrefl _ hfxy ** case refine'_2 \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s h : MonovaryOn f (g \u2218 \u2191\u03c3) \u2191s \u22a2 \u2211 i in s, f i \u2022 g i \u2264 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) ** convert h.sum_smul_comp_perm_le_sum_smul ((set_support_inv_eq _).subset.trans h\u03c3) using 1 ** case h.e'_3 \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b3 : LinearOrderedRing \u03b1 inst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u03b1 \u03b2 inst\u271d : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 hfg : MonovaryOn f g \u2191s h\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s h : MonovaryOn f (g \u2218 \u2191\u03c3) \u2191s \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) ** simp_rw [Function.comp_apply, apply_inv_self] ** Qed", + "informal": "" + }, + { + "formal": "Subalgebra.topologicalClosure_star_comm ** R : Type u_1 A : Type u_2 B : Type u_3 inst\u271d\u00b9\u00b2 : CommSemiring R inst\u271d\u00b9\u00b9 : StarRing R inst\u271d\u00b9\u2070 : TopologicalSpace A inst\u271d\u2079 : Semiring A inst\u271d\u2078 : Algebra R A inst\u271d\u2077 : StarRing A inst\u271d\u2076 : StarModule R A inst\u271d\u2075 : TopologicalSemiring A inst\u271d\u2074 : ContinuousStar A inst\u271d\u00b3 : TopologicalSpace B inst\u271d\u00b2 : Semiring B inst\u271d\u00b9 : Algebra R B inst\u271d : StarRing B s : Subalgebra R A \u22a2 Subalgebra.topologicalClosure (star s) = star (Subalgebra.topologicalClosure s) ** 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))) ** R : Type u_1 A : Type u_2 B : Type u_3 inst\u271d\u00b9\u00b2 : CommSemiring R inst\u271d\u00b9\u00b9 : StarRing R inst\u271d\u00b9\u2070 : TopologicalSpace A inst\u271d\u2079 : Semiring A inst\u271d\u2078 : Algebra R A inst\u271d\u2077 : StarRing A inst\u271d\u2076 : StarModule R A inst\u271d\u2075 : TopologicalSemiring A inst\u271d\u2074 : ContinuousStar A inst\u271d\u00b3 : TopologicalSpace B inst\u271d\u00b2 : Semiring B inst\u271d\u00b9 : Algebra R B inst\u271d : StarRing B s : Subalgebra R A \u22a2 \u2200 (t : Subalgebra R A), Subalgebra.topologicalClosure (star t) \u2264 star (Subalgebra.topologicalClosure t) ** exact fun t => (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure)\n (isClosed_closure.preimage continuous_star) ** R : Type u_1 A : Type u_2 B : Type u_3 inst\u271d\u00b9\u00b2 : CommSemiring R inst\u271d\u00b9\u00b9 : StarRing R inst\u271d\u00b9\u2070 : TopologicalSpace A inst\u271d\u2079 : Semiring A inst\u271d\u2078 : Algebra R A inst\u271d\u2077 : StarRing A inst\u271d\u2076 : StarModule R A inst\u271d\u2075 : TopologicalSemiring A inst\u271d\u2074 : ContinuousStar A inst\u271d\u00b3 : TopologicalSpace B inst\u271d\u00b2 : Semiring B inst\u271d\u00b9 : Algebra R B inst\u271d : StarRing B s : Subalgebra R A this : \u2200 (t : Subalgebra R A), Subalgebra.topologicalClosure (star t) \u2264 star (Subalgebra.topologicalClosure t) \u22a2 star (Subalgebra.topologicalClosure s) \u2264 Subalgebra.topologicalClosure (star s) ** simpa only [star_star] using Subalgebra.star_mono (this (star s)) ** Qed", + "informal": "" + }, + { + "formal": "Metric.subsingleton_closedBall ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b3 : Type w inst\u271d : MetricSpace \u03b3 x\u271d : \u03b3 s : Set \u03b3 x : \u03b3 r : \u211d hr : r \u2264 0 \u22a2 Set.Subsingleton (closedBall x r) ** rcases hr.lt_or_eq with (hr | rfl) ** case inl \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b3 : Type w inst\u271d : MetricSpace \u03b3 x\u271d : \u03b3 s : Set \u03b3 x : \u03b3 r : \u211d hr\u271d : r \u2264 0 hr : r < 0 \u22a2 Set.Subsingleton (closedBall x r) ** rw [closedBall_eq_empty.2 hr] ** case inl \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b3 : Type w inst\u271d : MetricSpace \u03b3 x\u271d : \u03b3 s : Set \u03b3 x : \u03b3 r : \u211d hr\u271d : r \u2264 0 hr : r < 0 \u22a2 Set.Subsingleton \u2205 ** exact subsingleton_empty ** case inr \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b3 : Type w inst\u271d : MetricSpace \u03b3 x\u271d : \u03b3 s : Set \u03b3 x : \u03b3 hr : 0 \u2264 0 \u22a2 Set.Subsingleton (closedBall x 0) ** rw [closedBall_zero] ** case inr \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : PseudoMetricSpace \u03b1 \u03b3 : Type w inst\u271d : MetricSpace \u03b3 x\u271d : \u03b3 s : Set \u03b3 x : \u03b3 hr : 0 \u2264 0 \u22a2 Set.Subsingleton {x} ** exact subsingleton_singleton ** Qed", + "informal": "" + }, + { + "formal": "Int.coprime_of_sq_sum ** r s : \u2124 h2 : IsCoprime s r \u22a2 IsCoprime (r ^ 2 + s ^ 2) r ** rw [sq, sq] ** r s : \u2124 h2 : IsCoprime s r \u22a2 IsCoprime (r * r + s * s) r ** exact (IsCoprime.mul_left h2 h2).mul_add_left_left r ** Qed", + "informal": "" + }, + { + "formal": "Rat.neg_mkRat ** n : Int d : Nat \u22a2 -mkRat n d = mkRat (-n) d ** if z : d = 0 then simp [z] else simp [\u2190 normalize_eq_mkRat z, neg_normalize] ** n : Int d : Nat z : d = 0 \u22a2 -mkRat n d = mkRat (-n) d ** simp [z] ** n : Int d : Nat z : \u00acd = 0 \u22a2 -mkRat n d = mkRat (-n) d ** simp [\u2190 normalize_eq_mkRat z, neg_normalize] ** Qed", + "informal": "" + }, + { + "formal": "IsLocalization.lift_mem_adjoin_finsetIntegerMultiple ** R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S' x : S s : Finset S' hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s \u22a2 \u2203 m, m \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s) ** 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 ** case intro.mk.intro.intro R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S' x : S s : Finset S' hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s a : R ha : a \u2208 \u2191M e : { 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 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s) \u22a2 \u2203 m, m \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s) ** refine' \u27e8\u27e8a, ha\u27e9, _\u27e9 ** case intro.mk.intro.intro R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S' x : S s : Finset S' hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s a : R ha : a \u2208 \u2191M e : { 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 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s) \u22a2 { val := a, property := ha } \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s) ** simpa only [Submonoid.smul_def, algebraMap_smul] using e ** R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S' x : S s : Finset S' hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s \u22a2 Submonoid.map (algebraMap R S) M \u2264 (Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)).toSubsemiring.toSubmonoid ** rintro _ \u27e8a, _, rfl\u27e9 ** case intro.intro R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S' x : S s : Finset S' hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s a : R left\u271d : a \u2208 \u2191M \u22a2 \u2191(algebraMap R S) a \u2208 (Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)).toSubsemiring.toSubmonoid ** exact Subalgebra.algebraMap_mem _ a ** Qed", + "informal": "" + }, + { + "formal": "QuaternionAlgebra.self_add_star ** S : Type u_1 T : Type u_2 R : Type u_3 inst\u271d : CommRing R c\u2081 c\u2082 r x y z : R a b c : \u210d[R,c\u2081,c\u2082] \u22a2 a + star a = 2 * \u2191a.re ** simp only [self_add_star', two_mul, coe_add] ** Qed", + "informal": "" + }, + { + "formal": "DihedralGroup.exponent ** n : \u2115 \u22a2 Monoid.exponent (DihedralGroup n) = lcm n 2 ** rcases eq_zero_or_neZero n with (rfl | hn) ** case inr n : \u2115 hn : NeZero n \u22a2 Monoid.exponent (DihedralGroup n) = lcm n 2 ** apply Nat.dvd_antisymm ** case inl \u22a2 Monoid.exponent (DihedralGroup 0) = lcm 0 2 ** exact Monoid.exponent_eq_zero_of_order_zero orderOf_r_one ** case inr.a n : \u2115 hn : NeZero n \u22a2 Monoid.exponent (DihedralGroup n) \u2223 lcm n 2 ** apply Monoid.exponent_dvd_of_forall_pow_eq_one ** case inr.a.hG n : \u2115 hn : NeZero n \u22a2 \u2200 (g : DihedralGroup n), g ^ lcm n 2 = 1 ** rintro (m | m) ** case inr.a.hG.r n : \u2115 hn : NeZero n m : ZMod n \u22a2 r m ^ lcm n 2 = 1 ** rw [\u2190 orderOf_dvd_iff_pow_eq_one, orderOf_r] ** case inr.a.hG.r n : \u2115 hn : NeZero n m : ZMod n \u22a2 n / Nat.gcd n (ZMod.val m) \u2223 lcm n 2 ** refine' Nat.dvd_trans \u27e8gcd n m.val, _\u27e9 (dvd_lcm_left n 2) ** case inr.a.hG.r n : \u2115 hn : NeZero n m : ZMod n \u22a2 n = n / Nat.gcd n (ZMod.val m) * gcd n (ZMod.val m) ** exact (Nat.div_mul_cancel (Nat.gcd_dvd_left n m.val)).symm ** case inr.a.hG.sr n : \u2115 hn : NeZero n m : ZMod n \u22a2 sr m ^ lcm n 2 = 1 ** rw [\u2190 orderOf_dvd_iff_pow_eq_one, orderOf_sr] ** case inr.a.hG.sr n : \u2115 hn : NeZero n m : ZMod n \u22a2 2 \u2223 lcm n 2 ** exact dvd_lcm_right n 2 ** case inr.a n : \u2115 hn : NeZero n \u22a2 lcm n 2 \u2223 Monoid.exponent (DihedralGroup n) ** apply lcm_dvd ** case inr.a.hab n : \u2115 hn : NeZero n \u22a2 n \u2223 Monoid.exponent (DihedralGroup n) ** convert Monoid.order_dvd_exponent (r (1 : ZMod n)) ** case h.e'_3 n : \u2115 hn : NeZero n \u22a2 n = orderOf (r 1) ** exact orderOf_r_one.symm ** case inr.a.hcb n : \u2115 hn : NeZero n \u22a2 2 \u2223 Monoid.exponent (DihedralGroup n) ** convert Monoid.order_dvd_exponent (sr (0 : ZMod n)) ** case h.e'_3 n : \u2115 hn : NeZero n \u22a2 2 = orderOf (sr 0) ** exact (orderOf_sr 0).symm ** Qed", + "informal": "" + }, + { + "formal": "AddCircle.le_add_order_smul_norm_of_isOfFinAddOrder ** p : \u211d hp : Fact (0 < p) u : AddCircle p hu : IsOfFinAddOrder u hu' : u \u2260 0 \u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016 ** obtain \u27e8n, hn\u27e9 := exists_norm_eq_of_isOfFinAddOrder hu ** case intro p : \u211d hp : Fact (0 < p) u : AddCircle p hu : IsOfFinAddOrder u hu' : u \u2260 0 n : \u2115 hn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u)) \u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016 ** replace hu : (addOrderOf u : \u211d) \u2260 0 ** case intro p : \u211d hp : Fact (0 < p) u : AddCircle p hu' : u \u2260 0 n : \u2115 hn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u)) hu : \u2191(addOrderOf u) \u2260 0 \u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016 ** conv_lhs => rw [\u2190 mul_one p] ** case intro p : \u211d hp : Fact (0 < p) u : AddCircle p hu' : u \u2260 0 n : \u2115 hn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u)) hu : \u2191(addOrderOf u) \u2260 0 \u22a2 p * 1 \u2264 addOrderOf u \u2022 \u2016u\u2016 ** 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] ** case intro p : \u211d hp : Fact (0 < p) u : AddCircle p hu' : u \u2260 0 n : \u2115 hn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u)) hu : \u2191(addOrderOf u) \u2260 0 \u22a2 n \u2260 0 ** contrapose! hu' ** case intro p : \u211d hp : Fact (0 < p) u : AddCircle p n : \u2115 hn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u)) hu : \u2191(addOrderOf u) \u2260 0 hu' : n = 0 \u22a2 u = 0 ** simpa only [hu', Nat.cast_zero, zero_div, mul_zero, norm_eq_zero] using hn ** case hu p : \u211d hp : Fact (0 < p) u : AddCircle p hu : IsOfFinAddOrder u hu' : u \u2260 0 n : \u2115 hn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u)) \u22a2 \u2191(addOrderOf u) \u2260 0 ** norm_cast ** case hu p : \u211d hp : Fact (0 < p) u : AddCircle p hu : IsOfFinAddOrder u hu' : u \u2260 0 n : \u2115 hn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u)) \u22a2 \u00acaddOrderOf u = 0 ** exact (addOrderOf_pos_iff.mpr hu).ne' ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.variance_zero ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc\u271d \u03bc : Measure \u03a9 \u22a2 variance 0 \u03bc = 0 ** simp only [variance, evariance_zero, ENNReal.zero_toReal] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.addHaar_ball_mul ** E : Type u_1 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : MeasurableSpace E inst\u271d\u2076 : BorelSpace E inst\u271d\u2075 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u2074 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F s\u271d : Set E inst\u271d : Nontrivial E x : E r : \u211d hr : 0 \u2264 r s : \u211d \u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s) ** rcases hr.eq_or_lt with (rfl | h) ** case inl E : Type u_1 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : MeasurableSpace E inst\u271d\u2076 : BorelSpace E inst\u271d\u2075 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u2074 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F s\u271d : Set E inst\u271d : Nontrivial E x : E s : \u211d hr : 0 \u2264 0 \u22a2 \u2191\u2191\u03bc (ball x (0 * s)) = ENNReal.ofReal (0 ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s) ** simp only [zero_pow (finrank_pos (K := \u211d) (V := E)), measure_empty, zero_mul,\n ENNReal.ofReal_zero, ball_zero] ** case inr E : Type u_1 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : MeasurableSpace E inst\u271d\u2076 : BorelSpace E inst\u271d\u2075 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u2074 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F s\u271d : Set E inst\u271d : Nontrivial E x : E r : \u211d hr : 0 \u2264 r s : \u211d h : 0 < r \u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s) ** exact addHaar_ball_mul_of_pos \u03bc x h s ** Qed", + "informal": "" + }, + { + "formal": "Basis.coord_repr_symm ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b\u271d b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x : M b : Basis \u03b9 R M i : \u03b9 f : \u03b9 \u2192\u2080 R \u22a2 \u2191(coord b i) (\u2191(LinearEquiv.symm b.repr) f) = \u2191f i ** simp only [repr_symm_apply, coord_apply, repr_total] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.map_subtype_le ** R : Type u_1 R\u2081 : Type u_2 R\u2082 : Type u_3 R\u2083 : Type u_4 R\u2084 : Type u_5 S : Type u_6 K : Type u_7 K\u2082 : Type u_8 M : Type u_9 M' : Type u_10 M\u2081 : Type u_11 M\u2082 : Type u_12 M\u2083 : Type u_13 M\u2084 : Type u_14 N : Type u_15 N\u2082 : Type u_16 \u03b9 : Type u_17 V : Type u_18 V\u2082 : Type u_19 inst\u271d\u2075 : Semiring R inst\u271d\u2074 : Semiring R\u2082 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid M\u2082 inst\u271d\u00b9 : Module R M inst\u271d : Module R\u2082 M\u2082 p p'\u271d : Submodule R M q : Submodule R\u2082 M\u2082 \u03c4\u2081\u2082 : R \u2192+* R\u2082 F : Type u_20 sc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082 p' : Submodule R { x // x \u2208 p } \u22a2 map (Submodule.subtype p) p' \u2264 p ** simpa using (map_le_range : map p.subtype p' \u2264 range p.subtype) ** Qed", + "informal": "" + }, + { + "formal": "FiniteField.exists_nonsquare ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 \u22a2 \u2203 a, \u00acIsSquare a ** let sq : F \u2192 F := fun x => x ^ 2 ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 \u22a2 \u2203 a, \u00acIsSquare a ** 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] ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 h : \u00acFunction.Injective sq \u22a2 \u2203 a, \u00acIsSquare a ** rw [Finite.injective_iff_surjective] at h ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 h : \u00acFunction.Surjective sq \u22a2 \u2203 a, \u00acIsSquare a ** simp_rw [IsSquare, \u2190 pow_two, @eq_comm _ _ (_ ^ 2)] ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 h : \u00acFunction.Surjective sq \u22a2 \u2203 a, \u00ac\u2203 r, r ^ 2 = a ** unfold Function.Surjective at h ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 h : \u00ac\u2200 (b : F), \u2203 a, sq a = b \u22a2 \u2203 a, \u00ac\u2203 r, r ^ 2 = a ** push_neg at h \u22a2 ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 h : \u2203 b, \u2200 (a : F), sq a \u2260 b \u22a2 \u2203 a, \u2200 (r : F), r ^ 2 \u2260 a ** exact h ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 \u22a2 \u00acFunction.Injective sq ** simp only [Function.Injective, not_forall, exists_prop] ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 \u22a2 \u2203 x x_1, x ^ 2 = x_1 ^ 2 \u2227 \u00acx = x_1 ** refine' \u27e8-1, 1, _, Ring.neg_one_ne_one_of_char_ne_two hF\u27e9 ** K : Type u_1 R : Type u_2 F : Type u_3 inst\u271d\u00b9 : Field F inst\u271d : Finite F hF : ringChar F \u2260 2 sq : F \u2192 F := fun x => x ^ 2 \u22a2 (-1) ^ 2 = 1 ^ 2 ** simp only [one_pow, neg_one_sq] ** Qed", + "informal": "" + }, + { + "formal": "ZMod.val_one_eq_one_mod ** n : \u2115 \u22a2 val 1 = 1 % n ** rw [\u2190 Nat.cast_one, val_nat_cast] ** Qed", + "informal": "" + }, + { + "formal": "GeneralizedContinuedFraction.of_convergence_epsilon ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u22a2 \u2200 (\u03b5 : K), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5 ** intro \u03b5 \u03b5_pos ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5 ** rcases (exists_nat_gt (1 / \u03b5) : \u2203 N' : \u2115, 1 / \u03b5 < N') with \u27e8N', one_div_\u03b5_lt_N'\u27e9 ** case intro K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5 ** let N := max N' 5 ** case intro K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5 ** exists N ** case intro K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 \u22a2 \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5 ** intro n n_ge_N ** case intro K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N \u22a2 |v - convergents (of v) n| < \u03b5 ** let g := of v ** case intro K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v \u22a2 |v - convergents (of v) n| < \u03b5 ** cases' Decidable.em (g.TerminatedAt n) with terminated_at_n not_terminated_at_n ** case intro.inl K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v terminated_at_n : TerminatedAt g n \u22a2 |v - convergents (of v) n| < \u03b5 ** have : v = g.convergents n := of_correctness_of_terminatedAt terminated_at_n ** case intro.inl K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v terminated_at_n : TerminatedAt g n this : v = convergents g n \u22a2 |v - convergents (of v) n| < \u03b5 ** have : v - g.convergents n = 0 := sub_eq_zero.mpr this ** case intro.inl K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v terminated_at_n : TerminatedAt g n this\u271d : v = convergents g n this : v - convergents g n = 0 \u22a2 |v - convergents (of v) n| < \u03b5 ** rw [this] ** case intro.inl K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v terminated_at_n : TerminatedAt g n this\u271d : v = convergents g n this : v - convergents g n = 0 \u22a2 |0| < \u03b5 ** exact_mod_cast \u03b5_pos ** case intro.inr K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n \u22a2 |v - convergents (of v) n| < \u03b5 ** let B := g.denominators n ** case intro.inr K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n \u22a2 |v - convergents (of v) n| < \u03b5 ** let nB := g.denominators (n + 1) ** case intro.inr K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) \u22a2 |v - convergents (of v) n| < \u03b5 ** have abs_v_sub_conv_le : |v - g.convergents n| \u2264 1 / (B * nB) :=\n abs_sub_convergents_le not_terminated_at_n ** case intro.inr K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) \u22a2 |v - convergents (of v) n| < \u03b5 ** suffices : 1 / (B * nB) < \u03b5 ** case intro.inr K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) this : 1 / (B * nB) < \u03b5 \u22a2 |v - convergents (of v) n| < \u03b5 case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) \u22a2 1 / (B * nB) < \u03b5 ** exact lt_of_le_of_lt abs_v_sub_conv_le this ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) \u22a2 1 / (B * nB) < \u03b5 ** 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) ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB \u22a2 1 / (B * nB) < \u03b5 ** 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) ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B \u22a2 1 / (B * nB) < \u03b5 ** have zero_lt_B : 0 < B := B_ineq.trans_lt' $ by exact_mod_cast fib_pos.2 n.succ_pos ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B \u22a2 1 / (B * nB) < \u03b5 ** have nB_pos : 0 < nB := nB_ineq.trans_lt' $ by exact_mod_cast fib_pos.2 $ succ_pos _ ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB \u22a2 1 / (B * nB) < \u03b5 ** have zero_lt_mul_conts : 0 < B * nB := by positivity ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB \u22a2 1 / (B * nB) < \u03b5 ** suffices : 1 < \u03b5 * (B * nB) ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB this : 1 < \u03b5 * (B * nB) \u22a2 1 / (B * nB) < \u03b5 case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB \u22a2 1 < \u03b5 * (B * nB) ** exact (div_lt_iff zero_lt_mul_conts).mpr this ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB \u22a2 1 < \u03b5 * (B * nB) ** 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 ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 1 < \u03b5 * (B * nB) ** suffices : \u03b5 * n \u2264 \u03b5 * (B * nB) ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n this : \u03b5 * \u2191n \u2264 \u03b5 * (B * nB) \u22a2 1 < \u03b5 * (B * nB) case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB) ** exact lt_of_lt_of_le one_lt_\u03b5_mul_N this ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB) ** suffices : (n : K) \u2264 B * nB ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n this : \u2191n \u2264 B * nB \u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB) case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 \u2191n \u2264 B * nB ** exact (mul_le_mul_left \u03b5_pos).mpr this ** case this K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 \u2191n \u2264 B * nB ** 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 ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B \u22a2 0 < \u2191(fib (n + 1)) ** exact_mod_cast fib_pos.2 n.succ_pos ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B \u22a2 0 < \u2191(fib (n + 2)) ** exact_mod_cast fib_pos.2 $ succ_pos _ ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB \u22a2 0 < B * nB ** positivity ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB \u22a2 1 < \u03b5 * \u2191n ** have one_lt_\u03b5_mul_N' : 1 < \u03b5 * (N' : K) := (div_lt_iff' \u03b5_pos).mp one_div_\u03b5_lt_N' ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N' \u22a2 1 < \u03b5 * \u2191n ** have : (N' : K) \u2264 N := by exact_mod_cast le_max_left _ _ ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N' this : \u2191N' \u2264 \u2191N \u22a2 1 < \u03b5 * \u2191n ** have : \u03b5 * N' \u2264 \u03b5 * n :=\n (mul_le_mul_left \u03b5_pos).mpr (le_trans this (by exact_mod_cast n_ge_N)) ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N' this\u271d : \u2191N' \u2264 \u2191N this : \u03b5 * \u2191N' \u2264 \u03b5 * \u2191n \u22a2 1 < \u03b5 * \u2191n ** exact lt_of_lt_of_le one_lt_\u03b5_mul_N' this ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N' \u22a2 \u2191N' \u2264 \u2191N ** exact_mod_cast le_max_left _ _ ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N' this : \u2191N' \u2264 \u2191N \u22a2 \u2191N \u2264 \u2191n ** exact_mod_cast n_ge_N ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 \u2191n \u2264 \u2191(fib n) ** exact_mod_cast le_fib_self <| le_trans (le_max_right N' 5) n_ge_N ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 \u2191(fib n) \u2264 \u2191(fib (n + 1)) ** exact_mod_cast fib_le_fib_succ ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 \u2191(fib (n + 1)) \u2264 \u2191(fib (n + 1)) * \u2191(fib (n + 1)) ** exact_mod_cast (fib (n + 1)).le_mul_self ** K : Type u_1 v : K inst\u271d\u00b2 : LinearOrderedField K inst\u271d\u00b9 : FloorRing K inst\u271d : Archimedean K \u03b5 : K \u03b5_pos : \u03b5 > 0 N' : \u2115 one_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N' N : \u2115 := max N' 5 n : \u2115 n_ge_N : n \u2265 N g : GeneralizedContinuedFraction K := of v not_terminated_at_n : \u00acTerminatedAt g n B : K := denominators g n nB : K := denominators g (n + 1) abs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB) nB_ineq : \u2191(fib (n + 2)) \u2264 nB B_ineq : \u2191(fib (n + 1)) \u2264 B zero_lt_B : 0 < B nB_pos : 0 < nB zero_lt_mul_conts : 0 < B * nB one_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n \u22a2 \u2191(fib (n + 1)) \u2264 \u2191(fib (n + 2)) ** exact_mod_cast fib_le_fib_succ ** Qed", + "informal": "" + }, + { + "formal": "WithTop.image_coe_Ici ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ici a = Ico \u2191a \u22a4 ** rw [\u2190 preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio] ** Qed", + "informal": "" + }, + { + "formal": "Nat.floor_eq_zero ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedSemiring \u03b1 inst\u271d : FloorSemiring \u03b1 a : \u03b1 n : \u2115 \u22a2 \u230aa\u230b\u208a = 0 \u2194 a < 1 ** rw [\u2190 lt_one_iff, \u2190 @cast_one \u03b1] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedSemiring \u03b1 inst\u271d : FloorSemiring \u03b1 a : \u03b1 n : \u2115 \u22a2 \u230aa\u230b\u208a < 1 \u2194 a < \u21911 ** exact floor_lt' Nat.one_ne_zero ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Localization.Construction.morphismProperty_is_top ** C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G : C \u2964 D hG : MorphismProperty.IsInvertedBy W G P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P \u22a2 P = \u22a4 ** funext X Y f ** case h.h.h C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G : C \u2964 D hG : MorphismProperty.IsInvertedBy W G P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y \u22a2 P f = \u22a4 f ** ext ** case h.h.h.a C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G : C \u2964 D hG : MorphismProperty.IsInvertedBy W G P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y \u22a2 P f \u2194 \u22a4 f ** constructor ** case h.h.h.a.mp C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G : C \u2964 D hG : MorphismProperty.IsInvertedBy W G P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y \u22a2 P f \u2192 \u22a4 f ** intro ** case h.h.h.a.mp C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G : C \u2964 D hG : MorphismProperty.IsInvertedBy W G P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : P f \u22a2 \u22a4 f ** apply MorphismProperty.top_apply ** case h.h.h.a.mpr C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G : C \u2964 D hG : MorphismProperty.IsInvertedBy W G P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y \u22a2 \u22a4 f \u2192 P f ** intro ** case h.h.h.a.mpr C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G : C \u2964 D hG : MorphismProperty.IsInvertedBy W G P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f \u22a2 P f ** let G : _ \u2964 W.Localization := Quotient.functor _ ** case h.h.h.a.mpr C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) \u22a2 P f ** haveI : Full G := Quotient.fullFunctor _ ** case h.h.h.a.mpr C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G \u22a2 P f ** 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) ** case h.h.h.a.mpr C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G \u22a2 \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f) ** intros X\u2081 X\u2082 p ** case h.h.h.a.mpr C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082 : Paths (LocQuiver W) p : X\u2081 \u27f6 X\u2082 \u22a2 P (G.map p) ** induction' p with X\u2082 X\u2083 p g hp ** C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this\u271d : Full G this : \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f) \u22a2 P f ** rcases X with \u27e8\u27e8X\u27e9\u27e9 ** case mk.mk C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P Y : MorphismProperty.Localization W G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this\u271d : Full G this : \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f) X : C f : { as := { obj := X } } \u27f6 Y a\u271d : \u22a4 f \u22a2 P f ** rcases Y with \u27e8\u27e8Y\u27e9\u27e9 ** case mk.mk.mk.mk C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this\u271d : Full G this : \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f) X Y : C f : { as := { obj := X } } \u27f6 { as := { obj := Y } } a\u271d : \u22a4 f \u22a2 P f ** simpa only [Functor.image_preimage] using this _ _ (G.preimage f) ** case h.h.h.a.mpr.nil C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082 : Paths (LocQuiver W) \u22a2 P (G.map Quiver.Path.nil) ** simpa only [Functor.map_id] using hP\u2081 (\ud835\udfd9 X\u2081.obj) ** case h.h.h.a.mpr.cons C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W) p : Quiver.Path X\u2081 X\u2082 g : X\u2082 \u27f6 X\u2083 hp : P (G.map p) \u22a2 P (G.map (Quiver.Path.cons p g)) ** let p' : X\u2081 \u27f6X\u2082 := p ** case h.h.h.a.mpr.cons C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W) p : Quiver.Path X\u2081 X\u2082 g : X\u2082 \u27f6 X\u2083 hp : P (G.map p) p' : X\u2081 \u27f6 X\u2082 := p \u22a2 P (G.map (Quiver.Path.cons p g)) ** rw [show p'.cons g = p' \u226b Quiver.Hom.toPath g by rfl, G.map_comp] ** case h.h.h.a.mpr.cons C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W) p : Quiver.Path X\u2081 X\u2082 g : X\u2082 \u27f6 X\u2083 hp : P (G.map p) p' : X\u2081 \u27f6 X\u2082 := p \u22a2 P (G.map p' \u226b G.map (Quiver.Hom.toPath g)) ** refine' hP\u2083 _ _ hp _ ** case h.h.h.a.mpr.cons C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W) p : Quiver.Path X\u2081 X\u2082 g : X\u2082 \u27f6 X\u2083 hp : P (G.map p) p' : X\u2081 \u27f6 X\u2082 := p \u22a2 P (G.map (Quiver.Hom.toPath g)) ** rcases g with (g | \u27e8g, hg\u27e9) ** C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W) p : Quiver.Path X\u2081 X\u2082 g : X\u2082 \u27f6 X\u2083 hp : P (G.map p) p' : X\u2081 \u27f6 X\u2082 := p \u22a2 Quiver.Path.cons p' g = p' \u226b Quiver.Hom.toPath g ** rfl ** case h.h.h.a.mpr.cons.inl C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W) p : Quiver.Path X\u2081 X\u2082 hp : P (G.map p) p' : X\u2081 \u27f6 X\u2082 := p g : X\u2082.obj \u27f6 X\u2083.obj \u22a2 P (G.map (Quiver.Hom.toPath (Sum.inl g))) ** apply hP\u2081 ** case h.h.h.a.mpr.cons.inr.mk C : Type uC inst\u271d\u00b9 : Category.{uC', uC} C W : MorphismProperty C D : Type uD inst\u271d : Category.{uD', uD} D G\u271d : C \u2964 D hG : MorphismProperty.IsInvertedBy W G\u271d P : MorphismProperty (MorphismProperty.Localization W) hP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f) hP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw) hP\u2083 : MorphismProperty.StableUnderComposition P X Y : MorphismProperty.Localization W f : X \u27f6 Y a\u271d : \u22a4 f G : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W) this : Full G X\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W) p : Quiver.Path X\u2081 X\u2082 hp : P (G.map p) p' : X\u2081 \u27f6 X\u2082 := p g : X\u2083.obj \u27f6 X\u2082.obj hg : W g \u22a2 P (G.map (Quiver.Hom.toPath (Sum.inr { val := g, property := hg }))) ** apply hP\u2082 ** Qed", + "informal": "" + }, + { + "formal": "Matrix.mul_inv_eq_iff_eq_mul_of_invertible ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : CommRing \u03b1 A\u271d B\u271d A B C : Matrix n n \u03b1 inst\u271d : Invertible A h : B * A\u207b\u00b9 = C \u22a2 B = C * A ** rw [\u2190 h, inv_mul_cancel_right_of_invertible] ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u00b3 : Fintype n inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : CommRing \u03b1 A\u271d B\u271d A B C : Matrix n n \u03b1 inst\u271d : Invertible A h : B = C * A \u22a2 B * A\u207b\u00b9 = C ** rw [h, mul_inv_cancel_right_of_invertible] ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.comapDomain'_single ** \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 \u03ba : Type u_1 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : DecidableEq \u03ba inst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i) h : \u03ba \u2192 \u03b9 h' : \u03b9 \u2192 \u03ba hh' : Function.LeftInverse h' h k : \u03ba x : \u03b2 (h k) \u22a2 comapDomain' h hh' (single (h k) x) = single k x ** ext i ** case h \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 \u03ba : Type u_1 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : DecidableEq \u03ba inst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i) h : \u03ba \u2192 \u03b9 h' : \u03b9 \u2192 \u03ba hh' : Function.LeftInverse h' h k : \u03ba x : \u03b2 (h k) i : \u03ba \u22a2 \u2191(comapDomain' h hh' (single (h k) x)) i = \u2191(single k x) i ** rw [comapDomain'_apply] ** case h \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 \u03ba : Type u_1 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : DecidableEq \u03ba inst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i) h : \u03ba \u2192 \u03b9 h' : \u03b9 \u2192 \u03ba hh' : Function.LeftInverse h' h k : \u03ba x : \u03b2 (h k) i : \u03ba \u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i ** obtain rfl | hik := Decidable.eq_or_ne i k ** case h.inl \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 \u03ba : Type u_1 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : DecidableEq \u03ba inst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i) h : \u03ba \u2192 \u03b9 h' : \u03b9 \u2192 \u03ba hh' : Function.LeftInverse h' h i : \u03ba x : \u03b2 (h i) \u22a2 \u2191(single (h i) x) (h i) = \u2191(single i x) i ** rw [single_eq_same, single_eq_same] ** case h.inr \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 \u03ba : Type u_1 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : DecidableEq \u03ba inst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i) h : \u03ba \u2192 \u03b9 h' : \u03b9 \u2192 \u03ba hh' : Function.LeftInverse h' h k : \u03ba x : \u03b2 (h k) i : \u03ba hik : i \u2260 k \u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i ** rw [single_eq_of_ne hik.symm, single_eq_of_ne (hh'.injective.ne hik.symm)] ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Language.card_withConstants ** L : Language L' : Language M : Type w inst\u271d : Structure L M \u03b1 : Type w' \u22a2 card (L[[\u03b1]]) = lift.{w', max u v} (card L) + lift.{max u v, w'} #\u03b1 ** rw [withConstants, card_sum, card_constantsOn] ** Qed", + "informal": "" + }, + { + "formal": "iInf_pair ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a\u271d b\u271d : \u03b1 f : \u03b2 \u2192 \u03b1 a b : \u03b2 \u22a2 \u2a05 x \u2208 {a, b}, f x = f a \u2293 f b ** rw [iInf_insert, iInf_singleton] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.stoppedProcess_eq_of_mem_finset ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u22a2 stoppedProcess u \u03c4 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) ** ext \u03c9 ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 \u22a2 stoppedProcess u \u03c4 n \u03c9 = (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 ** rw [Pi.add_apply, Finset.sum_apply] ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 \u22a2 stoppedProcess u \u03c4 n \u03c9 = 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 ** cases' le_or_lt n (\u03c4 \u03c9) with h h ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 \u22a2 stoppedProcess u \u03c4 n \u03c9 = 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 ** rw [stoppedProcess_eq_of_le h, Set.indicator_of_mem, Finset.sum_eq_zero, add_zero] ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 \u22a2 \u2200 (x : \u03b9), x \u2208 Finset.filter (fun x => x < n) s \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = x} (u x) \u03c9 = 0 ** intro m hm ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 m : \u03b9 hm : m \u2208 Finset.filter (fun x => x < n) s \u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = m} (u m) \u03c9 = 0 ** refine' Set.indicator_of_not_mem _ _ ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 m : \u03b9 hm : m \u2208 Finset.filter (fun x => x < n) s \u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m} ** rw [Finset.mem_filter] at hm ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 m : \u03b9 hm : m \u2208 s \u2227 m < n \u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m} ** exact (hm.2.trans_le h).ne' ** case h.inl.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 \u22a2 \u03c9 \u2208 {a | n \u2264 \u03c4 a} ** exact h ** case h.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 stoppedProcess u \u03c4 n \u03c9 = 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 ** rw [stoppedProcess_eq_of_ge (le_of_lt h), Finset.sum_eq_single_of_mem (\u03c4 \u03c9)] ** case h.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < 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 ** rw [Set.indicator_of_not_mem, zero_add, Set.indicator_of_mem] ** case h.inr.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u03c9 \u2208 {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9} ** exact rfl ** case h.inr.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u00ac\u03c9 \u2208 {a | n \u2264 \u03c4 a} ** exact not_le.2 h ** case h.inr.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u03c4 \u03c9 \u2208 Finset.filter (fun x => x < n) s ** rw [Finset.mem_filter] ** case h.inr.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u03c4 \u03c9 \u2208 s \u2227 \u03c4 \u03c9 < n ** exact \u27e8hbdd \u03c9 h, h\u27e9 ** case h.inr.h\u2080 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < 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 ** intro b _ hneq ** case h.inr.h\u2080 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n b : \u03b9 a\u271d : b \u2208 Finset.filter (fun x => x < n) s hneq : b \u2260 \u03c4 \u03c9 \u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0 ** rw [Set.indicator_of_not_mem] ** case h.inr.h\u2080.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n b : \u03b9 a\u271d : b \u2208 Finset.filter (fun x => x < n) s hneq : b \u2260 \u03c4 \u03c9 \u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = b} ** exact hneq.symm ** Qed", + "informal": "" + }, + { + "formal": "Int.not_lt ** a b : Int \u22a2 \u00aca < b \u2194 b \u2264 a ** rw [\u2190 Int.not_le, Decidable.not_not] ** Qed", + "informal": "" + }, + { + "formal": "Mathlib.Tactic.Group.zpow_trick ** G : Type u_1 inst\u271d : Group G a b : G n m : \u2124 \u22a2 a * b ^ n * b ^ m = a * b ^ (n + m) ** rw [mul_assoc, \u2190 zpow_add] ** Qed", + "informal": "" + }, + { + "formal": "WittVector.StandardOneDimIsocrystal.frobenius_apply ** p : \u2115 inst\u271d\u2074 : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b3 : CommRing k inst\u271d\u00b2 : IsDomain k inst\u271d\u00b9 : CharP k p inst\u271d : PerfectRing k p m : \u2124 x : StandardOneDimIsocrystal p k m \u22a2 \u2191\u03a6(p, k) x = \u2191p ^ m \u2022 \u2191\u03c6(p, k) x ** erw [smul_eq_mul] ** p : \u2115 inst\u271d\u2074 : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b3 : CommRing k inst\u271d\u00b2 : IsDomain k inst\u271d\u00b9 : CharP k p inst\u271d : PerfectRing k p m : \u2124 x : StandardOneDimIsocrystal p k m \u22a2 \u2191\u03a6(p, k) x = \u2191p ^ m * \u2191\u03c6(p, k) x ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Subsemiring.mem_iSup_of_directed ** R : Type u S\u271d : Type v T : Type w inst\u271d\u00b2 : NonAssocSemiring R M : Submonoid R inst\u271d\u00b9 : NonAssocSemiring S\u271d inst\u271d : NonAssocSemiring T \u03b9 : Sort u_1 h\u03b9 : Nonempty \u03b9 S : \u03b9 \u2192 Subsemiring R hS : Directed (fun x x_1 => x \u2264 x_1) S x : R \u22a2 x \u2208 \u2a06 i, S i \u2194 \u2203 i, x \u2208 S i ** refine' \u27e8_, fun \u27e8i, hi\u27e9 => (SetLike.le_def.1 <| le_iSup S i) hi\u27e9 ** R : Type u S\u271d : Type v T : Type w inst\u271d\u00b2 : NonAssocSemiring R M : Submonoid R inst\u271d\u00b9 : NonAssocSemiring S\u271d inst\u271d : NonAssocSemiring T \u03b9 : Sort u_1 h\u03b9 : Nonempty \u03b9 S : \u03b9 \u2192 Subsemiring R hS : Directed (fun x x_1 => x \u2264 x_1) S x : R \u22a2 x \u2208 \u2a06 i, S i \u2192 \u2203 i, x \u2208 S i ** 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) ** R : Type u S\u271d : Type v T : Type w inst\u271d\u00b2 : NonAssocSemiring R M : Submonoid R inst\u271d\u00b9 : NonAssocSemiring S\u271d inst\u271d : NonAssocSemiring T \u03b9 : Sort u_1 h\u03b9 : Nonempty \u03b9 S : \u03b9 \u2192 Subsemiring R hS : Directed (fun x x_1 => x \u2264 x_1) S x : R U : Subsemiring R := Subsemiring.mk' (\u22c3 i, \u2191(S i)) (\u2a06 i, (S i).toSubmonoid) (_ : \u2191(\u2a06 i, (S i).toSubmonoid) = \u22c3 i, \u2191(S i).toSubmonoid) (\u2a06 i, toAddSubmonoid (S i)) (_ : \u2191(\u2a06 i, toAddSubmonoid (S i)) = \u22c3 i, \u2191(toAddSubmonoid (S i))) \u22a2 x \u2208 \u2a06 i, S i \u2192 \u2203 i, x \u2208 S i ** suffices h : \u2a06 i, S i \u2264 U by simpa using @h x ** R : Type u S\u271d : Type v T : Type w inst\u271d\u00b2 : NonAssocSemiring R M : Submonoid R inst\u271d\u00b9 : NonAssocSemiring S\u271d inst\u271d : NonAssocSemiring T \u03b9 : Sort u_1 h\u03b9 : Nonempty \u03b9 S : \u03b9 \u2192 Subsemiring R hS : Directed (fun x x_1 => x \u2264 x_1) S x : R U : Subsemiring R := Subsemiring.mk' (\u22c3 i, \u2191(S i)) (\u2a06 i, (S i).toSubmonoid) (_ : \u2191(\u2a06 i, (S i).toSubmonoid) = \u22c3 i, \u2191(S i).toSubmonoid) (\u2a06 i, toAddSubmonoid (S i)) (_ : \u2191(\u2a06 i, toAddSubmonoid (S i)) = \u22c3 i, \u2191(toAddSubmonoid (S i))) \u22a2 \u2a06 i, S i \u2264 U ** exact iSup_le fun i x hx => Set.mem_iUnion.2 \u27e8i, hx\u27e9 ** R : Type u S\u271d : Type v T : Type w inst\u271d\u00b2 : NonAssocSemiring R M : Submonoid R inst\u271d\u00b9 : NonAssocSemiring S\u271d inst\u271d : NonAssocSemiring T \u03b9 : Sort u_1 h\u03b9 : Nonempty \u03b9 S : \u03b9 \u2192 Subsemiring R hS : Directed (fun x x_1 => x \u2264 x_1) S x : R U : Subsemiring R := Subsemiring.mk' (\u22c3 i, \u2191(S i)) (\u2a06 i, (S i).toSubmonoid) (_ : \u2191(\u2a06 i, (S i).toSubmonoid) = \u22c3 i, \u2191(S i).toSubmonoid) (\u2a06 i, toAddSubmonoid (S i)) (_ : \u2191(\u2a06 i, toAddSubmonoid (S i)) = \u22c3 i, \u2191(toAddSubmonoid (S i))) h : \u2a06 i, S i \u2264 U \u22a2 x \u2208 \u2a06 i, S i \u2192 \u2203 i, x \u2208 S i ** simpa using @h x ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.integral_abs_condexp_le ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : \u00acm \u2264 m0 \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** by_cases hfint : Integrable f \u03bc ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : \u00acm \u2264 m0 \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** simp_rw [condexp_of_not_le hm, Pi.zero_apply, abs_zero, integral_zero] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : \u00acm \u2264 m0 \u22a2 0 \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** exact integral_nonneg fun x => abs_nonneg _ ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** simp only [condexp_undef hfint, Pi.zero_apply, abs_zero, integral_const, Algebra.id.smul_eq_mul,\n mul_zero] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 0 \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc ** exact integral_nonneg fun x => abs_nonneg _ ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \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) ** rw [ENNReal.toReal_le_toReal] <;> simp_rw [\u2190 Real.norm_eq_abs, ofReal_norm_eq_coe_nnnorm] ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \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 ** rw [\u2190 snorm_one_eq_lintegral_nnnorm, \u2190 snorm_one_eq_lintegral_nnnorm] ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 snorm (fun a => (\u03bc[f|m]) a) 1 \u03bc \u2264 snorm (fun a => f a) 1 \u03bc ** exact snorm_one_condexp_le_snorm _ ** case pos.ha \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(\u03bc[f|m]) a\u2016\u208a \u2202\u03bc \u2260 \u22a4 ** exact ne_of_lt integrable_condexp.2 ** case pos.hb \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc \u2260 \u22a4 ** exact ne_of_lt hfint.2 ** case pos.hf \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 0 \u2264\u1d50[\u03bc] fun x => |f x| ** exact eventually_of_forall fun x => abs_nonneg _ ** case pos.hfm \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 AEStronglyMeasurable (fun x => |f x|) \u03bc ** simp_rw [\u2190 Real.norm_eq_abs] ** case pos.hfm \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 AEStronglyMeasurable (fun x => \u2016f x\u2016) \u03bc ** exact hfint.1.norm ** case pos.hf \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 0 \u2264\u1d50[\u03bc] fun x => |(\u03bc[f|m]) x| ** exact eventually_of_forall fun x => abs_nonneg _ ** case pos.hfm \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 AEStronglyMeasurable (fun x => |(\u03bc[f|m]) x|) \u03bc ** simp_rw [\u2190 Real.norm_eq_abs] ** case pos.hfm \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hm : m \u2264 m0 hfint : Integrable f \u22a2 AEStronglyMeasurable (fun x => \u2016(\u03bc[f|m]) x\u2016) \u03bc ** exact (stronglyMeasurable_condexp.mono hm).aestronglyMeasurable.norm ** Qed", + "informal": "" + }, + { + "formal": "PadicInt.p_nonnunit ** p : \u2115 hp : Fact (Nat.Prime p) \u22a2 \u2191p \u2208 nonunits \u2124_[p] ** have : (p : \u211d)\u207b\u00b9 < 1 := inv_lt_one <| by exact_mod_cast hp.1.one_lt ** p : \u2115 hp : Fact (Nat.Prime p) this : (\u2191p)\u207b\u00b9 < 1 \u22a2 \u2191p \u2208 nonunits \u2124_[p] ** rwa [\u2190 norm_p, \u2190 mem_nonunits] at this ** p : \u2115 hp : Fact (Nat.Prime p) \u22a2 1 < \u2191p ** exact_mod_cast hp.1.one_lt ** Qed", + "informal": "" + }, + { + "formal": "PMF.toMeasure_uniformOfFintype_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : Fintype \u03b1 inst\u271d\u00b9 : Nonempty \u03b1 s : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(toMeasure (uniformOfFintype \u03b1)) s = \u2191(Fintype.card \u2191s) / \u2191(Fintype.card \u03b1) ** simp [uniformOfFintype, hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : Fintype \u03b1 inst\u271d\u00b9 : Nonempty \u03b1 s : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 hs : MeasurableSet s \u22a2 \u2191(Finset.card (Finset.filter (fun x => x \u2208 s) Finset.univ)) / \u2191(Finset.card Finset.univ) = \u2191(Finset.card (Finset.filter (fun x => x \u2208 s) Finset.univ)) / \u2191(Fintype.card \u03b1) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "ONote.NFBelow.lt ** e : ONote n : \u2115+ a : ONote b : Ordinal.{0} h : NFBelow (ONote.oadd e n a) b \u22a2 repr e < b ** (cases' h with _ _ _ _ eb _ h\u2081 h\u2082 h\u2083; exact h\u2083) ** e : ONote n : \u2115+ a : ONote b : Ordinal.{0} h : NFBelow (ONote.oadd e n a) b \u22a2 repr e < b ** cases' h with _ _ _ _ eb _ h\u2081 h\u2082 h\u2083 ** case oadd' e : ONote n : \u2115+ a : ONote b eb : Ordinal.{0} h\u2081 : NFBelow e eb h\u2082 : NFBelow a (repr e) h\u2083 : repr e < b \u22a2 repr e < b ** exact h\u2083 ** Qed", + "informal": "" + }, + { + "formal": "Algebraic.cardinal_mk_le_mul ** R A : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : IsDomain A inst\u271d\u00b9 : Algebra R A inst\u271d : NoZeroSMulDivisors R A \u22a2 #{ x // IsAlgebraic R x } \u2264 #R[X] * \u2135\u2080 ** rw [\u2190 lift_id #_, \u2190 lift_id #R[X]] ** R A : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : IsDomain A inst\u271d\u00b9 : Algebra R A inst\u271d : NoZeroSMulDivisors R A \u22a2 lift.{u, u} #{ x // IsAlgebraic R x } \u2264 lift.{u, u} #R[X] * \u2135\u2080 ** exact cardinal_mk_lift_le_mul R A ** Qed", + "informal": "" + }, + { + "formal": "IsCoercive.ker_eq_bot ** V : Type u inst\u271d\u00b2 : NormedAddCommGroup V inst\u271d\u00b9 : InnerProductSpace \u211d V inst\u271d : CompleteSpace V B : V \u2192L[\u211d] V \u2192L[\u211d] \u211d coercive : IsCoercive B \u22a2 ker (continuousLinearMapOfBilin B) = \u22a5 ** rw [LinearMapClass.ker_eq_bot] ** V : Type u inst\u271d\u00b2 : NormedAddCommGroup V inst\u271d\u00b9 : InnerProductSpace \u211d V inst\u271d : CompleteSpace V B : V \u2192L[\u211d] V \u2192L[\u211d] \u211d coercive : IsCoercive B \u22a2 Function.Injective \u2191(continuousLinearMapOfBilin B) ** rcases coercive.antilipschitz with \u27e8_, _, antilipschitz\u27e9 ** case intro.intro V : Type u inst\u271d\u00b2 : NormedAddCommGroup V inst\u271d\u00b9 : InnerProductSpace \u211d V inst\u271d : CompleteSpace V B : V \u2192L[\u211d] V \u2192L[\u211d] \u211d coercive : IsCoercive B w\u271d : \u211d\u22650 left\u271d : 0 < w\u271d antilipschitz : AntilipschitzWith w\u271d \u2191(continuousLinearMapOfBilin B) \u22a2 Function.Injective \u2191(continuousLinearMapOfBilin B) ** exact antilipschitz.injective ** Qed", + "informal": "" + }, + { + "formal": "Set.mem_accumulate ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : \u03b1 \u2192 Set \u03b2 t : \u03b1 \u2192 Set \u03b3 inst\u271d : LE \u03b1 x : \u03b1 z : \u03b2 \u22a2 z \u2208 Accumulate s x \u2194 \u2203 y, y \u2264 x \u2227 z \u2208 s y ** simp_rw [accumulate_def, mem_iUnion\u2082, exists_prop] ** Qed", + "informal": "" + }, + { + "formal": "Stream'.WSeq.toSeq_ofSeq ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 \u22a2 toSeq \u2191s = s ** apply Subtype.eq ** case a \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 \u22a2 \u2191(toSeq \u2191s) = \u2191s ** funext n ** case a.h \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 n : \u2115 \u22a2 \u2191(toSeq \u2191s) n = \u2191s n ** dsimp [toSeq] ** case a.h \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 n : \u2115 \u22a2 Computation.get (get? (\u2191s) n) = \u2191s n ** apply get_eq_of_mem ** case a.h.a \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 n : \u2115 \u22a2 \u2191s n \u2208 get? (\u2191s) n ** rw [get?_ofSeq] ** case a.h.a \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 n : \u2115 \u22a2 \u2191s n \u2208 Computation.pure (Seq.get? s n) ** apply ret_mem ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.lintegral_iInf_ae ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e h_meas : \u2200 (n : \u2115), Measurable (f n) h_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n h_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4 fn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc fn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc h_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a n : \u2115 a : \u03b1 h : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a \u22a2 f n a \u2264 f 0 a ** induction' n with n ih ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e h_meas : \u2200 (n : \u2115), Measurable (f n) h_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n h_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4 fn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc fn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc h_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a a : \u03b1 h : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a \u22a2 f Nat.zero a \u2264 f 0 a ** exact le_rfl ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e h_meas : \u2200 (n : \u2115), Measurable (f n) h_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n h_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4 fn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc fn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc h_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a a : \u03b1 h : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a n : \u2115 ih : f n a \u2264 f 0 a \u22a2 f (Nat.succ n) a \u2264 f 0 a ** exact le_trans (h n) ih ** Qed", + "informal": "" + }, + { + "formal": "CliffordAlgebra.forall_mul_self_eq_iff ** R : Type u_1 inst\u271d\u2076 : CommRing R M : Type u_2 inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M Q : QuadraticForm R M n : \u2115 A\u271d : Type u_3 inst\u271d\u00b3 : Semiring A\u271d inst\u271d\u00b2 : Algebra R A\u271d A : Type u_4 inst\u271d\u00b9 : Ring A inst\u271d : Algebra R A h2 : IsUnit 2 f : M \u2192\u2097[R] A \u22a2 (\u2200 (x : M), \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)) \u2194 LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R A) f) f + LinearMap.comp (LinearMap.compl\u2082 (LinearMap.flip (LinearMap.mul R A)) f) f = LinearMap.compr\u2082 (\u2191BilinForm.toLin (QuadraticForm.polarBilin Q)) (Algebra.linearMap R A) ** simp_rw [FunLike.ext_iff] ** R : Type u_1 inst\u271d\u2076 : CommRing R M : Type u_2 inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M Q : QuadraticForm R M n : \u2115 A\u271d : Type u_3 inst\u271d\u00b3 : Semiring A\u271d inst\u271d\u00b2 : Algebra R A\u271d A : Type u_4 inst\u271d\u00b9 : Ring A inst\u271d : Algebra R A h2 : IsUnit 2 f : M \u2192\u2097[R] A \u22a2 (\u2200 (x : M), \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)) \u2194 \u2200 (x x_1 : M), \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R A) f) f + LinearMap.comp (LinearMap.compl\u2082 (LinearMap.flip (LinearMap.mul R A)) f) f) x) x_1 = \u2191(\u2191(LinearMap.compr\u2082 (\u2191BilinForm.toLin (QuadraticForm.polarBilin Q)) (Algebra.linearMap R A)) x) x_1 ** refine \u27e8mul_add_swap_eq_polar_of_forall_mul_self_eq _, fun h x => ?_\u27e9 ** R : Type u_1 inst\u271d\u2076 : CommRing R M : Type u_2 inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M Q : QuadraticForm R M n : \u2115 A\u271d : Type u_3 inst\u271d\u00b3 : Semiring A\u271d inst\u271d\u00b2 : Algebra R A\u271d A : Type u_4 inst\u271d\u00b9 : Ring A inst\u271d : Algebra R A h2 : IsUnit 2 f : M \u2192\u2097[R] A h : \u2200 (x x_1 : M), \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R A) f) f + LinearMap.comp (LinearMap.compl\u2082 (LinearMap.flip (LinearMap.mul R A)) f) f) x) x_1 = \u2191(\u2191(LinearMap.compr\u2082 (\u2191BilinForm.toLin (QuadraticForm.polarBilin Q)) (Algebra.linearMap R A)) x) x_1 x : M \u22a2 \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x) ** change \u2200 x y : M, f x * f y + f y * f x = algebraMap R A (QuadraticForm.polar Q x y) at h ** R : Type u_1 inst\u271d\u2076 : CommRing R M : Type u_2 inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M Q : QuadraticForm R M n : \u2115 A\u271d : Type u_3 inst\u271d\u00b3 : Semiring A\u271d inst\u271d\u00b2 : Algebra R A\u271d A : Type u_4 inst\u271d\u00b9 : Ring A inst\u271d : Algebra R A h2 : IsUnit 2 f : M \u2192\u2097[R] A x : M h : \u2200 (x y : M), \u2191f x * \u2191f y + \u2191f y * \u2191f x = \u2191(algebraMap R A) (QuadraticForm.polar (\u2191Q) x y) \u22a2 \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x) ** apply h2.mul_left_cancel ** R : Type u_1 inst\u271d\u2076 : CommRing R M : Type u_2 inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M Q : QuadraticForm R M n : \u2115 A\u271d : Type u_3 inst\u271d\u00b3 : Semiring A\u271d inst\u271d\u00b2 : Algebra R A\u271d A : Type u_4 inst\u271d\u00b9 : Ring A inst\u271d : Algebra R A h2 : IsUnit 2 f : M \u2192\u2097[R] A x : M h : \u2200 (x y : M), \u2191f x * \u2191f y + \u2191f y * \u2191f x = \u2191(algebraMap R A) (QuadraticForm.polar (\u2191Q) x y) \u22a2 2 * (\u2191f x * \u2191f x) = 2 * \u2191(algebraMap R A) (\u2191Q x) ** rw [two_mul, two_mul, h x x, QuadraticForm.polar_self, two_mul, map_add] ** Qed", + "informal": "" + }, + { + "formal": "Ideal.map_isPrime_of_surjective ** R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I \u22a2 IsPrime (map f I) ** refine' \u27e8fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _\u27e9 ** case refine'_1 R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I h : map f I = \u22a4 \u22a2 \u22a4 \u2264 I ** replace h := congr_arg (comap f) h ** case refine'_1 R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I h : comap f (map f I) = comap f \u22a4 \u22a2 \u22a4 \u2264 I ** rw [comap_map_of_surjective _ hf, comap_top] at h ** case refine'_1 R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I h : I \u2294 comap f \u22a5 = \u22a4 \u22a2 \u22a4 \u2264 I ** exact h \u25b8 sup_le (le_of_eq rfl) hk ** case refine'_2 R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I x y : S \u22a2 x * y \u2208 map f I \u2192 x \u2208 map f I \u2228 y \u2208 map f I ** refine' fun hxy => (hf x).recOn fun a ha => (hf y).recOn fun b hb => _ ** case refine'_2 R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I x y : S hxy : x * y \u2208 map f I a : R ha : \u2191f a = x b : R hb : \u2191f b = y \u22a2 x \u2208 map f I \u2228 y \u2208 map f I ** rw [\u2190 ha, \u2190 hb, \u2190 _root_.map_mul f, mem_map_iff_of_surjective _ hf] at hxy ** case refine'_2 R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I x y : S a : R ha : \u2191f a = x b : R hxy : \u2203 x, x \u2208 I \u2227 \u2191f x = \u2191f (a * b) hb : \u2191f b = y \u22a2 x \u2208 map f I \u2228 y \u2208 map f I ** rcases hxy with \u27e8c, hc, hc'\u27e9 ** case refine'_2.intro.intro R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I x y : S a : R ha : \u2191f a = x b : R hb : \u2191f b = y c : R hc : c \u2208 I hc' : \u2191f c = \u2191f (a * b) \u22a2 x \u2208 map f I \u2228 y \u2208 map f I ** rw [\u2190 sub_eq_zero, \u2190 map_sub] at hc' ** case refine'_2.intro.intro R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I x y : S a : R ha : \u2191f a = x b : R hb : \u2191f b = y c : R hc : c \u2208 I hc'\u271d : \u2191f c = \u2191f (a * b) hc' : \u2191f (c - a * b) = 0 \u22a2 x \u2208 map f I \u2228 y \u2208 map f I ** 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 ** case refine'_2.intro.intro R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I x y : S a : R ha : \u2191f a = x b : R hb : \u2191f b = y c : R hc : c \u2208 I hc'\u271d : \u2191f c = \u2191f (a * b) hc' : \u2191f (c - a * b) = 0 this : a * b \u2208 I \u22a2 x \u2208 map f I \u2228 y \u2208 map f I ** 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 ** R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I x y : S a : R ha : \u2191f a = x b : R hb : \u2191f b = y c : R hc : c \u2208 I hc'\u271d : \u2191f c = \u2191f (a * b) hc' : \u2191f (c - a * b) = 0 \u22a2 a * b \u2208 I ** convert I.sub_mem hc (hk (hc' : c - a * b \u2208 RingHom.ker f)) using 1 ** case h.e'_4 R : Type u_1 S : Type u_2 F : Type u_3 inst\u271d\u00b9 : Ring R inst\u271d : Ring S rc : RingHomClass F R S f : F hf : Function.Surjective \u2191f I : Ideal R H : IsPrime I hk : RingHom.ker f \u2264 I x y : S a : R ha : \u2191f a = x b : R hb : \u2191f b = y c : R hc : c \u2208 I hc'\u271d : \u2191f c = \u2191f (a * b) hc' : \u2191f (c - a * b) = 0 \u22a2 a * b = c - (c - a * b) ** abel ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.kernelSubobject_factors_iff ** C : Type u inst\u271d\u00b2 : Category.{v, u} C X Y Z : C inst\u271d\u00b9 : HasZeroMorphisms C f : X \u27f6 Y inst\u271d : HasKernel f W : C h : W \u27f6 X w : Factors (kernelSubobject f) h \u22a2 h \u226b f = 0 ** rw [\u2190 Subobject.factorThru_arrow _ _ w, Category.assoc, kernelSubobject_arrow_comp,\n comp_zero] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SimpleFunc.piecewise_compl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s\u1d9c f g : \u03b1 \u2192\u209b \u03b2 \u22a2 \u2191(piecewise s\u1d9c hs f g) = \u2191(piecewise s (_ : MeasurableSet s) g f) ** simp [hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s\u1d9c f g : \u03b1 \u2192\u209b \u03b2 \u22a2 Set.piecewise s\u1d9c \u2191f \u2191g = Set.piecewise s \u2191g \u2191f ** convert Set.piecewise_compl s f g ** Qed", + "informal": "" + }, + { + "formal": "Asymptotics.isLittleO_sum_range_of_tendsto_zero ** \u03b1 : Type u_1 inst\u271d : NormedAddCommGroup \u03b1 f : \u2115 \u2192 \u03b1 h : Tendsto f atTop (\ud835\udcdd 0) \u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n ** have := ((isLittleO_one_iff \u211d).2 h).sum_range fun i => zero_le_one ** \u03b1 : Type u_1 inst\u271d : NormedAddCommGroup \u03b1 f : \u2115 \u2192 \u03b1 h : Tendsto f atTop (\ud835\udcdd 0) this : 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 \u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n ** simp only [sum_const, card_range, Nat.smul_one_eq_coe] at this ** \u03b1 : Type u_1 inst\u271d : NormedAddCommGroup \u03b1 f : \u2115 \u2192 \u03b1 h : Tendsto f atTop (\ud835\udcdd 0) this : Tendsto (fun n => \u2191n) atTop atTop \u2192 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n \u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n ** exact this tendsto_nat_cast_atTop_atTop ** Qed", + "informal": "" + }, + { + "formal": "Complex.log_im_le_pi ** x : \u2102 \u22a2 (log x).im \u2264 \u03c0 ** simp only [log_im, arg_le_pi] ** Qed", + "informal": "" + }, + { + "formal": "List.card_cons_of_mem ** \u03b1 : Type u_1 \u03b2 : Sort ?u.12983 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 a : \u03b1 as : List \u03b1 h : a \u2208 as \u22a2 card (a :: as) = card as ** simp [card, h] ** Qed", + "informal": "" + }, + { + "formal": "PolynomialModule.comp_smul ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : AddCommGroup M inst\u271d\u00b9\u2070 : Module R M I : Ideal R S : Type u_3 inst\u271d\u2079 : CommSemiring S inst\u271d\u2078 : Algebra S R inst\u271d\u2077 : Module S M inst\u271d\u2076 : IsScalarTower S R M R' : Type u_4 M' : Type u_5 inst\u271d\u2075 : CommRing R' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R' M' inst\u271d\u00b2 : Algebra R R' inst\u271d\u00b9 : Module R M' inst\u271d : IsScalarTower R R' M' p p' : R[X] q : PolynomialModule R M \u22a2 \u2191(comp p) (p' \u2022 q) = Polynomial.comp p' p \u2022 \u2191(comp p) q ** rw [comp_apply, map_smul, eval_smul, Polynomial.comp, Polynomial.eval_map, comp_apply] ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : AddCommGroup M inst\u271d\u00b9\u2070 : Module R M I : Ideal R S : Type u_3 inst\u271d\u2079 : CommSemiring S inst\u271d\u2078 : Algebra S R inst\u271d\u2077 : Module S M inst\u271d\u2076 : IsScalarTower S R M R' : Type u_4 M' : Type u_5 inst\u271d\u2075 : CommRing R' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R' M' inst\u271d\u00b2 : Algebra R R' inst\u271d\u00b9 : Module R M' inst\u271d : IsScalarTower R R' M' p p' : R[X] q : PolynomialModule R M \u22a2 eval\u2082 (algebraMap R R[X]) p p' \u2022 \u2191(eval p) (\u2191(map R[X] (lsingle R 0)) q) = eval\u2082 C p p' \u2022 \u2191(eval p) (\u2191(map R[X] (lsingle R 0)) q) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "mul_le_of_one_le_left ** \u03b1 : Type u \u03b2 : Type u_1 inst\u271d : OrderedRing \u03b1 a b c d : \u03b1 hb : b \u2264 0 h : 1 \u2264 a \u22a2 a * b \u2264 b ** simpa only [one_mul] using mul_le_mul_of_nonpos_right h hb ** Qed", + "informal": "" + }, + { + "formal": "Units.inv_mul_cancel_right ** \u03b1 : Type u inst\u271d : Monoid \u03b1 a\u271d b\u271d c u : \u03b1\u02e3 a : \u03b1 b : \u03b1\u02e3 \u22a2 a * \u2191b\u207b\u00b9 * \u2191b = a ** rw [mul_assoc, inv_mul, mul_one] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.eval\u2082_algebraMap_X ** R\u271d : Type u S : Type v T : Type w A\u271d : Type z A' : Type u_1 B' : Type u_2 a b : R\u271d n : \u2115 inst\u271d\u2077 : CommSemiring A' inst\u271d\u2076 : Semiring B' inst\u271d\u2075 : CommSemiring R\u271d p\u271d q r : R\u271d[X] inst\u271d\u2074 : Semiring A\u271d inst\u271d\u00b3 : Algebra R\u271d A\u271d R : Type u_3 A : Type u_4 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : Semiring A inst\u271d : Algebra R A p : R[X] f : R[X] \u2192\u2090[R] A \u22a2 eval\u2082 (algebraMap R A) (\u2191f X) p = \u2191f p ** conv_rhs => rw [\u2190 Polynomial.sum_C_mul_X_pow_eq p] ** R\u271d : Type u S : Type v T : Type w A\u271d : Type z A' : Type u_1 B' : Type u_2 a b : R\u271d n : \u2115 inst\u271d\u2077 : CommSemiring A' inst\u271d\u2076 : Semiring B' inst\u271d\u2075 : CommSemiring R\u271d p\u271d q r : R\u271d[X] inst\u271d\u2074 : Semiring A\u271d inst\u271d\u00b3 : Algebra R\u271d A\u271d R : Type u_3 A : Type u_4 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : Semiring A inst\u271d : Algebra R A p : R[X] f : R[X] \u2192\u2090[R] A \u22a2 eval\u2082 (algebraMap R A) (\u2191f X) p = \u2191f (sum p fun n a => \u2191C a * X ^ n) ** simp only [eval\u2082_eq_sum, sum_def] ** R\u271d : Type u S : Type v T : Type w A\u271d : Type z A' : Type u_1 B' : Type u_2 a b : R\u271d n : \u2115 inst\u271d\u2077 : CommSemiring A' inst\u271d\u2076 : Semiring B' inst\u271d\u2075 : CommSemiring R\u271d p\u271d q r : R\u271d[X] inst\u271d\u2074 : Semiring A\u271d inst\u271d\u00b3 : Algebra R\u271d A\u271d R : Type u_3 A : Type u_4 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : Semiring A inst\u271d : Algebra R A p : R[X] f : R[X] \u2192\u2090[R] A \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) ** simp only [f.map_sum, f.map_mul, f.map_pow, eq_intCast, map_intCast] ** R\u271d : Type u S : Type v T : Type w A\u271d : Type z A' : Type u_1 B' : Type u_2 a b : R\u271d n : \u2115 inst\u271d\u2077 : CommSemiring A' inst\u271d\u2076 : Semiring B' inst\u271d\u2075 : CommSemiring R\u271d p\u271d q r : R\u271d[X] inst\u271d\u2074 : Semiring A\u271d inst\u271d\u00b3 : Algebra R\u271d A\u271d R : Type u_3 A : Type u_4 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : Semiring A inst\u271d : Algebra R A p : R[X] f : R[X] \u2192\u2090[R] A \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 ** simp [Polynomial.C_eq_algebraMap] ** Qed", + "informal": "" + }, + { + "formal": "Subring.mem_map ** R : Type u S : Type v T : Type w inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : Ring S inst\u271d : Ring T s\u271d : Subring R f : R \u2192+* S s : Subring R y : S \u22a2 (\u2203 x x_1, \u2191f x = y) \u2194 \u2203 x, x \u2208 s \u2227 \u2191f x = y ** simp ** Qed", + "informal": "" + }, + { + "formal": "TensorProduct.directSum_symm_lof_tmul ** R : Type u inst\u271d\u00b9\u2070 : CommRing R \u03b9\u2081 : Type v\u2081 \u03b9\u2082 : Type v\u2082 inst\u271d\u2079 : DecidableEq \u03b9\u2081 inst\u271d\u2078 : DecidableEq \u03b9\u2082 M\u2081 : \u03b9\u2081 \u2192 Type w\u2081 M\u2081' : Type w\u2081' M\u2082 : \u03b9\u2082 \u2192 Type w\u2082 M\u2082' : Type w\u2082' inst\u271d\u2077 : (i\u2081 : \u03b9\u2081) \u2192 AddCommGroup (M\u2081 i\u2081) inst\u271d\u2076 : AddCommGroup M\u2081' inst\u271d\u2075 : (i\u2082 : \u03b9\u2082) \u2192 AddCommGroup (M\u2082 i\u2082) inst\u271d\u2074 : AddCommGroup M\u2082' inst\u271d\u00b3 : (i\u2081 : \u03b9\u2081) \u2192 Module R (M\u2081 i\u2081) inst\u271d\u00b2 : Module R M\u2081' inst\u271d\u00b9 : (i\u2082 : \u03b9\u2082) \u2192 Module R (M\u2082 i\u2082) inst\u271d : Module R M\u2082' i\u2081 : \u03b9\u2081 m\u2081 : M\u2081 i\u2081 i\u2082 : \u03b9\u2082 m\u2082 : M\u2082 i\u2082 \u22a2 \u2191(LinearEquiv.symm (TensorProduct.directSum R M\u2081 M\u2082)) (\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)) = \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 ** rw [LinearEquiv.symm_apply_eq, directSum_lof_tmul_lof] ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.condCount_add_compl_eq ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u\u271d u t : Set \u03a9 hs : Set.Finite s \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 ** 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] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u\u271d u t : Set \u03a9 hs : Set.Finite s this : \u2191\u2191(condCount s) t = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c) \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 ** rw [this] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u\u271d u t : Set \u03a9 hs : Set.Finite s this : \u2191\u2191(condCount s) t = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c) \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 \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c) ** simp [condCount_inter_self hs] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u\u271d u t : Set \u03a9 hs : Set.Finite s \u22a2 \u2191\u2191(condCount s) t = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Function.Commute.iterate_pos_eq_iff_map_eq ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 f g : \u03b1 \u2192 \u03b1 h : Commute f g hf : Monotone f hg : StrictMono g x : \u03b1 n : \u2115 hn : 0 < n \u22a2 f^[n] x = g^[n] x \u2194 f x = g x ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Fin.castIso_to_equiv ** n m : \u2115 h : n = m \u22a2 (castIso h).toEquiv = Equiv.cast (_ : Fin n = Fin m) ** subst h ** n : \u2115 \u22a2 (castIso (_ : n = n)).toEquiv = Equiv.cast (_ : Fin n = Fin n) ** simp ** Qed", + "informal": "" + }, + { + "formal": "LieSubmodule.bot_lie ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \u22a2 \u2045\u22a5, N\u2046 = \u22a5 ** suffices \u2045(\u22a5 : LieIdeal R L), N\u2046 \u2264 \u22a5 by exact le_bot_iff.mp this ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \u22a2 \u2045\u22a5, N\u2046 \u2264 \u22a5 ** rw [lieIdeal_oper_eq_span, lieSpan_le] ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \u22a2 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2286 \u2191\u22a5 ** rintro m \u27e8\u27e8x, hx\u27e9, n, hn\u27e9 ** case intro.mk.intro R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 m : M x : L hx : x \u2208 \u22a5 n : { x // x \u2208 N } hn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m \u22a2 m \u2208 \u2191\u22a5 ** rw [\u2190 hn] ** case intro.mk.intro R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 m : M x : L hx : x \u2208 \u22a5 n : { x // x \u2208 N } hn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m \u22a2 \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 \u2208 \u2191\u22a5 ** change x \u2208 (\u22a5 : LieIdeal R L) at hx ** case intro.mk.intro R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 m : M x : L n : { x // x \u2208 N } hx : x \u2208 \u22a5 hn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m \u22a2 \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 \u2208 \u2191\u22a5 ** rw [mem_bot] at hx ** case intro.mk.intro R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 m : M x : L n : { x // x \u2208 N } hx\u271d : x \u2208 \u22a5 hx : x = 0 hn : \u2045\u2191{ val := x, property := hx\u271d }, \u2191n\u2046 = m \u22a2 \u2045\u2191{ val := x, property := hx\u271d }, \u2191n\u2046 \u2208 \u2191\u22a5 ** simp [hx] ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 this : \u2045\u22a5, N\u2046 \u2264 \u22a5 \u22a2 \u2045\u22a5, N\u2046 = \u22a5 ** exact le_bot_iff.mp this ** Qed", + "informal": "" + }, + { + "formal": "QuadraticForm.associated_left_inverse ** S : Type u_1 T : Type u_2 R : Type u_3 M : Type u_4 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : Algebra S R inst\u271d : Invertible 2 B\u2081 : BilinForm R M Q : QuadraticForm R M h : BilinForm.IsSymm B\u2081 x y : M \u22a2 bilin (\u2191(associatedHom S) (toQuadraticForm B\u2081)) x y = bilin B\u2081 x y ** rw [associated_toQuadraticForm, h.eq x y, \u2190 two_mul, \u2190 mul_assoc, invOf_mul_self,\n one_mul] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.ndinter_cons_of_not_mem ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s\u271d : Multiset \u03b1 a : \u03b1 s t : Multiset \u03b1 h : \u00aca \u2208 t \u22a2 ndinter (a ::\u2098 s) t = ndinter s t ** simp [ndinter, h] ** Qed", + "informal": "" + }, + { + "formal": "NormOneClass.nontrivial ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b1 : Type u_5 inst\u271d\u00b2 : SeminormedAddCommGroup \u03b1 inst\u271d\u00b9 : One \u03b1 inst\u271d : NormOneClass \u03b1 \u22a2 \u20160\u2016 \u2260 \u20161\u2016 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_range_succ ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 f : \u2115 \u2192 \u03b2 n : \u2115 \u22a2 \u220f x in range (n + 1), f x = (\u220f x in range n, f x) * f n ** simp only [mul_comm, prod_range_succ_comm] ** Qed", + "informal": "" + }, + { + "formal": "one_le_div_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedField \u03b1 a b c d : \u03b1 n : \u2124 \u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b ** rcases lt_trichotomy b 0 with (hb | rfl | hb) ** case inl \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedField \u03b1 a b c d : \u03b1 n : \u2124 hb : b < 0 \u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b ** simp [hb, hb.not_lt, one_le_div_of_neg] ** case inr.inl \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedField \u03b1 a c d : \u03b1 n : \u2124 \u22a2 1 \u2264 a / 0 \u2194 0 < 0 \u2227 0 \u2264 a \u2228 0 < 0 \u2227 a \u2264 0 ** simp [lt_irrefl, zero_lt_one.not_le, zero_lt_one] ** case inr.inr \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedField \u03b1 a b c d : \u03b1 n : \u2124 hb : 0 < b \u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b ** simp [hb, hb.not_lt, one_le_div] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.vecMulVec_cons ** \u03b1 : Type u o n m : \u2115 m' : Type u\u2098 n' : Type u\u2099 o' : Type u\u2092 a b : \u2115 inst\u271d : NonUnitalNonAssocSemiring \u03b1 v : m' \u2192 \u03b1 x : \u03b1 w : Fin n \u2192 \u03b1 \u22a2 vecMulVec v (vecCons x w) = fun i => v i \u2022 vecCons x w ** ext i j ** case a.h \u03b1 : Type u o n m : \u2115 m' : Type u\u2098 n' : Type u\u2099 o' : Type u\u2092 a b : \u2115 inst\u271d : NonUnitalNonAssocSemiring \u03b1 v : m' \u2192 \u03b1 x : \u03b1 w : Fin n \u2192 \u03b1 i : m' j : Fin (Nat.succ n) \u22a2 vecMulVec v (vecCons x w) i j = (v i \u2022 vecCons x w) j ** rw [vecMulVec_apply, Pi.smul_apply, smul_eq_mul] ** Qed", + "informal": "" + }, + { + "formal": "Rat.num_lt_succ_floor_mul_den ** q : \u211a \u22a2 q.num < (\u230aq\u230b + 1) * \u2191q.den ** suffices (q.num : \u211a) < (\u230aq\u230b + 1) * q.den by exact_mod_cast this ** q : \u211a \u22a2 \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den ** 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] ** q : \u211a \u22a2 \u2191q.num < (q - fract q + 1) * \u2191q.den ** 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] ** q : \u211a \u22a2 \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den ** suffices 0 < (1 - fract q) * q.den by\n rw [\u2190 sub_lt_iff_lt_add']\n simpa ** q : \u211a \u22a2 0 < (1 - fract q) * \u2191q.den ** 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] ** q : \u211a this : 0 < 1 - fract q \u22a2 0 < (1 - fract q) * \u2191q.den ** exact mul_pos this (by exact_mod_cast q.pos) ** q : \u211a this : \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den \u22a2 q.num < (\u230aq\u230b + 1) * \u2191q.den ** exact_mod_cast this ** q : \u211a this : \u2191q.num < (q - fract q + 1) * \u2191q.den \u22a2 \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den ** have : (\u230aq\u230b : \u211a) = q - fract q := eq_sub_of_add_eq <| floor_add_fract q ** q : \u211a this\u271d : \u2191q.num < (q - fract q + 1) * \u2191q.den this : \u2191\u230aq\u230b = q - fract q \u22a2 \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den ** rwa [this] ** q : \u211a this : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den \u22a2 \u2191q.num < (q - fract q + 1) * \u2191q.den ** 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 ** q : \u211a this\u271d : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den this : (q - fract q + 1) * \u2191q.den = \u2191q.num + (1 - fract q) * \u2191q.den \u22a2 \u2191q.num < (q - fract q + 1) * \u2191q.den ** rwa [this] ** q : \u211a this : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den \u22a2 (q - fract q + 1) * \u2191q.den = \u2191q.num + (1 - fract q) * \u2191q.den ** 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 ** q : \u211a this : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den \u22a2 (q - fract q + 1) * \u2191q.den = (q + (1 - fract q)) * \u2191q.den ** ring ** q : \u211a this : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den \u22a2 (q + (1 - fract q)) * \u2191q.den = q * \u2191q.den + (1 - fract q) * \u2191q.den ** rw [add_mul] ** q : \u211a this : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den \u22a2 q * \u2191q.den + (1 - fract q) * \u2191q.den = \u2191q.num + (1 - fract q) * \u2191q.den ** simp ** q : \u211a this : 0 < (1 - fract q) * \u2191q.den \u22a2 \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den ** rw [\u2190 sub_lt_iff_lt_add'] ** q : \u211a this : 0 < (1 - fract q) * \u2191q.den \u22a2 \u2191q.num - \u2191q.num < (1 - fract q) * \u2191q.den ** simpa ** q : \u211a \u22a2 0 < 1 - fract q ** have : fract q < 1 := fract_lt_one q ** q : \u211a this : fract q < 1 \u22a2 0 < 1 - fract q ** have : 0 + fract q < 1 := by simp [this] ** q : \u211a this\u271d : fract q < 1 this : 0 + fract q < 1 \u22a2 0 < 1 - fract q ** rwa [lt_sub_iff_add_lt] ** q : \u211a this : fract q < 1 \u22a2 0 + fract q < 1 ** simp [this] ** q : \u211a this : 0 < 1 - fract q \u22a2 0 < \u2191q.den ** exact_mod_cast q.pos ** Qed", + "informal": "" + }, + { + "formal": "Filter.map\u2082_pure_left ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 m : \u03b1 \u2192 \u03b2 \u2192 \u03b3 f f\u2081 f\u2082 : Filter \u03b1 g g\u2081 g\u2082 : Filter \u03b2 h\u271d h\u2081 h\u2082 : Filter \u03b3 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 u\u271d : Set \u03b3 v : Set \u03b4 a : \u03b1 b : \u03b2 c : \u03b3 u : Set \u03b3 h : u \u2208 map (fun b => m a b) g \u22a2 image2 m {a} ((fun b => m a b) \u207b\u00b9' u) \u2286 u ** rw [image2_singleton_left, image_subset_iff] ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.prod_eq_zero ** \u03b1 \u03b2 : Type u \u03b9 : Type u_1 f : \u03b9 \u2192 Cardinal.{u} \u22a2 prod f = 0 \u2194 \u2203 i, f i = 0 ** lift f to \u03b9 \u2192 Type u using fun _ => trivial ** case intro \u03b1 \u03b2 : Type u \u03b9 : Type u_1 f : \u03b9 \u2192 Type u \u22a2 (prod fun i => #(f i)) = 0 \u2194 \u2203 i, (fun i => #(f i)) i = 0 ** simp only [mk_eq_zero_iff, \u2190 mk_pi, isEmpty_pi] ** Qed", + "informal": "" + }, + { + "formal": "parallelepiped_comp_equiv ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 \u22a2 parallelepiped (v \u2218 \u2191e) = parallelepiped v ** simp only [parallelepiped] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 \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 ** let K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a : \u03b9' => \u211d) e ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this : Icc 0 1 = \u2191K '' Icc 0 1 \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 ** rw [this, \u2190 image_comp] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this : Icc 0 1 = \u2191K '' Icc 0 1 \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 ** congr 1 with x ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this : Icc 0 1 = \u2191K '' Icc 0 1 x : E \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 ** have := fun z : \u03b9' \u2192 \u211d => e.symm.sum_comp fun i => z i \u2022 v (e i) ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this\u271d : Icc 0 1 = \u2191K '' Icc 0 1 x : E this : \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) \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 ** simp_rw [Equiv.apply_symm_apply] at this ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this\u271d : Icc 0 1 = \u2191K '' Icc 0 1 x : E this : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 x : \u03b9, z (\u2191e.symm x) \u2022 v x = \u2211 x : \u03b9', z x \u2022 v (\u2191e x) \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 ** simp_rw [Function.comp_apply, mem_image, mem_Icc, Equiv.piCongrLeft'_apply, this] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e \u22a2 Icc 0 1 = \u2191K '' Icc 0 1 ** rw [\u2190 Equiv.preimage_eq_iff_eq_image] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e \u22a2 \u2191K \u207b\u00b9' Icc 0 1 = Icc 0 1 ** ext x ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e x : \u03b9' \u2192 \u211d \u22a2 x \u2208 \u2191K \u207b\u00b9' Icc 0 1 \u2194 x \u2208 Icc 0 1 ** simp only [mem_preimage, mem_Icc, Pi.le_def, Pi.zero_apply, Equiv.piCongrLeft'_apply,\n Pi.one_apply] ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e x : \u03b9' \u2192 \u211d \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 ** 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 ** case h.refine'_1 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e x : \u03b9' \u2192 \u211d h : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1 i : \u03b9' \u22a2 0 \u2264 x i ** simpa only [Equiv.symm_apply_apply] using h.1 (e i) ** case h.refine'_2 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e x : \u03b9' \u2192 \u211d h : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1 i : \u03b9' \u22a2 x i \u2264 1 ** simpa only [Equiv.symm_apply_apply] using h.2 (e i) ** Qed", + "informal": "" + }, + { + "formal": "nhdsWithin_pi_eq ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b1\u271d \u03b9 : Type u_5 \u03b1 : \u03b9 \u2192 Type u_6 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i) I : Set \u03b9 hI : Set.Finite I s : (i : \u03b9) \u2192 Set (\u03b1 i) x : (i : \u03b9) \u2192 \u03b1 i \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)) ** simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, \u2190 iInf_principal_finite hI, comap_inf,\n comap_principal, eval] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b1\u271d \u03b9 : Type u_5 \u03b1 : \u03b9 \u2192 Type u_6 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i) I : Set \u03b9 hI : Set.Finite I s : (i : \u03b9) \u2192 Set (\u03b1 i) x : (i : \u03b9) \u2192 \u03b1 i \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) = (\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 \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i)) ** rw [iInf_split _ fun i => i \u2208 I, inf_right_comm] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b1\u271d \u03b9 : Type u_5 \u03b1 : \u03b9 \u2192 Type u_6 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i) I : Set \u03b9 hI : Set.Finite I s : (i : \u03b9) \u2192 Set (\u03b1 i) x : (i : \u03b9) \u2192 \u03b1 i \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 \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun f => f i) (\ud835\udcdd (x i)) = (\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 \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i)) ** simp only [iInf_inf_eq] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.tendsto_sub ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e h : \u22a4 \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4 \u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u22a4, \u22a4)) (\ud835\udcdd (\u22a4 - \u22a4)) ** simp only at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b\u271d c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e b : \u211d\u22650 x\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4 \u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u22a4, \u2191b)) (\ud835\udcdd (\u22a4 - \u2191b)) ** rw [top_sub_coe, tendsto_nhds_top_iff_nnreal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b\u271d c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e b : \u211d\u22650 x\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4 \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 ** 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 => ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b\u271d c d : \u211d\u22650\u221e r p q : \u211d\u22650 x\u271d\u00b9 y\u271d z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e b : \u211d\u22650 x\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4 x : \u211d\u22650 y : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e hy : \u2191(b + 1 + x) < y.1 \u2227 y.2 \u2264 \u2191(b + 1) \u22a2 \u2191x < y.1 - y.2 ** rw [lt_tsub_iff_left] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b\u271d c d : \u211d\u22650\u221e r p q : \u211d\u22650 x\u271d\u00b9 y\u271d z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e b : \u211d\u22650 x\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4 x : \u211d\u22650 y : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e hy : \u2191(b + 1 + x) < y.1 \u2227 y.2 \u2264 \u2191(b + 1) \u22a2 y.2 + \u2191x < y.1 ** calc y.2 + x \u2264 \u2191(b + 1) + x := add_le_add_right hy.2 _\n_ < y.1 := hy.1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e a : \u211d\u22650 x\u271d : \u2191a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4 \u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u2191a, \u22a4)) (\ud835\udcdd (\u2191a - \u22a4)) ** rw [sub_top] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e a : \u211d\u22650 x\u271d : \u2191a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4 \u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u2191a, \u22a4)) (\ud835\udcdd 0) ** refine (tendsto_pure.2 ?_).mono_right (pure_le_nhds _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e a : \u211d\u22650 x\u271d : \u2191a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4 \u22a2 \u2200\u1da0 (x : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u2191a, \u22a4), x.1 - x.2 = 0 ** 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 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d b\u271d c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e a b : \u211d\u22650 x\u271d : \u2191a \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4 \u22a2 Tendsto (fun a => a.1 - a.2) (\ud835\udcdd (a, b)) (\ud835\udcdd (a - b)) ** exact continuous_sub.tendsto (a, b) ** Qed", + "informal": "" + }, + { + "formal": "Option.orElse_none ** \u03b1 : Type u_1 x : Option \u03b1 \u22a2 (HOrElse.hOrElse x fun x => none) = x ** cases x <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.sum_measure_preimage_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 s : Finset \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (y : \u03b2), y \u2208 s \u2192 MeasurableSet (f \u207b\u00b9' {y}) \u22a2 \u2211 b in s, \u2191\u2191\u03bc (f \u207b\u00b9' {b}) = \u2191\u2191\u03bc (f \u207b\u00b9' \u2191s) ** simp only [\u2190 measure_biUnion_finset (pairwiseDisjoint_fiber f s) hf,\n Finset.set_biUnion_preimage_singleton] ** Qed", + "informal": "" + }, + { + "formal": "Int.natAbs_bit1_nonneg ** a : \u2124 h : 0 \u2264 a \u22a2 natAbs (bit0 a + 1) = bit0 (natAbs a) + natAbs 1 ** rw [Int.natAbs_add_nonneg (Int.bit0_nonneg h) (le_of_lt Int.zero_lt_one), Int.natAbs_bit0] ** Qed", + "informal": "" + }, + { + "formal": "Pell.pellZd_succ ** a : \u2115 a1 : 1 < a n : \u2115 \u22a2 pellZd a1 (n + 1) = pellZd a1 n * { re := \u2191a, im := 1 } ** simp [Zsqrtd.ext] ** Qed", + "informal": "" + }, + { + "formal": "Bool.ite_eq_true_distrib ** c : Prop inst\u271d : Decidable c a b : Bool \u22a2 ((if c then a else b) = true) = if c then a = true else b = true ** by_cases c <;> simp [*] ** Qed", + "informal": "" + }, + { + "formal": "iSup_emptyset ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b2 \u2192 \u03b1 \u22a2 \u2a06 x \u2208 \u2205, f x = \u22a5 ** simp ** Qed", + "informal": "" + }, + { + "formal": "List.forall\u2082_eq_eq_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop \u22a2 (Forall\u2082 fun x x_1 => x = x_1) = Eq ** funext a b ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a b : List \u03b1 \u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b = (a = b) ** apply propext ** case h.h.a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a b : List \u03b1 \u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b \u2194 a = b ** constructor ** case h.h.a.mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a b : List \u03b1 \u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b \u2192 a = b ** intro h ** case h.h.a.mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a b : List \u03b1 h : Forall\u2082 (fun x x_1 => x = x_1) a b \u22a2 a = b ** induction h ** case h.h.a.mp.cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a b : List \u03b1 a\u271d\u00b2 b\u271d : \u03b1 l\u2081\u271d l\u2082\u271d : List \u03b1 a\u271d\u00b9 : a\u271d\u00b2 = b\u271d a\u271d : Forall\u2082 (fun x x_1 => x = x_1) l\u2081\u271d l\u2082\u271d a_ih\u271d : l\u2081\u271d = l\u2082\u271d \u22a2 a\u271d\u00b2 :: l\u2081\u271d = b\u271d :: l\u2082\u271d ** simp only [*] ** case h.h.a.mp.nil \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a b : List \u03b1 \u22a2 [] = [] ** rfl ** case h.h.a.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a b : List \u03b1 \u22a2 a = b \u2192 Forall\u2082 (fun x x_1 => x = x_1) a b ** rintro rfl ** case h.h.a.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a : List \u03b1 \u22a2 Forall\u2082 (fun x x_1 => x = x_1) a a ** exact forall\u2082_refl _ ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.setToFun_add_left' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' hT'' : DominatedFinMeasAdditive \u03bc T'' C'' h_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s f : \u03b1 \u2192 E \u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' hT'' : DominatedFinMeasAdditive \u03bc T'' C'' h_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f ** simp_rw [setToFun_eq _ hf, L1.setToL1_add_left' hT hT' hT'' h_add] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' hT'' : DominatedFinMeasAdditive \u03bc T'' C'' h_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f ** simp_rw [setToFun_undef _ hf, add_zero] ** Qed", + "informal": "" + }, + { + "formal": "AdjoinRoot.Minpoly.toAdjoin.apply_X ** R : Type u S : Type v K : Type w inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S inst\u271d : Algebra R S x : S \u22a2 \u2191(toAdjoin R x) (\u2191(mk (minpoly R x)) X) = { val := x, property := (_ : x \u2208 adjoin R {x}) } ** simp [toAdjoin] ** Qed", + "informal": "" + }, + { + "formal": "Complex.continuous_circleTransform ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous (circleTransform R z w f) ** apply_rules [Continuous.smul, continuous_const] ** case hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun x => deriv (circleMap z R) x case hg.hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9 case hg.hg.hg E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun x => f (circleMap z R x) ** simp_rw [deriv_circleMap] ** case hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun x => circleMap 0 R x * I case hg.hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9 case hg.hg.hg E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun x => f (circleMap z R x) ** apply_rules [Continuous.mul, continuous_circleMap 0 R, continuous_const] ** case hg.hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9 ** apply continuous_circleMap_inv hw ** case hg.hg.hg E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun x => f (circleMap z R x) ** apply ContinuousOn.comp_continuous hf (continuous_circleMap z R) ** case hg.hg.hg E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 \u2200 (x : \u211d), circleMap z R x \u2208 sphere z R ** exact fun _ => (circleMap_mem_sphere _ hR.le) _ ** Qed", + "informal": "" + }, + { + "formal": "Algebra.adjoin_eq_range ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R inst\u271d\u00b9 : Algebra R S\u2081 inst\u271d : CommSemiring S\u2082 f : \u03c3 \u2192 S\u2081 s : Set S\u2081 \u22a2 Algebra.adjoin R s = AlgHom.range (aeval Subtype.val) ** rw [\u2190 Algebra.adjoin_range_eq_range_aeval, Subtype.range_coe] ** Qed", + "informal": "" + }, + { + "formal": "CommRingCat.pushoutCocone_pt ** R A B : CommRingCat f : R \u27f6 A g : R \u27f6 B \u22a2 CommRingCat ** letI := f.toAlgebra ** R A B : CommRingCat f : R \u27f6 A g : R \u27f6 B this : Algebra \u2191R \u2191A := RingHom.toAlgebra f \u22a2 CommRingCat ** letI := g.toAlgebra ** R A B : CommRingCat f : R \u27f6 A g : R \u27f6 B this\u271d : Algebra \u2191R \u2191A := RingHom.toAlgebra f this : Algebra \u2191R \u2191B := RingHom.toAlgebra g \u22a2 CommRingCat ** exact CommRingCat.of (A \u2297[R] B) ** Qed", + "informal": "" + }, + { + "formal": "List.scanl_cons ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 a : \u03b1 l : List \u03b1 \u22a2 scanl f b (a :: l) = [b] ++ scanl f (f b a) l ** simp only [scanl, eq_self_iff_true, singleton_append, and_self_iff] ** Qed", + "informal": "" + }, + { + "formal": "aux1 ** A : Type u_1 inst\u271d\u00b9 : NonUnitalNonAssocRing A inst\u271d : IsCommJordan A a b c : A \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 = \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 + \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 + (\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 + \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) + (\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 + \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) ** rw [add_lie, add_lie] ** A : Type u_1 inst\u271d\u00b9 : NonUnitalNonAssocRing A inst\u271d : IsCommJordan A a b c : A \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 + \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 + \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 = \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 + \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 + (\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 + \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) + (\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 + \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) ** iterate 15 rw [lie_add] ** A : Type u_1 inst\u271d\u00b9 : NonUnitalNonAssocRing A inst\u271d : IsCommJordan A a b c : A \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 + \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 + (\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 + \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) + (\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 + \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) = \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 + \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 + (\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 + \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) + (\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 + \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) ** rw [lie_add] ** Qed", + "informal": "" + }, + { + "formal": "Std.RBNode.mem_insert_of_mem ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** 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) ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t p : Path \u03b1 e : zoom (cmp v) t Path.root = (nil, p) \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_nil ht e ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t p : Path \u03b1 e : zoom (cmp v) t Path.root = (nil, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** simp [\u2190 mem_toList, h\u2081] at h ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n p : Path \u03b1 e : zoom (cmp v) t Path.root = (nil, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** simp [\u2190 mem_toList, h\u2082] ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n p : Path \u03b1 e : zoom (cmp v) t Path.root = (nil, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d \u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq ** cases h <;> simp [*] ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_node ht e ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** simp [\u2190 mem_toList, h\u2081] at h ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** simp [\u2190 mem_toList, h\u2082] ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d \u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq ** rcases h with h|h|h <;> simp [*] ** case inr.inl \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' = v\u271d \u22a2 (v\u271d \u2208 w\u271d\u00b9 \u2228 v\u271d = v \u2228 v\u271d \u2208 w\u271d) \u2228 cmp v v\u271d = Ordering.eq ** exact .inr (Path.zoom_zoomed\u2081 e) ** Qed", + "informal": "" + }, + { + "formal": "Order.pred_le_iff_eq_or_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : PredOrder \u03b1 a b : \u03b1 \u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b ** by_cases ha : IsMin a ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : PredOrder \u03b1 a b : \u03b1 ha : IsMin a \u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b ** rw [ha.pred_eq, or_iff_right_of_imp ge_of_eq] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : PredOrder \u03b1 a b : \u03b1 ha : \u00acIsMin a \u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b ** rw [\u2190 pred_lt_iff_of_not_isMin ha, le_iff_eq_or_lt, eq_comm] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.linearIndependent_powers_iff_aeval ** R : Type u S : Type u_1 \u03c3 : Type v M : Type w inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M f : M \u2192\u2097[R] M v : M \u22a2 (LinearIndependent R fun n => \u2191(f ^ n) v) \u2194 \u2200 (p : R[X]), \u2191(\u2191(aeval f) p) v = 0 \u2192 p = 0 ** rw [linearIndependent_iff] ** R : Type u S : Type u_1 \u03c3 : Type v M : Type w inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M f : M \u2192\u2097[R] M v : M \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 \u2200 (p : R[X]), \u2191(\u2191(aeval f) p) v = 0 \u2192 p = 0 ** simp only [Finsupp.total_apply, aeval_endomorphism, forall_iff_forall_finsupp, Sum, support,\n coeff, ofFinsupp_eq_zero] ** R : Type u S : Type u_1 \u03c3 : Type v M : Type w inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M f : M \u2192\u2097[R] M v : M \u22a2 (\u2200 (l : \u2115 \u2192\u2080 R), (Finsupp.sum l fun i a => a \u2022 \u2191(f ^ i) v) = 0 \u2192 l = 0) \u2194 \u2200 (q : AddMonoidAlgebra R \u2115), (sum { toFinsupp := q } fun n b => b \u2022 \u2191(f ^ n) v) = 0 \u2192 q = 0 ** exact Iff.rfl ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Paths.lift_spec ** V : Type u\u2081 inst\u271d\u00b9 : Quiver V C : Type u_1 inst\u271d : Category.{u_2, u_1} C \u03c6 : V \u2964q C \u22a2 of \u22d9q (lift \u03c6).toPrefunctor = \u03c6 ** fapply Prefunctor.ext ** case h_obj V : Type u\u2081 inst\u271d\u00b9 : Quiver V C : Type u_1 inst\u271d : Category.{u_2, u_1} C \u03c6 : V \u2964q C \u22a2 \u2200 (X : V), (of \u22d9q (lift \u03c6).toPrefunctor).obj X = \u03c6.obj X ** rintro X ** case h_obj V : Type u\u2081 inst\u271d\u00b9 : Quiver V C : Type u_1 inst\u271d : Category.{u_2, u_1} C \u03c6 : V \u2964q C X : V \u22a2 (of \u22d9q (lift \u03c6).toPrefunctor).obj X = \u03c6.obj X ** rfl ** case h_map V : Type u\u2081 inst\u271d\u00b9 : Quiver V C : Type u_1 inst\u271d : Category.{u_2, u_1} C \u03c6 : V \u2964q C \u22a2 \u2200 (X Y : V) (f : X \u27f6 Y), (of \u22d9q (lift \u03c6).toPrefunctor).map f = Eq.recOn (_ : \u03c6.obj Y = (of \u22d9q (lift \u03c6).toPrefunctor).obj Y) (Eq.recOn (_ : \u03c6.obj X = (of \u22d9q (lift \u03c6).toPrefunctor).obj X) (\u03c6.map f)) ** rintro X Y f ** case h_map V : Type u\u2081 inst\u271d\u00b9 : Quiver V C : Type u_1 inst\u271d : Category.{u_2, u_1} C \u03c6 : V \u2964q C X Y : V f : X \u27f6 Y \u22a2 (of \u22d9q (lift \u03c6).toPrefunctor).map f = Eq.recOn (_ : \u03c6.obj Y = (of \u22d9q (lift \u03c6).toPrefunctor).obj Y) (Eq.recOn (_ : \u03c6.obj X = (of \u22d9q (lift \u03c6).toPrefunctor).obj X) (\u03c6.map f)) ** rcases \u03c6 with \u27e8\u03c6o, \u03c6m\u27e9 ** case h_map.mk V : Type u\u2081 inst\u271d\u00b9 : Quiver V C : Type u_1 inst\u271d : Category.{u_2, u_1} C X Y : V f : X \u27f6 Y \u03c6o : V \u2192 C \u03c6m : {X Y : V} \u2192 (X \u27f6 Y) \u2192 (\u03c6o X \u27f6 \u03c6o Y) \u22a2 (of \u22d9q (lift { obj := \u03c6o, map := \u03c6m }).toPrefunctor).map f = Eq.recOn (_ : { obj := \u03c6o, map := \u03c6m }.obj Y = (of \u22d9q (lift { obj := \u03c6o, map := \u03c6m }).toPrefunctor).obj Y) (Eq.recOn (_ : { obj := \u03c6o, map := \u03c6m }.obj X = (of \u22d9q (lift { obj := \u03c6o, map := \u03c6m }).toPrefunctor).obj X) ({ obj := \u03c6o, map := \u03c6m }.map f)) ** dsimp [lift, Quiver.Hom.toPath] ** case h_map.mk V : Type u\u2081 inst\u271d\u00b9 : Quiver V C : Type u_1 inst\u271d : Category.{u_2, u_1} C X Y : V f : X \u27f6 Y \u03c6o : V \u2192 C \u03c6m : {X Y : V} \u2192 (X \u27f6 Y) \u2192 (\u03c6o X \u27f6 \u03c6o Y) \u22a2 \ud835\udfd9 (\u03c6o X) \u226b \u03c6m f = \u03c6m f ** simp only [Category.id_comp] ** Qed", + "informal": "" + }, + { + "formal": "Nat.ofDigits_digits_append_digits ** n\u271d b m n : \u2115 \u22a2 ofDigits b (digits b n ++ digits b m) = n + b ^ List.length (digits b n) * m ** rw [ofDigits_append, ofDigits_digits, ofDigits_digits] ** Qed", + "informal": "" + }, + { + "formal": "PFunctor.M.isPath_cons' ** F : PFunctor.{u} X : Type u_1 f\u271d : X \u2192 \u2191F X xs : Path F a : F.A f : B F a \u2192 M F i : B F a \u22a2 IsPath ({ fst := a, snd := i } :: xs) (M.mk { fst := a, snd := f }) \u2192 IsPath xs (f i) ** generalize h : M.mk \u27e8a, f\u27e9 = x ** F : PFunctor.{u} X : Type u_1 f\u271d : X \u2192 \u2191F X xs : Path F a : F.A f : B F a \u2192 M F i : B F a x : M F h : M.mk { fst := a, snd := f } = x \u22a2 IsPath ({ fst := a, snd := i } :: xs) x \u2192 IsPath xs (f i) ** rintro (_ | \u27e8_, _, _, _, rfl, hp\u27e9) ** case cons F : PFunctor.{u} X : Type u_1 f\u271d\u00b9 : X \u2192 \u2191F X xs : Path F a : F.A f : B F a \u2192 M F i : B F a f\u271d : B F a \u2192 M F hp : IsPath xs (f\u271d i) h : M.mk { fst := a, snd := f } = M.mk { fst := a, snd := f\u271d } \u22a2 IsPath xs (f i) ** cases mk_inj h ** case cons.refl F : PFunctor.{u} X : Type u_1 f\u271d : X \u2192 \u2191F X xs : Path F a : F.A f : B F a \u2192 M F i : B F a hp : IsPath xs (f i) h : M.mk { fst := a, snd := f } = M.mk { fst := a, snd := f } \u22a2 IsPath xs (f i) ** exact hp ** Qed", + "informal": "" + }, + { + "formal": "Ideal.hasBasis_nhds_adic ** R : Type u_1 inst\u271d : CommRing R I : Ideal R x : R \u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n) ** letI := I.adicTopology ** R : Type u_1 inst\u271d : CommRing R I : Ideal R x : R this : TopologicalSpace R := adicTopology I \u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n) ** have := I.hasBasis_nhds_zero_adic.map fun y => x + y ** R : Type u_1 inst\u271d : CommRing R I : Ideal R x : R this\u271d : TopologicalSpace R := adicTopology I this : HasBasis (Filter.map (fun y => x + y) (\ud835\udcdd 0)) (fun _n => True) fun i => (fun y => x + y) '' \u2191(I ^ i) \u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n) ** rwa [map_add_left_nhds_zero x] at this ** Qed", + "informal": "" + }, + { + "formal": "SetTheory.PGame.lf_def ** x y : PGame \u22a2 x \u29cf y \u2194 (\u2203 i, (\u2200 (i' : LeftMoves x), moveLeft x i' \u29cf moveLeft y i) \u2227 \u2200 (j : RightMoves (moveLeft y i)), x \u29cf moveRight (moveLeft y i) j) \u2228 \u2203 j, (\u2200 (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i \u29cf y) \u2227 \u2200 (j' : RightMoves y), moveRight x j \u29cf moveRight y j' ** rw [lf_iff_exists_le] ** x y : PGame \u22a2 ((\u2203 i, x \u2264 moveLeft y i) \u2228 \u2203 j, moveRight x j \u2264 y) \u2194 (\u2203 i, (\u2200 (i' : LeftMoves x), moveLeft x i' \u29cf moveLeft y i) \u2227 \u2200 (j : RightMoves (moveLeft y i)), x \u29cf moveRight (moveLeft y i) j) \u2228 \u2203 j, (\u2200 (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i \u29cf y) \u2227 \u2200 (j' : RightMoves y), moveRight x j \u29cf moveRight y j' ** conv =>\n lhs\n simp only [le_iff_forall_lf] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.lintegral_mul_le_Lp_mul_Lq ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g \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) ** by_cases hf_zero : \u222b\u207b a, f a ^ p \u2202\u03bc = 0 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \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) ** by_cases hg_zero : \u222b\u207b a, g a ^ q \u2202\u03bc = 0 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 \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) ** by_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 \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) ** by_cases hg_top : \u222b\u207b a, g a ^ q \u2202\u03bc = \u22a4 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4 \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) ** exact ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top hpq hf hf_top hg_top hf_zero hg_zero ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \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) ** refine' Eq.trans_le _ (zero_le _) ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0 ** exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.nonneg hf hf_zero ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 \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) ** refine' Eq.trans_le _ (zero_le _) ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0 ** rw [mul_comm] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc = 0 ** exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.symm.nonneg hg hg_zero ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 \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) ** exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.pos hpq.symm.nonneg hf_top hg_zero ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4 \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) ** rw [mul_comm, mul_comm ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p))] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4 \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) ** exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.symm.pos hpq.nonneg hg_top hf_zero ** Qed", + "informal": "" + }, + { + "formal": "LieSubalgebra.mem_normalizer_iff ** R : Type u_1 L : Type u_2 M : Type u_3 M' : Type u_4 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M' inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : LieRingModule L M' inst\u271d : LieModule R L M' H : LieSubalgebra R L x : L \u22a2 x \u2208 normalizer H \u2194 \u2200 (y : L), y \u2208 H \u2192 \u2045x, y\u2046 \u2208 H ** rw [mem_normalizer_iff'] ** R : Type u_1 L : Type u_2 M : Type u_3 M' : Type u_4 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M' inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : LieRingModule L M' inst\u271d : LieModule R L M' H : LieSubalgebra R L x : L \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 ** refine' forall\u2082_congr fun y hy => _ ** R : Type u_1 L : Type u_2 M : Type u_3 M' : Type u_4 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M' inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : LieRingModule L M' inst\u271d : LieModule R L M' H : LieSubalgebra R L x y : L hy : y \u2208 H \u22a2 \u2045y, x\u2046 \u2208 H \u2194 \u2045x, y\u2046 \u2208 H ** rw [\u2190 lie_skew, neg_mem_iff (G := L)] ** Qed", + "informal": "" + }, + { + "formal": "Real.Angle.sign_coe_nonneg_of_nonneg_of_le_pi ** \u03b8 : \u211d h0 : 0 \u2264 \u03b8 hpi : \u03b8 \u2264 \u03c0 \u22a2 0 \u2264 sign \u2191\u03b8 ** rw [sign, sign_nonneg_iff] ** \u03b8 : \u211d h0 : 0 \u2264 \u03b8 hpi : \u03b8 \u2264 \u03c0 \u22a2 0 \u2264 sin \u2191\u03b8 ** exact sin_nonneg_of_nonneg_of_le_pi h0 hpi ** Qed", + "informal": "" + }, + { + "formal": "Colex.lt_singleton_iff_mem_lt ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 \u22a2 toColex s < toColex {r} \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r ** simp only [lt_def, mem_singleton, \u2190 and_assoc, exists_eq_right] ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 \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 ** constructor ** case mp \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 \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 ** intro t x hx ** case mp \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 t : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s x : \u03b1 hx : x \u2208 s \u22a2 x < r ** rw [\u2190 not_le] ** case mp \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 t : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s x : \u03b1 hx : x \u2208 s \u22a2 \u00acr \u2264 x ** intro h ** case mp \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 t : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s x : \u03b1 hx : x \u2208 s h : r \u2264 x \u22a2 False ** rcases lt_or_eq_of_le h with (h\u2081 | rfl) ** case mp.inl \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 t : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s x : \u03b1 hx : x \u2208 s h : r \u2264 x h\u2081 : r < x \u22a2 False ** exact ne_of_irrefl h\u2081 ((t.1 h\u2081).1 hx).symm ** case mp.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 t : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s hx : r \u2208 s h : r \u2264 r \u22a2 False ** exact t.2 hx ** case mpr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 \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 ** 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 ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 r : \u03b1 s : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 x < r \u22a2 \u00acr \u2208 s ** simpa using h r ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Walk.fst_mem_support_of_mem_edges ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' t u v w : V p : Walk G v w he : Quotient.mk (Sym2.Rel.setoid V) (t, u) \u2208 edges p \u22a2 t \u2208 support p ** obtain \u27e8d, hd, he\u27e9 := List.mem_map.mp he ** case intro.intro V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' t u v w : V p : Walk G v w he\u271d : Quotient.mk (Sym2.Rel.setoid V) (t, u) \u2208 edges p d : Dart G hd : d \u2208 darts p he : Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (t, u) \u22a2 t \u2208 support p ** rw [dart_edge_eq_mk'_iff'] at he ** case intro.intro V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' t u v w : V p : Walk G v w he\u271d : Quotient.mk (Sym2.Rel.setoid V) (t, u) \u2208 edges p d : Dart G hd : d \u2208 darts p he : d.toProd.1 = t \u2227 d.toProd.2 = u \u2228 d.toProd.1 = u \u2227 d.toProd.2 = t \u22a2 t \u2208 support p ** rcases he with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9) ** case intro.intro.inl.intro V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' v w : V p : Walk G v w d : Dart G hd : d \u2208 darts p he : Quotient.mk (Sym2.Rel.setoid V) (d.toProd.1, d.toProd.2) \u2208 edges p \u22a2 d.toProd.1 \u2208 support p ** exact dart_fst_mem_support_of_mem_darts _ hd ** case intro.intro.inr.intro V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' v w : V p : Walk G v w d : Dart G hd : d \u2208 darts p he : Quotient.mk (Sym2.Rel.setoid V) (d.toProd.2, d.toProd.1) \u2208 edges p \u22a2 d.toProd.2 \u2208 support p ** exact dart_snd_mem_support_of_mem_darts _ hd ** Qed", + "informal": "" + }, + { + "formal": "DirectSum.induction_on ** \u03b9 : Type v dec_\u03b9 : DecidableEq \u03b9 \u03b2 : \u03b9 \u2192 Type w inst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i) C : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop x : \u2a01 (i : \u03b9), \u03b2 i H_zero : C 0 H_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x) H_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y) \u22a2 C x ** apply DFinsupp.induction x H_zero ** \u03b9 : Type v dec_\u03b9 : DecidableEq \u03b9 \u03b2 : \u03b9 \u2192 Type w inst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i) C : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop x : \u2a01 (i : \u03b9), \u03b2 i H_zero : C 0 H_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x) H_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y) \u22a2 \u2200 (i : \u03b9) (b : (fun i => \u03b2 i) i) (f : \u03a0\u2080 (i : \u03b9), (fun i => \u03b2 i) i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 C f \u2192 C (DFinsupp.single i b + f) ** intro i b f h1 h2 ih ** \u03b9 : Type v dec_\u03b9 : DecidableEq \u03b9 \u03b2 : \u03b9 \u2192 Type w inst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i) C : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop x : \u2a01 (i : \u03b9), \u03b2 i H_zero : C 0 H_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x) H_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y) i : \u03b9 b : \u03b2 i f : \u03a0\u2080 (i : \u03b9), (fun i => \u03b2 i) i h1 : \u2191f i = 0 h2 : b \u2260 0 ih : C f \u22a2 C (DFinsupp.single i b + f) ** solve_by_elim ** Qed", + "informal": "" + }, + { + "formal": "Submodule.lt_of_le_of_finrank_lt_finrank ** K : Type u V : Type v inst\u271d\u2074 : Ring K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V V\u2082 : Type v' inst\u271d\u00b9 : AddCommGroup V\u2082 inst\u271d : Module K V\u2082 s t : Submodule K V le : s \u2264 t lt : finrank K { x // x \u2208 s } < finrank K { x // x \u2208 t } h : s = t \u22a2 finrank K { x // x \u2208 s } = finrank K { x // x \u2208 t } ** rw [h] ** Qed", + "informal": "" + }, + { + "formal": "unique_subtype_iff_exists_unique ** \u03b1 : Sort u_1 p : \u03b1 \u2192 Prop x\u271d\u00b9 : \u2203! a, p a a : \u03b1 ha : (fun a => p a) a he : \u2200 (y : \u03b1), (fun a => p a) y \u2192 y = a x\u271d : Subtype p b : \u03b1 hb : p b \u22a2 { val := b, property := hb } = default ** congr ** case e_val \u03b1 : Sort u_1 p : \u03b1 \u2192 Prop x\u271d\u00b9 : \u2203! a, p a a : \u03b1 ha : (fun a => p a) a he : \u2200 (y : \u03b1), (fun a => p a) y \u2192 y = a x\u271d : Subtype p b : \u03b1 hb : p b \u22a2 b = a ** exact he b hb ** Qed", + "informal": "" + }, + { + "formal": "List.rel_reverse ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 b\u271d : \u03b2 l\u2081\u271d : List \u03b1 l\u2082\u271d : List \u03b2 h\u2081 : R a\u271d b\u271d h\u2082 : Forall\u2082 R l\u2081\u271d l\u2082\u271d \u22a2 Forall\u2082 R (reverse (a\u271d :: l\u2081\u271d)) (reverse (b\u271d :: l\u2082\u271d)) ** simp only [reverse_cons] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R S : \u03b1 \u2192 \u03b2 \u2192 Prop P : \u03b3 \u2192 \u03b4 \u2192 Prop R\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d : \u03b1 b\u271d : \u03b2 l\u2081\u271d : List \u03b1 l\u2082\u271d : List \u03b2 h\u2081 : R a\u271d b\u271d h\u2082 : Forall\u2082 R l\u2081\u271d l\u2082\u271d \u22a2 Forall\u2082 R (reverse l\u2081\u271d ++ [a\u271d]) (reverse l\u2082\u271d ++ [b\u271d]) ** exact rel_append (rel_reverse h\u2082) (Forall\u2082.cons h\u2081 Forall\u2082.nil) ** Qed", + "informal": "" + }, + { + "formal": "Real.rpow_zero_pos ** x : \u211d \u22a2 0 < x ^ 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.evariance_mul ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 evariance (fun \u03c9 => c * X \u03c9) \u03bc = ENNReal.ofReal (c ^ 2) * evariance X \u03bc ** rw [evariance, evariance, \u2190 lintegral_const_mul' _ _ ENNReal.ofReal_lt_top.ne] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc = \u222b\u207b (a : \u03a9), ENNReal.ofReal (c ^ 2) * \u2191\u2016X a - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc ** congr ** case e_f \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \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 ** ext1 \u03c9 ** case e_f.h \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \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 ** rw [ENNReal.ofReal, \u2190 ENNReal.coe_pow, \u2190 ENNReal.coe_pow, \u2190 ENNReal.coe_mul] ** case e_f.h \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \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) ** congr ** case e_f.h.e_a \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \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 ** 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 _)] ** case e_f.h.e_a \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \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 ** conv_rhs => rw [\u2190 nnnorm_norm, norm_mul, norm_abs_eq_norm, \u2190 norm_mul, nnnorm_norm, mul_sub] ** case e_f.h.e_a \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \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 ** congr ** case e_f.h.e_a.e_a.e_a.e_a \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \u22a2 \u222b (x : \u03a9), c * X x \u2202\u03bc = c * \u222b (x : \u03a9), X x \u2202\u03bc ** rw [mul_comm] ** case e_f.h.e_a.e_a.e_a.e_a \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \u22a2 \u222b (x : \u03a9), c * X x \u2202\u03bc = (\u222b (x : \u03a9), X x \u2202\u03bc) * c ** simp_rw [\u2190 smul_eq_mul, \u2190 integral_smul_const, smul_eq_mul, mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "Nat.Prime.mod_two_eq_one_iff_ne_two ** p : \u2115 inst\u271d : Fact (Prime p) \u22a2 p % 2 = 1 \u2194 p \u2260 2 ** refine' \u27e8fun h hf => _, (Nat.Prime.eq_two_or_odd <| @Fact.out p.Prime _).resolve_left\u27e9 ** p : \u2115 inst\u271d : Fact (Prime p) h : p % 2 = 1 hf : p = 2 \u22a2 False ** rw [hf] at h ** p : \u2115 inst\u271d : Fact (Prime p) h : 2 % 2 = 1 hf : p = 2 \u22a2 False ** simp at h ** Qed", + "informal": "" + }, + { + "formal": "Set.offDiag_singleton ** \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 \u00d7 \u03b1 a\u271d a : \u03b1 \u22a2 offDiag {a} = \u2205 ** simp ** Qed", + "informal": "" + }, + { + "formal": "IsOpen.measure_eq_zero_iff ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : T2Space Y \u03bc \u03bd : Measure X inst\u271d : IsOpenPosMeasure \u03bc s U F : Set X x : X hU : IsOpen U \u22a2 \u2191\u2191\u03bc U = 0 \u2194 U = \u2205 ** simpa only [not_lt, nonpos_iff_eq_zero, not_nonempty_iff_eq_empty] using\n not_congr (hU.measure_pos_iff \u03bc) ** Qed", + "informal": "" + }, + { + "formal": "MonoidAlgebra.support_mul_single_subset ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b2 : Semiring k inst\u271d\u00b9 : DecidableEq G inst\u271d : Mul G f : MonoidAlgebra k G r : k a : G \u22a2 (f * single a r).support \u2286 image (fun x => x * a) f.support ** intro x hx ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b2 : Semiring k inst\u271d\u00b9 : DecidableEq G inst\u271d : Mul G f : MonoidAlgebra k G r : k a x : G hx : x \u2208 (f * single a r).support \u22a2 x \u2208 image (fun x => x * a) f.support ** contrapose hx ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b2 : Semiring k inst\u271d\u00b9 : DecidableEq G inst\u271d : Mul G f : MonoidAlgebra k G r : k a x : G hx : \u00acx \u2208 image (fun x => x * a) f.support \u22a2 \u00acx \u2208 (f * single a r).support ** 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 ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b2 : Semiring k inst\u271d\u00b9 : DecidableEq G inst\u271d : Mul G f : MonoidAlgebra k G r : k a x : G hx : \u00acx \u2208 image (fun x => x * a) f.support this : \u2200 (y : G), y * a = x \u2192 \u2191f y = 0 \u22a2 \u00acx \u2208 (f * single a r).support ** simp only [mem_support_iff, mul_apply, sum_single_index, zero_mul, ite_self, sum_zero,\n Classical.not_not] ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b2 : Semiring k inst\u271d\u00b9 : DecidableEq G inst\u271d : Mul G f : MonoidAlgebra k G r : k a x : G hx : \u00acx \u2208 image (fun x => x * a) f.support this : \u2200 (y : G), y * a = x \u2192 \u2191f y = 0 \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 ** 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]) ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b2 : Semiring k inst\u271d\u00b9 : DecidableEq G inst\u271d : Mul G f : MonoidAlgebra k G r : k a x : G hx : \u00acx \u2208 image (fun x => x * a) f.support \u22a2 \u2200 (y : G), y * a = x \u2192 \u2191f y = 0 ** simpa only [not_and', mem_image, mem_support_iff, exists_prop, not_exists,\n Classical.not_not] using hx ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b2 : Semiring k inst\u271d\u00b9 : DecidableEq G inst\u271d : Mul G f : MonoidAlgebra k G r : k a x : G hx : \u00acx \u2208 image (fun x => x * a) f.support this : \u2200 (y : G), y * a = x \u2192 \u2191f y = 0 \u22a2 \u2200 (x_1 : G), 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 ** simp (config := { contextual := true }) only [this, sum_single_index, ite_eq_right_iff,\n eq_self_iff_true, imp_true_iff, zero_mul] ** Qed", + "informal": "" + }, + { + "formal": "Complex.exp_bound ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n \u22a2 \u2191abs (cexp x - \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) \u2264 \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9) ** 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] ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n \u22a2 CauSeq.lim (cauSeqAbs (exp' x + -const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) \u2264 \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9) ** refine' lim_le (CauSeq.le_of_exists \u27e8n, fun j hj => _\u27e9) ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n \u22a2 \u2191(cauSeqAbs (exp' x + -const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) j \u2264 \u2191(const abs' (\u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9))) j ** simp_rw [\u2190 sub_eq_add_neg] ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n \u22a2 \u2191(cauSeqAbs (exp' x - const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) j \u2264 \u2191(const abs' (\u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9))) j ** 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) ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 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 \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9) ** rw [sum_range_sub_sum_range hj] ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n \u22a2 \u2191abs (\u2211 m in filter (fun k => n \u2264 k) (range j), x ^ m / \u2191(Nat.factorial m)) = \u2191abs (\u2211 m in filter (fun k => n \u2264 k) (range j), x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m))) ** refine' congr_arg abs (sum_congr rfl fun m hm => _) ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n m : \u2115 hm : m \u2208 filter (fun k => n \u2264 k) (range j) \u22a2 x ^ m / \u2191(Nat.factorial m) = x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m)) ** rw [mem_filter, mem_range] at hm ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n m : \u2115 hm : m < j \u2227 n \u2264 m \u22a2 x ^ m / \u2191(Nat.factorial m) = x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m)) ** rw [\u2190 mul_div_assoc, \u2190 pow_add, add_tsub_cancel_of_le hm.2] ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n \u22a2 \u2211 m in filter (fun k => n \u2264 k) (range j), \u2191abs (x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m))) \u2264 \u2211 m in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (1 / \u2191(Nat.factorial m)) ** simp_rw [map_mul, map_pow, map_div\u2080, abs_cast_nat] ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 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 \u2211 x_1 in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (1 / \u2191(Nat.factorial x_1)) ** gcongr ** case h.h.h x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n i\u271d : \u2115 a\u271d : i\u271d \u2208 filter (fun k => n \u2264 k) (range j) \u22a2 \u2191abs (x ^ (i\u271d - n)) \u2264 1 ** rw [abv_pow abs] ** case h.h.h x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n i\u271d : \u2115 a\u271d : i\u271d \u2208 filter (fun k => n \u2264 k) (range j) \u22a2 \u2191abs x ^ (i\u271d - n) \u2264 1 ** exact pow_le_one _ (abs.nonneg _) hx ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n \u22a2 \u2211 m in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (1 / \u2191(Nat.factorial m)) = \u2191abs x ^ n * \u2211 m in filter (fun k => n \u2264 k) (range j), 1 / \u2191(Nat.factorial m) ** simp [abs_mul, abv_pow abs, abs_div, mul_sum.symm] ** x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 n \u22a2 \u2191abs x ^ n * \u2211 m in filter (fun k => n \u2264 k) (range j), 1 / \u2191(Nat.factorial m) \u2264 \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9) ** gcongr ** case h x : \u2102 hx : \u2191abs x \u2264 1 n : \u2115 hn : 0 < n j : \u2115 hj : j \u2265 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 ** exact sum_div_factorial_le _ _ hn ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.restrict_finset_biUnion_congr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 s : Finset \u03b9 t : \u03b9 \u2192 Set \u03b1 \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) ** induction' s using Finset.induction_on with i s _ hs ** case insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 t : \u03b9 \u2192 Set \u03b1 i : \u03b9 s : Finset \u03b9 a\u271d : \u00aci \u2208 s hs : 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) \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 \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 restrict \u03bc (t i_1) = restrict \u03bd (t i_1) ** simp only [forall_eq_or_imp, iUnion_iUnion_eq_or_left, Finset.mem_insert] ** case insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 t : \u03b9 \u2192 Set \u03b1 i : \u03b9 s : Finset \u03b9 a\u271d : \u00aci \u2208 s hs : 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) \u22a2 restrict \u03bc (t i \u222a \u22c3 x \u2208 s, t x) = restrict \u03bd (t i \u222a \u22c3 x \u2208 s, t x) \u2194 restrict \u03bc (t i) = restrict \u03bd (t i) \u2227 \u2200 (a : \u03b9), a \u2208 s \u2192 restrict \u03bc (t a) = restrict \u03bd (t a) ** rw [restrict_union_congr, \u2190 hs] ** case empty \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 t : \u03b9 \u2192 Set \u03b1 \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) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Real.continuousAt_rpow_of_pos ** p : \u211d \u00d7 \u211d hp : 0 < p.2 \u22a2 ContinuousAt (fun p => p.1 ^ p.2) p ** cases' p with x y ** case mk x y : \u211d hp : 0 < (x, y).2 \u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y) ** dsimp only at hp ** case mk x y : \u211d hp : 0 < y \u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y) ** obtain hx | rfl := ne_or_eq x 0 ** case mk.inr y : \u211d hp : 0 < y \u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y) ** 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) ** case mk.inr y : \u211d hp : 0 < y A : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) \u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y) ** 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 ** case mk.inr y : \u211d hp : 0 < y A : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) B : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) \u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y) ** 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' ** case mk.inr y : \u211d hp : 0 < y A : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) B : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) C : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0) \u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y) ** 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 _)) ** case mk.inl x y : \u211d hp : 0 < y hx : x \u2260 0 \u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y) ** exact continuousAt_rpow_of_ne (x, y) hx ** y : \u211d hp : 0 < y A : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) B : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) \u22a2 Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0) ** rw [nhdsWithin_singleton, tendsto_pure, pure_prod, eventually_map] ** y : \u211d hp : 0 < y A : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) B : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) \u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd y, (0, a).1 ^ (0, a).2 = 0 ** exact (lt_mem_nhds hp).mono fun y hy => zero_rpow hy.ne' ** Qed", + "informal": "" + }, + { + "formal": "Relation.ReflTransGen.head_induction_on ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d b c d : \u03b1 P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop a : \u03b1 h : ReflTransGen r a b refl : P b (_ : ReflTransGen r b b) head : \u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b) \u22a2 P a h ** induction h ** case refl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d b c d a : \u03b1 P : (a_1 : \u03b1) \u2192 ReflTransGen r a_1 a \u2192 Prop refl : P a (_ : ReflTransGen r a a) head : \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) \u22a2 P a (_ : ReflTransGen r a a) case tail \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d\u00b2 b c d a b\u271d c\u271d : \u03b1 a\u271d\u00b9 : ReflTransGen r a b\u271d a\u271d : r b\u271d c\u271d a_ih\u271d : \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b\u271d \u2192 Prop}, P b\u271d (_ : ReflTransGen r b\u271d b\u271d) \u2192 (\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 P : (a : \u03b1) \u2192 ReflTransGen r a c\u271d \u2192 Prop refl : P c\u271d (_ : ReflTransGen r c\u271d c\u271d) head : \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) \u22a2 P a (_ : ReflTransGen r a c\u271d) ** case refl => exact refl ** case tail \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d\u00b2 b c d a b\u271d c\u271d : \u03b1 a\u271d\u00b9 : ReflTransGen r a b\u271d a\u271d : r b\u271d c\u271d a_ih\u271d : \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b\u271d \u2192 Prop}, P b\u271d (_ : ReflTransGen r b\u271d b\u271d) \u2192 (\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 P : (a : \u03b1) \u2192 ReflTransGen r a c\u271d \u2192 Prop refl : P c\u271d (_ : ReflTransGen r c\u271d c\u271d) head : \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) \u22a2 P a (_ : ReflTransGen r a c\u271d) ** 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) } ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d b c d a : \u03b1 P : (a_1 : \u03b1) \u2192 ReflTransGen r a_1 a \u2192 Prop refl : P a (_ : ReflTransGen r a a) head : \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) \u22a2 P a (_ : ReflTransGen r a a) ** exact refl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1 a\u271d : ReflTransGen r a b hbc : r b c ih : \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop}, P b (_ : ReflTransGen r b b) \u2192 (\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 P : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop refl : P c (_ : ReflTransGen r c c) head : \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) \u22a2 P a (_ : ReflTransGen r a c) ** refine @ih (\u03bb {a : \u03b1} (hab : ReflTransGen r a b) => P a (ReflTransGen.tail hab hbc)) ?_ ?_ ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1 a\u271d : ReflTransGen r a b hbc : r b c ih : \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop}, P b (_ : ReflTransGen r b b) \u2192 (\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 P : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop refl : P c (_ : ReflTransGen r c c) head : \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) \u22a2 (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r b b) case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1 a\u271d : ReflTransGen r a b hbc : r b c ih : \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop}, P b (_ : ReflTransGen r b b) \u2192 (\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 P : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop refl : P c (_ : ReflTransGen r c c) head : \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) \u22a2 \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 b), (fun {a} hab => P a (_ : ReflTransGen r a c)) h \u2192 (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r a b) ** { exact head hbc _ refl } ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1 a\u271d : ReflTransGen r a b hbc : r b c ih : \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop}, P b (_ : ReflTransGen r b b) \u2192 (\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 P : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop refl : P c (_ : ReflTransGen r c c) head : \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) \u22a2 \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 b), (fun {a} hab => P a (_ : ReflTransGen r a c)) h \u2192 (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r a b) ** { exact fun h1 h2 \u21a6 head h1 (h2.tail hbc) } ** Qed", + "informal": "" + }, + { + "formal": "Set.pairwise_disjoint_Ico_mul_zpow ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 \u22a2 Pairwise (Disjoint on fun n => Ico (a * b ^ n) (a * b ^ (n + 1))) ** simp_rw [Function.onFun, Set.disjoint_iff] ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 \u22a2 Pairwise fun x y => Ico (a * b ^ x) (a * b ^ (x + 1)) \u2229 Ico (a * b ^ y) (a * b ^ (y + 1)) \u2286 \u2205 ** intro m n hmn x hx ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 m n : \u2124 hmn : m \u2260 n x : \u03b1 hx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1)) \u22a2 x \u2208 \u2205 ** apply hmn ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 m n : \u2124 hmn : m \u2260 n x : \u03b1 hx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1)) \u22a2 m = n ** 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 ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 m n : \u2124 hmn : m \u2260 n x : \u03b1 hx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1)) hb : 1 < b \u22a2 m = n ** have i1 := hx.1.1.trans_lt hx.2.2 ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 m n : \u2124 hmn : m \u2260 n x : \u03b1 hx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1)) hb : 1 < b i1 : a * b ^ m < a * b ^ (n + 1) \u22a2 m = n ** have i2 := hx.2.1.trans_lt hx.1.2 ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 m n : \u2124 hmn : m \u2260 n x : \u03b1 hx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1)) hb : 1 < b i1 : a * b ^ m < a * b ^ (n + 1) i2 : a * b ^ n < a * b ^ (m + 1) \u22a2 m = n ** rw [mul_lt_mul_iff_left, zpow_lt_zpow_iff hb, Int.lt_add_one_iff] at i1 i2 ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 m n : \u2124 hmn : m \u2260 n x : \u03b1 hx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1)) hb : 1 < b i1 : m \u2264 n i2 : n \u2264 m \u22a2 m = n ** exact le_antisymm i1 i2 ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 m n : \u2124 hmn : m \u2260 n x : \u03b1 hx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1)) \u22a2 1 < b ** have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.trans_lt hx.1.2 ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 m n : \u2124 hmn : m \u2260 n x : \u03b1 hx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1)) this : a * b ^ m < a * b ^ (m + 1) \u22a2 1 < b ** rwa [mul_lt_mul_iff_left, \u2190 mul_one (b ^ m), zpow_add_one, mul_lt_mul_iff_left] at this ** Qed", + "informal": "" + }, + { + "formal": "lp.uniformContinuous_coe ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) \u22a2 UniformContinuous Subtype.val ** have hp : p \u2260 0 := (zero_lt_one.trans_le _i.elim).ne' ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) hp : p \u2260 0 \u22a2 UniformContinuous Subtype.val ** rw [uniformContinuous_pi] ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) hp : p \u2260 0 \u22a2 \u2200 (i : \u03b1), UniformContinuous fun x => \u2191x i ** intro i ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) hp : p \u2260 0 i : \u03b1 \u22a2 UniformContinuous fun x => \u2191x i ** rw [NormedAddCommGroup.uniformity_basis_dist.uniformContinuous_iff\n NormedAddCommGroup.uniformity_basis_dist] ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) hp : p \u2260 0 i : \u03b1 \u22a2 \u2200 (i_1 : \u211d), 0 < i_1 \u2192 \u2203 j, 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 < i_1} ** intro \u03b5 h\u03b5 ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) hp : p \u2260 0 i : \u03b1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 j, 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} ** refine' \u27e8\u03b5, h\u03b5, _\u27e9 ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) hp : p \u2260 0 i : \u03b1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \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} ** rintro f g (hfg : \u2016f - g\u2016 < \u03b5) ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) hp : p \u2260 0 i : \u03b1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 f g : { x // x \u2208 lp E p } hfg : \u2016f - g\u2016 < \u03b5 \u22a2 (\u2191f i, \u2191g i) \u2208 {p | \u2016p.1 - p.2\u2016 < \u03b5} ** have : \u2016f i - g i\u2016 \u2264 \u2016f - g\u2016 := norm_apply_le_norm hp (f - g) i ** \u03b1 : Type u_1 E : \u03b1 \u2192 Type u_2 p q : \u211d\u22650\u221e inst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i) _i : Fact (1 \u2264 p) hp : p \u2260 0 i : \u03b1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 f g : { x // x \u2208 lp E p } hfg : \u2016f - g\u2016 < \u03b5 this : \u2016\u2191f i - \u2191g i\u2016 \u2264 \u2016f - g\u2016 \u22a2 (\u2191f i, \u2191g i) \u2208 {p | \u2016p.1 - p.2\u2016 < \u03b5} ** exact this.trans_lt hfg ** Qed", + "informal": "" + }, + { + "formal": "WittVector.ghostFun_nat_cast ** p : \u2115 R : Type u_1 S : Type u_2 T : Type u_3 hp : Fact (Nat.Prime p) inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : CommRing S inst\u271d : CommRing T \u03b1 : Type u_4 \u03b2 : Type u_5 x y : \ud835\udd4e R i : \u2115 \u22a2 WittVector.ghostFun (Nat.unaryCast i) = \u2191i ** induction i <;>\n simp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.coe_nat] ** Qed", + "informal": "" + }, + { + "formal": "ContMDiffAt.div ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c H : Type u_2 inst\u271d\u00b9\u2078 : TopologicalSpace H E : Type u_3 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E I : ModelWithCorners \ud835\udd5c E H F : Type u_4 inst\u271d\u00b9\u2075 : NormedAddCommGroup F inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F J : ModelWithCorners \ud835\udd5c F F G : Type u_5 inst\u271d\u00b9\u00b3 : TopologicalSpace G inst\u271d\u00b9\u00b2 : ChartedSpace H G inst\u271d\u00b9\u00b9 : Group G inst\u271d\u00b9\u2070 : LieGroup I G E' : Type u_6 inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : NormedSpace \ud835\udd5c E' H' : Type u_7 inst\u271d\u2077 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M : Type u_8 inst\u271d\u2076 : TopologicalSpace M inst\u271d\u2075 : ChartedSpace H' M E'' : Type u_9 inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_10 inst\u271d\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M' : Type u_11 inst\u271d\u00b9 : TopologicalSpace M' inst\u271d : ChartedSpace H'' M' n : \u2115\u221e f g : M \u2192 G x\u2080 : M hf : ContMDiffAt I' I n f x\u2080 hg : ContMDiffAt I' I n g x\u2080 \u22a2 ContMDiffAt I' I n (fun x => f x / g x) x\u2080 ** simp_rw [div_eq_mul_inv] ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c H : Type u_2 inst\u271d\u00b9\u2078 : TopologicalSpace H E : Type u_3 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E I : ModelWithCorners \ud835\udd5c E H F : Type u_4 inst\u271d\u00b9\u2075 : NormedAddCommGroup F inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F J : ModelWithCorners \ud835\udd5c F F G : Type u_5 inst\u271d\u00b9\u00b3 : TopologicalSpace G inst\u271d\u00b9\u00b2 : ChartedSpace H G inst\u271d\u00b9\u00b9 : Group G inst\u271d\u00b9\u2070 : LieGroup I G E' : Type u_6 inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : NormedSpace \ud835\udd5c E' H' : Type u_7 inst\u271d\u2077 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M : Type u_8 inst\u271d\u2076 : TopologicalSpace M inst\u271d\u2075 : ChartedSpace H' M E'' : Type u_9 inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_10 inst\u271d\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M' : Type u_11 inst\u271d\u00b9 : TopologicalSpace M' inst\u271d : ChartedSpace H'' M' n : \u2115\u221e f g : M \u2192 G x\u2080 : M hf : ContMDiffAt I' I n f x\u2080 hg : ContMDiffAt I' I n g x\u2080 \u22a2 ContMDiffAt I' I n (fun x => f x * (g x)\u207b\u00b9) x\u2080 ** exact hf.mul hg.inv ** Qed", + "informal": "" + }, + { + "formal": "PNat.gcd_comm ** m n : \u2115+ \u22a2 gcd m n = gcd n m ** apply eq ** case a m n : \u2115+ \u22a2 \u2191(gcd m n) = \u2191(gcd n m) ** simp only [gcd_coe] ** case a m n : \u2115+ \u22a2 Nat.gcd \u2191m \u2191n = Nat.gcd \u2191n \u2191m ** apply Nat.gcd_comm ** Qed", + "informal": "" + }, + { + "formal": "Finset.filter_lt_eq_Ioi ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b3 : Preorder \u03b1 inst\u271d\u00b2 : LocallyFiniteOrderTop \u03b1 a : \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidablePred ((fun x x_1 => x < x_1) a) \u22a2 filter ((fun x x_1 => x < x_1) a) univ = Ioi a ** ext ** case a \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b3 : Preorder \u03b1 inst\u271d\u00b2 : LocallyFiniteOrderTop \u03b1 a : \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidablePred ((fun x x_1 => x < x_1) a) a\u271d : \u03b1 \u22a2 a\u271d \u2208 filter ((fun x x_1 => x < x_1) a) univ \u2194 a\u271d \u2208 Ioi a ** simp ** Qed", + "informal": "" + }, + { + "formal": "Matrix.transvection_mul_apply_of_ne ** n : Type u_1 p : Type u_2 R : Type u\u2082 \ud835\udd5c : Type u_3 inst\u271d\u2074 : Field \ud835\udd5c inst\u271d\u00b3 : DecidableEq n inst\u271d\u00b2 : DecidableEq p inst\u271d\u00b9 : CommRing R i j : n inst\u271d : Fintype n a b : n ha : a \u2260 i c : R M : Matrix n n R \u22a2 (transvection i j c * M) a b = M a b ** simp [transvection, Matrix.add_mul, ha] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.leadingCoeff_X_sub_C ** R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : Nontrivial R inst\u271d : Ring S r : S \u22a2 leadingCoeff (X - \u2191C r) = 1 ** rw [sub_eq_add_neg, \u2190 map_neg C r, leadingCoeff_X_add_C] ** Qed", + "informal": "" + }, + { + "formal": "Real.sq_lt ** x y : \u211d \u22a2 x ^ 2 < y \u2194 -sqrt y < x \u2227 x < sqrt y ** rw [\u2190 abs_lt, \u2190 sq_abs, lt_sqrt (abs_nonneg _)] ** Qed", + "informal": "" + }, + { + "formal": "Nat.Partrec.Code.evaln_map ** k : \u2115 c : Code n : \u2115 \u22a2 (Option.bind (Option.map (evaln k c) (List.get? (List.range k) n)) fun b => b) = evaln k c n ** by_cases kn : n < k ** case pos k : \u2115 c : Code n : \u2115 kn : n < k \u22a2 (Option.bind (Option.map (evaln k c) (List.get? (List.range k) n)) fun b => b) = evaln k c n ** simp [List.get?_range kn] ** case neg k : \u2115 c : Code n : \u2115 kn : \u00acn < k \u22a2 (Option.bind (Option.map (evaln k c) (List.get? (List.range k) n)) fun b => b) = evaln k c n ** rw [List.get?_len_le] ** case neg k : \u2115 c : Code n : \u2115 kn : \u00acn < k \u22a2 List.length (List.range k) \u2264 n ** simpa using kn ** case neg k : \u2115 c : Code n : \u2115 kn : \u00acn < k \u22a2 (Option.bind (Option.map (evaln k c) Option.none) fun b => b) = evaln k c n ** cases e : evaln k c n ** case neg.some k : \u2115 c : Code n : \u2115 kn : \u00acn < k val\u271d : \u2115 e : evaln k c n = some val\u271d \u22a2 (Option.bind (Option.map (evaln k c) Option.none) fun b => b) = some val\u271d ** exact kn.elim (evaln_bound e) ** case neg.none k : \u2115 c : Code n : \u2115 kn : \u00acn < k e : evaln k c n = Option.none \u22a2 (Option.bind (Option.map (evaln k c) Option.none) fun b => b) = Option.none ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Equiv.subset_image ** \u03b1\u271d : Sort u \u03b2\u271d : Sort v \u03b3 : Sort w \u03b1 : Type u_1 \u03b2 : Type u_2 e : \u03b1 \u2243 \u03b2 s : Set \u03b1 t : Set \u03b2 \u22a2 \u2191e.symm '' t \u2286 s \u2194 t \u2286 \u2191e '' s ** rw [image_subset_iff, e.image_eq_preimage] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.countP_eq_countP_filter_add ** case h.e'_3.h.e'_6 \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 p\u271d : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p\u271d s : Multiset \u03b1 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q l : List \u03b1 \u22a2 countP p (filter (fun a => \u00acq a) (Quot.mk Setoid.r l)) = List.countP (fun x => decide (p x)) (List.filter (fun a => decide \u00acdecide (q a) = true) l) ** simp [countP_filter] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.degree_int_cast_le ** R : Type u S : Type v a b c d : R n\u271d m : \u2115 inst\u271d : Ring R n : \u2124 \u22a2 natDegree \u2191n \u2264 Zero.zero ** simp ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.pderiv_C_mul ** R : Type u \u03c3 : Type v a a' a\u2081 a\u2082 : R s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R f : MvPolynomial \u03c3 R i : \u03c3 \u22a2 \u2191(pderiv i) (\u2191C a * f) = \u2191C a * \u2191(pderiv i) f ** rw [C_mul', Derivation.map_smul, C_mul'] ** Qed", + "informal": "" + }, + { + "formal": "gc_nhds ** \u03b1 : Type u \u03b2 : Type v a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 \u22a2 nhdsAdjoint a f \u2264 t \u2194 f \u2264 (fun t => \ud835\udcdd a) t ** rw [le_nhds_iff] ** \u03b1 : Type u \u03b2 : Type v a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 \u22a2 nhdsAdjoint a f \u2264 t \u2194 \u2200 (s : Set \u03b1), a \u2208 s \u2192 IsOpen s \u2192 s \u2208 f ** exact \u27e8fun H s hs has => H _ has hs, fun H s has hs => H _ hs has\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.isOpenImmersion_iff_stalk ** X Y Z : Scheme f\u271d : X \u27f6 Y g : Y \u27f6 Z f : X \u27f6 Y \u22a2 IsOpenImmersion f \u2194 OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x) ** constructor ** case mp X Y Z : Scheme f\u271d : X \u27f6 Y g : Y \u27f6 Z f : X \u27f6 Y \u22a2 IsOpenImmersion f \u2192 OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x) ** intro h ** case mp X Y Z : Scheme f\u271d : X \u27f6 Y g : Y \u27f6 Z f : X \u27f6 Y h : IsOpenImmersion f \u22a2 OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x) ** exact \u27e8h.1, inferInstance\u27e9 ** case mpr X Y Z : Scheme f\u271d : X \u27f6 Y g : Y \u27f6 Z f : X \u27f6 Y \u22a2 (OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)) \u2192 IsOpenImmersion f ** rintro \u27e8h\u2081, h\u2082\u27e9 ** case mpr.intro X Y Z : Scheme f\u271d : X \u27f6 Y g : Y \u27f6 Z f : X \u27f6 Y h\u2081 : OpenEmbedding \u2191f.val.base h\u2082 : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x) \u22a2 IsOpenImmersion f ** exact IsOpenImmersion.of_stalk_iso f h\u2081 ** Qed", + "informal": "" + }, + { + "formal": "Sum.isRight_eq_false ** \u03b1 : Type u_1 \u03b2 : Type u_2 x : \u03b1 \u2295 \u03b2 \u22a2 isRight x = false \u2194 isLeft x = true ** cases x <;> simp ** Qed", + "informal": "" + }, + { + "formal": "LocallyFinite.finite_nonempty_inter_compact ** \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Type u_1 \u03c0 : \u03b9\u271d \u2192 Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t : Set \u03b1 \u03b9 : Type u_3 f : \u03b9 \u2192 Set \u03b1 hf : LocallyFinite f s : Set \u03b1 hs : IsCompact s \u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)} ** choose U hxU hUf using hf ** \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Type u_1 \u03c0 : \u03b9\u271d \u2192 Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t : Set \u03b1 \u03b9 : Type u_3 f : \u03b9 \u2192 Set \u03b1 s : Set \u03b1 hs : IsCompact s U : \u03b1 \u2192 Set \u03b1 hxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x hUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)} \u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)} ** rcases hs.elim_nhds_subcover U fun x _ => hxU x with \u27e8t, -, hsU\u27e9 ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Type u_1 \u03c0 : \u03b9\u271d \u2192 Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t\u271d : Set \u03b1 \u03b9 : Type u_3 f : \u03b9 \u2192 Set \u03b1 s : Set \u03b1 hs : IsCompact s U : \u03b1 \u2192 Set \u03b1 hxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x hUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)} t : Finset \u03b1 hsU : s \u2286 \u22c3 x \u2208 t, U x \u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)} ** refine' (t.finite_toSet.biUnion fun x _ => hUf x).subset _ ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Type u_1 \u03c0 : \u03b9\u271d \u2192 Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t\u271d : Set \u03b1 \u03b9 : Type u_3 f : \u03b9 \u2192 Set \u03b1 s : Set \u03b1 hs : IsCompact s U : \u03b1 \u2192 Set \u03b1 hxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x hUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)} t : Finset \u03b1 hsU : s \u2286 \u22c3 x \u2208 t, U x \u22a2 {i | Set.Nonempty (f i \u2229 s)} \u2286 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)} ** rintro i \u27e8x, hx\u27e9 ** case intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Type u_1 \u03c0 : \u03b9\u271d \u2192 Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t\u271d : Set \u03b1 \u03b9 : Type u_3 f : \u03b9 \u2192 Set \u03b1 s : Set \u03b1 hs : IsCompact s U : \u03b1 \u2192 Set \u03b1 hxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x hUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)} t : Finset \u03b1 hsU : s \u2286 \u22c3 x \u2208 t, U x i : \u03b9 x : \u03b1 hx : x \u2208 f i \u2229 s \u22a2 i \u2208 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)} ** rcases mem_iUnion\u2082.1 (hsU hx.2) with \u27e8c, hct, hcx\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Type u_1 \u03c0 : \u03b9\u271d \u2192 Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t\u271d : Set \u03b1 \u03b9 : Type u_3 f : \u03b9 \u2192 Set \u03b1 s : Set \u03b1 hs : IsCompact s U : \u03b1 \u2192 Set \u03b1 hxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x hUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)} t : Finset \u03b1 hsU : s \u2286 \u22c3 x \u2208 t, U x i : \u03b9 x : \u03b1 hx : x \u2208 f i \u2229 s c : \u03b1 hct : c \u2208 t hcx : x \u2208 U c \u22a2 i \u2208 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)} ** exact mem_biUnion hct \u27e8x, hx.1, hcx\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "ModuleCat.ExtendScalars.map'_id ** R : Type u\u2081 S : Type u\u2082 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S f : R \u2192+* S M\u271d M : ModuleCat R x : \u2191(obj' f M) \u22a2 \u2191(map' f (\ud835\udfd9 M)) x = \u2191(\ud835\udfd9 (obj' f M)) x ** dsimp only [map'] ** R : Type u\u2081 S : Type u\u2082 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S f : R \u2192+* S M\u271d M : ModuleCat R x : \u2191(obj' f M) \u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) x = \u2191(\ud835\udfd9 (obj' f M)) x ** rw [ModuleCat.id_apply] ** R : Type u\u2081 S : Type u\u2082 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S f : R \u2192+* S M\u271d M : ModuleCat R x : \u2191(obj' f M) \u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) x = x ** induction' x using TensorProduct.induction_on with _ _ m s ihx ihy ** case zero R : Type u\u2081 S : Type u\u2082 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S f : R \u2192+* S M\u271d M : ModuleCat R \u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) 0 = 0 ** rw [map_zero] ** case tmul R : Type u\u2081 S : Type u\u2082 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S f : R \u2192+* S M\u271d M : ModuleCat R x\u271d : \u2191((restrictScalars f).obj (mk S)) y\u271d : \u2191M \u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) (x\u271d \u2297\u209c[R] y\u271d) = x\u271d \u2297\u209c[R] y\u271d ** erw [@LinearMap.baseChange_tmul R S M M _ _ (_), ModuleCat.id_apply] ** case add R : Type u\u2081 S : Type u\u2082 inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S f : R \u2192+* S M\u271d M : ModuleCat R m s : \u2191((restrictScalars f).obj (mk S)) \u2297[R] \u2191M ihx : \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) m = m ihy : \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) s = s \u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) (m + s) = m + s ** rw [map_add, ihx, ihy] ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.sup_eq_closure ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : Group G' inst\u271d\u00b9 : Group G'' A : Type u_4 inst\u271d : AddGroup A H\u271d K : Subgroup G k : Set G H H' : Subgroup G \u22a2 H \u2294 H' = closure (\u2191H \u222a \u2191H') ** simp_rw [closure_union, closure_eq] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.StructuredArrow.projectSubobject_factors ** C : Type u\u2081 inst\u271d\u00b3 : Category.{v, u\u2081} C D : Type u\u2082 inst\u271d\u00b2 : Category.{v, u\u2082} D S : D T : C \u2964 D inst\u271d\u00b9 : HasLimits C inst\u271d : PreservesLimits T A P : StructuredArrow S T f : P \u27f6 A hf : Mono f \u22a2 (P.hom \u226b T.map (Subobject.underlyingIso (MonoOver.arrow (MonoOver.mk' f)).right).inv) \u226b T.map (Subobject.arrow (projectSubobject (Subobject.mk f))) = A.hom ** dsimp ** C : Type u\u2081 inst\u271d\u00b3 : Category.{v, u\u2081} C D : Type u\u2082 inst\u271d\u00b2 : Category.{v, u\u2082} D S : D T : C \u2964 D inst\u271d\u00b9 : HasLimits C inst\u271d : PreservesLimits T A P : StructuredArrow S T f : P \u27f6 A hf : Mono f \u22a2 (P.hom \u226b T.map (Subobject.underlyingIso f.right).inv) \u226b T.map (Subobject.arrow (Subobject.mk f.right)) = A.hom ** simp [\u2190 T.map_comp] ** Qed", + "informal": "" + }, + { + "formal": "uniformContinuous_sInf_dom ** \u03b1 : Type ua \u03b2 : Type ub \u03b3 : Type uc \u03b4 : Type ud \u03b9 : Sort u_1 f : \u03b1 \u2192 \u03b2 u\u2081 : Set (UniformSpace \u03b1) u\u2082 : UniformSpace \u03b2 u : UniformSpace \u03b1 h\u2081 : u \u2208 u\u2081 hf : UniformContinuous f \u22a2 UniformContinuous f ** delta UniformContinuous ** \u03b1 : Type ua \u03b2 : Type ub \u03b3 : Type uc \u03b4 : Type ud \u03b9 : Sort u_1 f : \u03b1 \u2192 \u03b2 u\u2081 : Set (UniformSpace \u03b1) u\u2082 : UniformSpace \u03b2 u : UniformSpace \u03b1 h\u2081 : u \u2208 u\u2081 hf : UniformContinuous f \u22a2 Tendsto (fun x => (f x.1, f x.2)) (\ud835\udce4 \u03b1) (\ud835\udce4 \u03b2) ** rw [sInf_eq_iInf', iInf_uniformity] ** \u03b1 : Type ua \u03b2 : Type ub \u03b3 : Type uc \u03b4 : Type ud \u03b9 : Sort u_1 f : \u03b1 \u2192 \u03b2 u\u2081 : Set (UniformSpace \u03b1) u\u2082 : UniformSpace \u03b2 u : UniformSpace \u03b1 h\u2081 : u \u2208 u\u2081 hf : UniformContinuous f \u22a2 Tendsto (fun x => (f x.1, f x.2)) (\u2a05 i, \ud835\udce4 \u03b1) (\ud835\udce4 \u03b2) ** exact tendsto_iInf' \u27e8u, h\u2081\u27e9 hf ** Qed", + "informal": "" + }, + { + "formal": "Function.Antiperiodic.int_odd_mul_antiperiodic ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f g : \u03b1 \u2192 \u03b2 c c\u2081 c\u2082 x\u271d : \u03b1 inst\u271d\u00b9 : Ring \u03b1 inst\u271d : InvolutiveNeg \u03b2 h : Antiperiodic f c n : \u2124 x : \u03b1 \u22a2 f (x + (\u2191n * (2 * c) + c)) = -f x ** rw [\u2190 add_assoc, h, h.periodic.int_mul] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.MonoOver.factors_congr ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C X\u271d Y\u271d Z : C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D X : C f g : MonoOver X Y : C h : Y \u27f6 X e : f \u2245 g x\u271d : Factors f h u : Y \u27f6 f.obj.left hu : u \u226b arrow f = h \u22a2 (u \u226b ((forget X).map e.hom).left) \u226b arrow g = h ** simp [hu] ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C X\u271d Y\u271d Z : C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D X : C f g : MonoOver X Y : C h : Y \u27f6 X e : f \u2245 g x\u271d : Factors g h u : Y \u27f6 g.obj.left hu : u \u226b arrow g = h \u22a2 (u \u226b ((forget X).map e.inv).left) \u226b arrow f = h ** simp [hu] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.hitting_eq_hitting_of_exists ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 h' : \u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s \u22a2 hitting u s n m\u2081 \u03c9 = hitting u s n m\u2082 \u03c9 ** simp only [hitting, if_pos h'] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 h' : \u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s \u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) = 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 ** obtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h' ** case intro.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 j : \u03b9 hj\u2081 : j \u2208 Set.Icc n m\u2081 hj\u2082 : u j \u03c9 \u2208 s \u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) = 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 ** rw [if_pos] ** case intro.intro.hc \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 j : \u03b9 hj\u2081 : j \u2208 Set.Icc n m\u2081 hj\u2082 : u j \u03c9 \u2208 s \u22a2 \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s ** exact \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.trans h\u27e9, hj\u2082\u27e9 ** case intro.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 j : \u03b9 hj\u2081 : j \u2208 Set.Icc n m\u2081 hj\u2082 : u j \u03c9 \u2208 s \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}) ** 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))) ** case intro.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 j : \u03b9 hj\u2081 : j \u2208 Set.Icc n m\u2081 hj\u2082 : u j \u03c9 \u2208 s \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}) ** refine' le_csInf \u27e8j, Set.Icc_subset_Icc_right h hj\u2081, hj\u2082\u27e9 fun i hi => _ ** case intro.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 j : \u03b9 hj\u2081 : j \u2208 Set.Icc n m\u2081 hj\u2082 : u j \u03c9 \u2208 s i : \u03b9 hi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s} \u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i ** by_cases hi' : i \u2264 m\u2081 ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 j : \u03b9 hj\u2081 : j \u2208 Set.Icc n m\u2081 hj\u2082 : u j \u03c9 \u2208 s i : \u03b9 hi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s} hi' : i \u2264 m\u2081 \u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i ** exact csInf_le bddBelow_Icc.inter_of_left \u27e8\u27e8hi.1.1, hi'\u27e9, hi.2\u27e9 ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 j : \u03b9 hj\u2081 : j \u2208 Set.Icc n m\u2081 hj\u2082 : u j \u03c9 \u2208 s i : \u03b9 hi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s} hi' : \u00aci \u2264 m\u2081 \u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i ** change j \u2208 {i | u i \u03c9 \u2208 s} at hj\u2082 ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 m\u2081 m\u2082 : \u03b9 h : m\u2081 \u2264 m\u2082 j : \u03b9 hj\u2081 : j \u2208 Set.Icc n m\u2081 i : \u03b9 hi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s} hi' : \u00aci \u2264 m\u2081 hj\u2082 : j \u2208 {i | u i \u03c9 \u2208 s} \u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i ** 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')) ** Qed", + "informal": "" + }, + { + "formal": "Filter.HasBasis.inf_basis_neBot_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 l l' : Filter \u03b1 p : \u03b9 \u2192 Prop s : \u03b9 \u2192 Set \u03b1 t : Set \u03b1 i : \u03b9 p' : \u03b9' \u2192 Prop s' : \u03b9' \u2192 Set \u03b1 i' : \u03b9' hl : HasBasis l p s hl' : HasBasis l' p' s' \u22a2 (\u2200 {i : PProd \u03b9 \u03b9'}, p i.fst \u2227 p' i.snd \u2192 Set.Nonempty (s i.fst \u2229 s' i.snd)) \u2194 \u2200 \u2983i : \u03b9\u2984, p i \u2192 \u2200 \u2983i' : \u03b9'\u2984, p' i' \u2192 Set.Nonempty (s i \u2229 s' i') ** simp [@forall_swap _ \u03b9'] ** Qed", + "informal": "" + }, + { + "formal": "GaloisConnection.l_iSup ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x \u03ba : \u03b9 \u2192 Sort u_1 a a\u2081 a\u2082 : \u03b1 b b\u2081 b\u2082 : \u03b2 inst\u271d\u00b9 : CompleteLattice \u03b1 inst\u271d : CompleteLattice \u03b2 l : \u03b1 \u2192 \u03b2 u : \u03b2 \u2192 \u03b1 gc : GaloisConnection l u f : \u03b9 \u2192 \u03b1 \u22a2 IsLUB (range (l \u2218 f)) (l (iSup f)) ** rw [range_comp, \u2190 sSup_range] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x \u03ba : \u03b9 \u2192 Sort u_1 a a\u2081 a\u2082 : \u03b1 b b\u2081 b\u2082 : \u03b2 inst\u271d\u00b9 : CompleteLattice \u03b1 inst\u271d : CompleteLattice \u03b2 l : \u03b1 \u2192 \u03b2 u : \u03b2 \u2192 \u03b1 gc : GaloisConnection l u f : \u03b9 \u2192 \u03b1 \u22a2 IsLUB (l '' range f) (l (sSup (range f))) ** exact gc.isLUB_l_image (isLUB_sSup _) ** Qed", + "informal": "" + }, + { + "formal": "Real.dimH_ball_pi_fin ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2075 : EMetricSpace X inst\u271d\u2074 : EMetricSpace Y C K r\u271d : \u211d\u22650 f : X \u2192 Y s t : Set X E : Type u_4 inst\u271d\u00b3 : Fintype \u03b9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 x : Fin n \u2192 \u211d r : \u211d hr : 0 < r \u22a2 dimH (Metric.ball x r) = \u2191n ** rw [dimH_ball_pi x hr, Fintype.card_fin] ** Qed", + "informal": "" + }, + { + "formal": "one_le_one_div ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedSemifield \u03b1 a b c d e : \u03b1 m n : \u2124 h1 : 0 < a h2 : a \u2264 1 \u22a2 1 \u2264 1 / a ** rwa [le_one_div (@zero_lt_one \u03b1 _ _ _ _ _) h1, one_div_one] ** Qed", + "informal": "" + }, + { + "formal": "Std.Range.numElems_stop_le_start ** start stop step : Nat h : { start := start, stop := stop, step := step }.stop \u2264 { start := start, stop := stop, step := step }.start \u22a2 numElems { start := start, stop := stop, step := step } = 0 ** simp [numElems] ** start stop step : Nat h : { start := start, stop := stop, step := step }.stop \u2264 { start := start, stop := stop, step := step }.start \u22a2 (if step = 0 then if stop \u2264 start then 0 else stop else (stop - start + step - 1) / step) = 0 ** split <;> simp_all ** case inr start stop step : Nat h : stop \u2264 start h\u271d : \u00acstep = 0 \u22a2 (stop - start + step - 1) / step = 0 ** apply Nat.div_eq_of_lt ** case inr.h\u2080 start stop step : Nat h : stop \u2264 start h\u271d : \u00acstep = 0 \u22a2 stop - start + step - 1 < step ** simp [Nat.sub_eq_zero_of_le h] ** case inr.h\u2080 start stop step : Nat h : stop \u2264 start h\u271d : \u00acstep = 0 \u22a2 step - 1 < step ** exact Nat.pred_lt \u2039_\u203a ** Qed", + "informal": "" + }, + { + "formal": "one_add_mul_self_lt_rpow_one_add ** s : \u211d hs : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p \u22a2 1 + p * s < (1 + s) ^ p ** rcases eq_or_lt_of_le hs with (rfl | hs) ** case inr s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p hs : -1 < s \u22a2 1 + p * s < (1 + s) ^ p ** have hs1 : 0 < 1 + s := by linarith ** case inr s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p hs : -1 < s hs1 : 0 < 1 + s \u22a2 1 + p * s < (1 + s) ^ p ** cases' le_or_lt (1 + p * s) 0 with hs2 hs2 ** case inr.inr s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s \u22a2 1 + p * s < (1 + s) ^ p ** rw [rpow_def_of_pos hs1, \u2190 exp_log hs2] ** case inr.inr s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s \u22a2 rexp (log (1 + p * s)) < rexp (log (1 + s) * p) ** apply exp_strictMono ** case inr.inr.a s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s \u22a2 log (1 + p * s) < log (1 + s) * p ** have hp : 0 < p := by positivity ** case inr.inr.a s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p \u22a2 log (1 + p * s) < log (1 + s) * p ** have hs3 : 1 + s \u2260 1 := by contrapose! hs'; linarith ** case inr.inr.a s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 \u22a2 log (1 + p * s) < log (1 + s) * p ** have hs4 : 1 + p * s \u2260 1 := by contrapose! hs'; nlinarith ** case inr.inr.a s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 \u22a2 log (1 + p * s) < log (1 + s) * p ** cases' lt_or_gt_of_ne hs' with hs' hs' ** case inl p : \u211d hp : 1 < p hs : -1 \u2264 -1 hs' : -1 \u2260 0 \u22a2 1 + p * -1 < (1 + -1) ^ p ** have : p \u2260 0 := by positivity ** case inl p : \u211d hp : 1 < p hs : -1 \u2264 -1 hs' : -1 \u2260 0 this : p \u2260 0 \u22a2 1 + p * -1 < (1 + -1) ^ p ** simpa [zero_rpow this] ** p : \u211d hp : 1 < p hs : -1 \u2264 -1 hs' : -1 \u2260 0 \u22a2 p \u2260 0 ** positivity ** s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p hs : -1 < s \u22a2 0 < 1 + s ** linarith ** case inr.inl s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 1 + p * s \u2264 0 \u22a2 1 + p * s < (1 + s) ^ p ** exact hs2.trans_lt (rpow_pos_of_pos hs1 _) ** s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s \u22a2 0 < p ** positivity ** s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p \u22a2 1 + s \u2260 1 ** contrapose! hs' ** s : \u211d hs\u271d : -1 \u2264 s p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs' : 1 + s = 1 \u22a2 s = 0 ** linarith ** s : \u211d hs\u271d : -1 \u2264 s hs' : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 \u22a2 1 + p * s \u2260 1 ** contrapose! hs' ** s : \u211d hs\u271d : -1 \u2264 s p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs' : 1 + p * s = 1 \u22a2 s = 0 ** nlinarith ** case inr.inr.a.inl s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s < 0 \u22a2 log (1 + p * s) < log (1 + s) * p ** rw [\u2190 div_lt_iff hp, \u2190 div_lt_div_right_of_neg hs'] ** case inr.inr.a.inl s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s < 0 \u22a2 log (1 + s) / s < log (1 + p * s) / p / s ** haveI : (1 : \u211d) \u2208 Ioi 0 := zero_lt_one ** case inr.inr.a.inl s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s < 0 this : 1 \u2208 Ioi 0 \u22a2 log (1 + s) / s < log (1 + p * s) / p / s ** convert strictConcaveOn_log_Ioi.secant_strict_mono this hs2 hs1 hs4 hs3 _ using 1 ** case h.e'_3 s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s < 0 this : 1 \u2208 Ioi 0 \u22a2 log (1 + s) / s = (log (1 + s) - log 1) / (1 + s - 1) ** field_simp ** case h.e'_4 s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s < 0 this : 1 \u2208 Ioi 0 \u22a2 log (1 + p * s) / p / s = (log (1 + p * s) - log 1) / (1 + p * s - 1) ** field_simp ** case inr.inr.a.inl s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s < 0 this : 1 \u2208 Ioi 0 \u22a2 1 + p * s < 1 + s ** nlinarith ** case inr.inr.a.inr s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s > 0 \u22a2 log (1 + p * s) < log (1 + s) * p ** rw [\u2190 div_lt_iff hp, \u2190 div_lt_div_right hs'] ** case inr.inr.a.inr s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s > 0 \u22a2 log (1 + p * s) / p / s < log (1 + s) / s ** haveI : (1 : \u211d) \u2208 Ioi 0 := zero_lt_one ** case inr.inr.a.inr s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s > 0 this : 1 \u2208 Ioi 0 \u22a2 log (1 + p * s) / p / s < log (1 + s) / s ** convert strictConcaveOn_log_Ioi.secant_strict_mono this hs1 hs2 hs3 hs4 _ using 1 ** case h.e'_3 s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s > 0 this : 1 \u2208 Ioi 0 \u22a2 log (1 + p * s) / p / s = (log (1 + p * s) - log 1) / (1 + p * s - 1) ** field_simp ** case h.e'_4 s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s > 0 this : 1 \u2208 Ioi 0 \u22a2 log (1 + s) / s = (log (1 + s) - log 1) / (1 + s - 1) ** field_simp ** case inr.inr.a.inr s : \u211d hs\u271d : -1 \u2264 s hs'\u271d : s \u2260 0 p : \u211d hp\u271d : 1 < p hs : -1 < s hs1 : 0 < 1 + s hs2 : 0 < 1 + p * s hp : 0 < p hs3 : 1 + s \u2260 1 hs4 : 1 + p * s \u2260 1 hs' : s > 0 this : 1 \u2208 Ioi 0 \u22a2 1 + s < 1 + p * s ** nlinarith ** Qed", + "informal": "" + }, + { + "formal": "spectrum.smul_mem_smul_iff ** R : Type u A : Type v inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : Ring A inst\u271d : Algebra R A a : A s : R r : R\u02e3 \u22a2 r \u2022 s \u2208 \u03c3 (r \u2022 a) \u2194 s \u2208 \u03c3 a ** simp only [mem_iff, not_iff_not, Algebra.algebraMap_eq_smul_one, smul_assoc, \u2190 smul_sub,\n isUnit_smul_iff] ** Qed", + "informal": "" + }, + { + "formal": "Class.sInter_empty ** \u22a2 \u22c2\u2080 \u2205 = univ ** rw [sInter, classToCong_empty, Set.sInter_empty, univ] ** Qed", + "informal": "" + }, + { + "formal": "HasLineDerivAt.smul ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type u_3 inst\u271d\u00b9 : AddCommGroup E inst\u271d : Module \ud835\udd5c E f : E \u2192 F s : Set E x v : E f' : F h : HasLineDerivAt \ud835\udd5c f f' x v c : \ud835\udd5c \u22a2 HasLineDerivAt \ud835\udd5c f (c \u2022 f') x (c \u2022 v) ** simp only [\u2190 hasLineDerivWithinAt_univ] at h \u22a2 ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type u_3 inst\u271d\u00b9 : AddCommGroup E inst\u271d : Module \ud835\udd5c E f : E \u2192 F s : Set E x v : E f' : F c : \ud835\udd5c h : HasLineDerivWithinAt \ud835\udd5c f f' univ x v \u22a2 HasLineDerivWithinAt \ud835\udd5c f (c \u2022 f') univ x (c \u2022 v) ** exact HasLineDerivWithinAt.smul h c ** Qed", + "informal": "" + }, + { + "formal": "SimplexCategory.skeletal ** X Y : SimplexCategory x\u271d : IsIsomorphic X Y I : X \u2245 Y \u22a2 X = Y ** suffices Fintype.card (Fin (X.len + 1)) = Fintype.card (Fin (Y.len + 1)) by\n ext\n simpa ** X Y : SimplexCategory x\u271d : IsIsomorphic X Y I : X \u2245 Y \u22a2 Fintype.card (Fin (len X + 1)) = Fintype.card (Fin (len Y + 1)) ** apply Fintype.card_congr ** case f X Y : SimplexCategory x\u271d : IsIsomorphic X Y I : X \u2245 Y \u22a2 Fin (len X + 1) \u2243 Fin (len Y + 1) ** exact ((skeletalFunctor \u22d9 forget NonemptyFinLinOrd).mapIso I).toEquiv ** X Y : SimplexCategory x\u271d : IsIsomorphic X Y I : X \u2245 Y this : Fintype.card (Fin (len X + 1)) = Fintype.card (Fin (len Y + 1)) \u22a2 X = Y ** ext ** case a X Y : SimplexCategory x\u271d : IsIsomorphic X Y I : X \u2245 Y this : Fintype.card (Fin (len X + 1)) = Fintype.card (Fin (len Y + 1)) \u22a2 len X = len Y ** simpa ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_pow_mul_natDegree ** R : Type u S : Type v a b c d : R n\u271d m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] n : \u2115 \u22a2 coeff (p ^ n) (n * natDegree p) = leadingCoeff p ^ n ** induction' n with i hi ** case zero R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] \u22a2 coeff (p ^ Nat.zero) (Nat.zero * natDegree p) = leadingCoeff p ^ Nat.zero ** simp ** case succ R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i \u22a2 coeff (p ^ Nat.succ i) (Nat.succ i * natDegree p) = leadingCoeff p ^ Nat.succ i ** rw [pow_succ', pow_succ', Nat.succ_mul] ** case succ R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i \u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p ** by_cases hp1 : p.leadingCoeff ^ i = 0 ** case pos R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : leadingCoeff p ^ i = 0 \u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p ** rw [hp1, zero_mul] ** case pos R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : leadingCoeff p ^ i = 0 \u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0 ** by_cases hp2 : p ^ i = 0 ** case pos R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : leadingCoeff p ^ i = 0 hp2 : p ^ i = 0 \u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0 ** rw [hp2, zero_mul, coeff_zero] ** case neg R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : leadingCoeff p ^ i = 0 hp2 : \u00acp ^ i = 0 \u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0 ** apply coeff_eq_zero_of_natDegree_lt ** case neg.h R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : leadingCoeff p ^ i = 0 hp2 : \u00acp ^ i = 0 \u22a2 natDegree (p ^ i * p) < i * natDegree p + natDegree p ** 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 ** case neg.h R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : leadingCoeff p ^ i = 0 hp2 : \u00acp ^ i = 0 h1 : natDegree (p ^ i) < i * natDegree p \u22a2 natDegree (p ^ i * p) < i * natDegree p + natDegree p ** 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 _ ** R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : leadingCoeff p ^ i = 0 hp2 : \u00acp ^ i = 0 \u22a2 natDegree (p ^ i) < i * natDegree p ** refine lt_of_le_of_ne natDegree_pow_le fun h => hp2 ?_ ** R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : leadingCoeff p ^ i = 0 hp2 : \u00acp ^ i = 0 h : natDegree (p ^ i) = i * natDegree p \u22a2 p ^ i = 0 ** rw [\u2190 h, hp1] at hi ** R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (natDegree (p ^ i)) = 0 hp1 : leadingCoeff p ^ i = 0 hp2 : \u00acp ^ i = 0 h : natDegree (p ^ i) = i * natDegree p \u22a2 p ^ i = 0 ** exact leadingCoeff_eq_zero.mp hi ** case neg R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : \u00acleadingCoeff p ^ i = 0 \u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p ** rw [\u2190 natDegree_pow' hp1, \u2190 leadingCoeff_pow' hp1] ** case neg R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p\u271d q : R[X] \u03b9 : Type u_1 p : R[X] i : \u2115 hi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i hp1 : \u00acleadingCoeff p ^ i = 0 \u22a2 coeff (p ^ i * p) (natDegree (p ^ i) + natDegree p) = leadingCoeff (p ^ i) * leadingCoeff p ** exact coeff_mul_degree_add_degree _ _ ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.toFinsupp_X_pow ** R : Type u a b : R m n\u271d : \u2115 inst\u271d : Semiring R p q : R[X] n : \u2115 \u22a2 (X ^ n).toFinsupp = fun\u2080 | n => 1 ** rw [X_pow_eq_monomial, toFinsupp_monomial] ** Qed", + "informal": "" + }, + { + "formal": "Rat.uniformContinuous_abs ** \u03b5 : \u211d \u03b50 : \u03b5 > 0 a\u271d b\u271d : \u211a h : dist a\u271d b\u271d < \u03b5 \u22a2 dist |a\u271d| |b\u271d| \u2264 dist a\u271d b\u271d ** simpa [Rat.dist_eq] using abs_abs_sub_abs_le_abs_sub _ _ ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.centralMoment_one ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc \u22a2 centralMoment X 1 \u03bc = 0 ** by_cases h_int : Integrable X \u03bc ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : Integrable X \u22a2 centralMoment X 1 \u03bc = 0 ** rw [centralMoment_one' h_int] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : Integrable X \u22a2 (1 - ENNReal.toReal (\u2191\u2191\u03bc Set.univ)) * \u222b (x : \u03a9), X x \u2202\u03bc = 0 ** simp only [measure_univ, ENNReal.one_toReal, sub_self, zero_mul] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X \u22a2 centralMoment X 1 \u03bc = 0 ** simp only [centralMoment, Pi.sub_apply, pow_one] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X \u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0 ** 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 _) ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X this : \u00acIntegrable fun x => X x - integral \u03bc X \u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0 ** rw [integral_undef this] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X \u22a2 \u00acIntegrable fun x => X x - integral \u03bc X ** refine' fun h_sub => h_int _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X \u22a2 Integrable X ** have h_add : X = (fun x => X x - integral \u03bc X) + fun _ => integral \u03bc X := by ext1 x; simp ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X h_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X \u22a2 Integrable X ** rw [h_add] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X h_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X \u22a2 Integrable ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X) ** exact h_sub.add (integrable_const _) ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X \u22a2 X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X ** ext1 x ** case h \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X x : \u03a9 \u22a2 X x = ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X) x ** simp ** Qed", + "informal": "" + }, + { + "formal": "Disjoint.map ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 \u03b4 : Type u_6 inst\u271d\u2074 : Lattice \u03b1 inst\u271d\u00b3 : BoundedOrder \u03b1 inst\u271d\u00b2 : Lattice \u03b2 inst\u271d\u00b9 : BoundedOrder \u03b2 inst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 f : F a b : \u03b1 h : Disjoint a b \u22a2 Disjoint (\u2191f a) (\u2191f b) ** rw [disjoint_iff, \u2190 map_inf, h.eq_bot, map_bot] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.zpow_neg_one ** n' : Type u_1 inst\u271d\u00b2 : DecidableEq n' inst\u271d\u00b9 : Fintype n' R : Type u_2 inst\u271d : CommRing R A : M \u22a2 A ^ (-1) = A\u207b\u00b9 ** convert DivInvMonoid.zpow_neg' 0 A ** case h.e'_3.h.e'_3 n' : Type u_1 inst\u271d\u00b2 : DecidableEq n' inst\u271d\u00b9 : Fintype n' R : Type u_2 inst\u271d : CommRing R A : M \u22a2 A = DivInvMonoid.zpow (\u2191(Nat.succ 0)) A ** simp only [zpow_one, Int.ofNat_zero, Int.ofNat_succ, zpow_eq_pow, zero_add] ** Qed", + "informal": "" + }, + { + "formal": "Filter.HasBasis.equicontinuousAt_iff_left ** \u03b9 : Type u_1 \u03ba\u271d : Type u_2 X : Type u_3 Y : Type u_4 Z : Type u_5 \u03b1 : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 \ud835\udcd5 : Type u_9 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : TopologicalSpace Z inst\u271d\u00b2 : UniformSpace \u03b1 inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u03ba : Type u_10 p : \u03ba \u2192 Prop s : \u03ba \u2192 Set X F : \u03b9 \u2192 X \u2192 \u03b1 x\u2080 : X hX : HasBasis (\ud835\udcdd x\u2080) p s \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 ** rw [equicontinuousAt_iff_continuousAt, ContinuousAt,\n hX.tendsto_iff (UniformFun.hasBasis_nhds \u03b9 \u03b1 _)] ** \u03b9 : Type u_1 \u03ba\u271d : Type u_2 X : Type u_3 Y : Type u_4 Z : Type u_5 \u03b1 : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 \ud835\udcd5 : Type u_9 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : TopologicalSpace Z inst\u271d\u00b2 : UniformSpace \u03b1 inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u03ba : Type u_10 p : \u03ba \u2192 Prop s : \u03ba \u2192 Set X F : \u03b9 \u2192 X \u2192 \u03b1 x\u2080 : X hX : HasBasis (\ud835\udcdd x\u2080) p s \u22a2 (\u2200 (ib : Set (\u03b1 \u00d7 \u03b1)), ib \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 ia, p ia \u2227 \u2200 (x : X), x \u2208 s ia \u2192 (\u2191UniformFun.ofFun \u2218 Function.swap F) x \u2208 {g | ((\u2191UniformFun.ofFun \u2218 Function.swap F) x\u2080, g) \u2208 UniformFun.gen \u03b9 \u03b1 ib}) \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 ** simp only [Function.comp_apply, mem_setOf_eq, exists_prop] ** \u03b9 : Type u_1 \u03ba\u271d : Type u_2 X : Type u_3 Y : Type u_4 Z : Type u_5 \u03b1 : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 \ud835\udcd5 : Type u_9 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : TopologicalSpace Z inst\u271d\u00b2 : UniformSpace \u03b1 inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u03ba : Type u_10 p : \u03ba \u2192 Prop s : \u03ba \u2192 Set X F : \u03b9 \u2192 X \u2192 \u03b1 x\u2080 : X hX : HasBasis (\ud835\udcdd x\u2080) p s \u22a2 (\u2200 (ib : Set (\u03b1 \u00d7 \u03b1)), ib \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 ia, p ia \u2227 \u2200 (x : X), x \u2208 s ia \u2192 (\u2191UniformFun.ofFun (Function.swap F x\u2080), \u2191UniformFun.ofFun (Function.swap F x)) \u2208 UniformFun.gen \u03b9 \u03b1 ib) \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 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Sheaf.isSheaf_of_isLimit ** C : Type u inst\u271d\u00b3 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b2 : Category.{max v u, w} D K : Type z inst\u271d\u00b9 : SmallCategory K inst\u271d : HasLimitsOfShape K D F : K \u2964 Sheaf J D E : Cone (F \u22d9 sheafToPresheaf J D) hE : IsLimit E \u22a2 Presheaf.IsSheaf J E.pt ** rw [Presheaf.isSheaf_iff_multifork] ** C : Type u inst\u271d\u00b3 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b2 : Category.{max v u, w} D K : Type z inst\u271d\u00b9 : SmallCategory K inst\u271d : HasLimitsOfShape K D F : K \u2964 Sheaf J D E : Cone (F \u22d9 sheafToPresheaf J D) hE : IsLimit E \u22a2 \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S E.pt)) ** intro X S ** C : Type u inst\u271d\u00b3 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b2 : Category.{max v u, w} D K : Type z inst\u271d\u00b9 : SmallCategory K inst\u271d : HasLimitsOfShape K D F : K \u2964 Sheaf J D E : Cone (F \u22d9 sheafToPresheaf J D) hE : IsLimit E X : C S : GrothendieckTopology.Cover J X \u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S E.pt)) ** exact \u27e8isLimitMultiforkOfIsLimit _ _ hE _ _\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "isNoetherian_top_iff ** R : Type u_1 M : Type u_2 P : Type u_3 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid P inst\u271d\u00b9 : Module R M inst\u271d : Module R P \u22a2 IsNoetherian R { x // x \u2208 \u22a4 } \u2194 IsNoetherian R M ** constructor <;> intro h ** case mp R : Type u_1 M : Type u_2 P : Type u_3 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid P inst\u271d\u00b9 : Module R M inst\u271d : Module R P h : IsNoetherian R { x // x \u2208 \u22a4 } \u22a2 IsNoetherian R M ** exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (\u22a4 : Submodule R M) rfl) ** case mpr R : Type u_1 M : Type u_2 P : Type u_3 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid P inst\u271d\u00b9 : Module R M inst\u271d : Module R P h : IsNoetherian R M \u22a2 IsNoetherian R { x // x \u2208 \u22a4 } ** exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (\u22a4 : Submodule R M) rfl).symm ** Qed", + "informal": "" + }, + { + "formal": "Int.coe_leastOfBdd_eq ** P : \u2124 \u2192 Prop inst\u271d : DecidablePred P b b' : \u2124 Hb : \u2200 (z : \u2124), P z \u2192 b \u2264 z Hb' : \u2200 (z : \u2124), P z \u2192 b' \u2264 z Hinh : \u2203 z, P z \u22a2 \u2191(leastOfBdd b Hb Hinh) = \u2191(leastOfBdd b' Hb' Hinh) ** rcases leastOfBdd b Hb Hinh with \u27e8n, hn, h2n\u27e9 ** case mk.intro P : \u2124 \u2192 Prop inst\u271d : DecidablePred P b b' : \u2124 Hb : \u2200 (z : \u2124), P z \u2192 b \u2264 z Hb' : \u2200 (z : \u2124), P z \u2192 b' \u2264 z Hinh : \u2203 z, P z n : \u2124 hn : P n h2n : \u2200 (z : \u2124), P z \u2192 n \u2264 z \u22a2 \u2191{ val := n, property := (_ : P n \u2227 \u2200 (z : \u2124), P z \u2192 n \u2264 z) } = \u2191(leastOfBdd b' Hb' Hinh) ** rcases leastOfBdd b' Hb' Hinh with \u27e8n', hn', h2n'\u27e9 ** case mk.intro.mk.intro P : \u2124 \u2192 Prop inst\u271d : DecidablePred P b b' : \u2124 Hb : \u2200 (z : \u2124), P z \u2192 b \u2264 z Hb' : \u2200 (z : \u2124), P z \u2192 b' \u2264 z Hinh : \u2203 z, P z n : \u2124 hn : P n h2n : \u2200 (z : \u2124), P z \u2192 n \u2264 z n' : \u2124 hn' : P n' h2n' : \u2200 (z : \u2124), P z \u2192 n' \u2264 z \u22a2 \u2191{ val := n, property := (_ : P n \u2227 \u2200 (z : \u2124), P z \u2192 n \u2264 z) } = \u2191{ val := n', property := (_ : P n' \u2227 \u2200 (z : \u2124), P z \u2192 n' \u2264 z) } ** exact le_antisymm (h2n _ hn') (h2n' _ hn) ** Qed", + "informal": "" + }, + { + "formal": "Holor.slice_eq ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 x y : Holor \u03b1 (d :: ds) h : slice x = slice y t : HolorIndex (d :: ds) i : \u2115 is : List \u2115 hiis : \u2191t = i :: is \u22a2 Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds) ** rw [\u2190 hiis] ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 x y : Holor \u03b1 (d :: ds) h : slice x = slice y t : HolorIndex (d :: ds) i : \u2115 is : List \u2115 hiis : \u2191t = i :: is \u22a2 Forall\u2082 (fun x x_1 => x < x_1) (\u2191t) (d :: ds) ** exact t.2 ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 x y : Holor \u03b1 (d :: ds) h : slice x = slice y t : HolorIndex (d :: ds) i : \u2115 is : List \u2115 hiis : \u2191t = i :: is hiisdds : Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds) hid : i < d hisds : Forall\u2082 (fun x x_1 => x < x_1) is ds \u22a2 slice x i hid { val := is, property := hisds } = slice y i hid { val := is, property := hisds } ** rw [h] ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.mem_sup ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 s \u2294 t \u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x ** rw [sup_eq_closure] at h ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 closure (\u2191s \u222a \u2191t) \u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x ** refine Subgroup.closure_induction h ?_ ?_ ?_ ?_ ** case refine_1 G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 closure (\u2191s \u222a \u2191t) \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 ** rintro y (h | h) ** case refine_1.inl G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h\u271d : x \u2208 closure (\u2191s \u222a \u2191t) y : C h : y \u2208 \u2191s \u22a2 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = y ** exact \u27e8y, h, 1, t.one_mem, by simp\u27e9 ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h\u271d : x \u2208 closure (\u2191s \u222a \u2191t) y : C h : y \u2208 \u2191s \u22a2 y * 1 = y ** simp ** case refine_1.inr G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h\u271d : x \u2208 closure (\u2191s \u222a \u2191t) y : C h : y \u2208 \u2191t \u22a2 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = y ** exact \u27e81, s.one_mem, y, h, by simp\u27e9 ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h\u271d : x \u2208 closure (\u2191s \u222a \u2191t) y : C h : y \u2208 \u2191t \u22a2 1 * y = y ** simp ** case refine_2 G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 closure (\u2191s \u222a \u2191t) \u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = 1 ** exact \u27e81, s.one_mem, 1, \u27e8t.one_mem, mul_one 1\u27e9\u27e9 ** case refine_3 G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 closure (\u2191s \u222a \u2191t) \u22a2 \u2200 (x y : C), (\u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x) \u2192 (\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 ** rintro _ _ \u27e8y\u2081, hy\u2081, z\u2081, hz\u2081, rfl\u27e9 \u27e8y\u2082, hy\u2082, z\u2082, hz\u2082, rfl\u27e9 ** case refine_3.intro.intro.intro.intro.intro.intro.intro.intro G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 closure (\u2191s \u222a \u2191t) y\u2081 : C hy\u2081 : y\u2081 \u2208 s z\u2081 : C hz\u2081 : z\u2081 \u2208 t y\u2082 : C hy\u2082 : y\u2082 \u2208 s z\u2082 : C hz\u2082 : z\u2082 \u2208 t \u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = y\u2081 * z\u2081 * (y\u2082 * z\u2082) ** exact \u27e8_, mul_mem hy\u2081 hy\u2082, _, mul_mem hz\u2081 hz\u2082, by simp [mul_assoc, mul_left_comm]\u27e9 ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 closure (\u2191s \u222a \u2191t) y\u2081 : C hy\u2081 : y\u2081 \u2208 s z\u2081 : C hz\u2081 : z\u2081 \u2208 t y\u2082 : C hy\u2082 : y\u2082 \u2208 s z\u2082 : C hz\u2082 : z\u2082 \u2208 t \u22a2 y\u2081 * y\u2082 * (z\u2081 * z\u2082) = y\u2081 * z\u2081 * (y\u2082 * z\u2082) ** simp [mul_assoc, mul_left_comm] ** case refine_4 G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 closure (\u2191s \u222a \u2191t) \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 ** rintro _ \u27e8y, hy, z, hz, rfl\u27e9 ** case refine_4.intro.intro.intro.intro G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C h : x \u2208 closure (\u2191s \u222a \u2191t) y : C hy : y \u2208 s z : C hz : z \u2208 t \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 ** exact \u27e8_, inv_mem hy, _, inv_mem hz, mul_comm z y \u25b8 (mul_inv_rev z y).symm\u27e9 ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C x : C \u22a2 (\u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x) \u2192 x \u2208 s \u2294 t ** rintro \u27e8y, hy, z, hz, rfl\u27e9 ** case intro.intro.intro.intro G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A N : Type u_5 inst\u271d\u00b9 : Group N C : Type u_6 inst\u271d : CommGroup C s t : Subgroup C y : C hy : y \u2208 s z : C hz : z \u2208 t \u22a2 y * z \u2208 s \u2294 t ** exact mul_mem_sup hy hz ** Qed", + "informal": "" + }, + { + "formal": "abs_le_abs_of_nonpos ** \u03b1 : Type u_1 inst\u271d\u00b3 : AddGroup \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 a b c : \u03b1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 ha : a \u2264 0 hab : b \u2264 a \u22a2 |a| \u2264 |b| ** rw [abs_of_nonpos ha, abs_of_nonpos (hab.trans ha)] ** \u03b1 : Type u_1 inst\u271d\u00b3 : AddGroup \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 a b c : \u03b1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 ha : a \u2264 0 hab : b \u2264 a \u22a2 -a \u2264 -b ** exact neg_le_neg_iff.mpr hab ** Qed", + "informal": "" + }, + { + "formal": "Nat.dvd_left_iff_eq ** a b m\u271d n\u271d\u00b9 k m n\u271d : \u2115 h : m = n\u271d n : \u2115 \u22a2 n \u2223 m \u2194 n \u2223 n\u271d ** rw [h] ** Qed", + "informal": "" + }, + { + "formal": "EReal.coe_coe_sign ** x : SignType \u22a2 \u2191\u2191x = \u2191x ** cases x <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "PadicInt.lift_unique ** p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 g : R \u2192+* \u2124_[p] hg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n \u22a2 lift f_compat = g ** ext1 r ** case a p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 g : R \u2192+* \u2124_[p] hg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n r : R \u22a2 \u2191(lift f_compat) r = \u2191g r ** apply eq_of_forall_dist_le ** case a.h p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 g : R \u2192+* \u2124_[p] hg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n r : R \u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5 ** intro \u03b5 h\u03b5 ** case a.h p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 g : R \u2192+* \u2124_[p] hg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n r : R \u03b5 : \u211d h\u03b5 : \u03b5 > 0 \u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5 ** obtain \u27e8n, hn\u27e9 := exists_pow_neg_lt p h\u03b5 ** case a.h.intro p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 g : R \u2192+* \u2124_[p] hg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n r : R \u03b5 : \u211d h\u03b5 : \u03b5 > 0 n : \u2115 hn : \u2191p ^ (-\u2191n) < \u03b5 \u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5 ** apply le_trans _ (le_of_lt hn) ** p : \u2115 hp_prime : Fact (Nat.Prime p) R : Type u_1 inst\u271d : NonAssocSemiring R f : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k) f_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 g : R \u2192+* \u2124_[p] hg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n r : R \u03b5 : \u211d h\u03b5 : \u03b5 > 0 n : \u2115 hn : \u2191p ^ (-\u2191n) < \u03b5 \u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u2191p ^ (-\u2191n) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "RegularExpression.one_rmatch_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch 1 x = true \u2194 x = [] ** induction x <;> simp [rmatch, matchEpsilon, *] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.RanIsSheafOfCoverLifting.gluedSection_isAmalgamation ** C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x \u22a2 IsAmalgamation x (gluedSection hu \u2131 hS hx) ** intro V fV hV ** C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV \u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV.op (gluedSection hu \u2131 hS hx) = x fV hV ** refine limit.hom_ext (\u03bb (W : StructuredArrow (op V) G.op) => ?_) ** C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV W : StructuredArrow (op V) G.op \u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV.op (gluedSection hu \u2131 hS hx) \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W = x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W ** simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection] ** C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV W : StructuredArrow (op V) G.op \u22a2 limit.lift (StructuredArrow.map fV.op \u22d9 Ran.diagram G.op \u2131.val (op U)) (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)) \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W = x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W ** erw [limit.lift_\u03c0] ** C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV W : StructuredArrow (op V) G.op \u22a2 (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)).\u03c0.app W = x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W ** symm ** C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV W : StructuredArrow (op V) G.op \u22a2 x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W = (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)).\u03c0.app W ** convert helper hu \u2131 hS hx _ (x fV hV) _ _ using 1 ** case convert_3 C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV W : StructuredArrow (op V) G.op \u22a2 \u2200 {V' : C} {fV_1 : G.obj V' \u27f6 V} (hV_1 : S.arrows (fV_1 \u226b fV)), x fV hV \u226b ((ran G.op).obj \u2131.val).map fV_1.op = x (fV_1 \u226b fV) hV_1 ** intro V' fV' hV' ** case convert_3 C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV W : StructuredArrow (op V) G.op V' : C fV' : G.obj V' \u27f6 V hV' : S.arrows (fV' \u226b fV) \u22a2 x fV hV \u226b ((ran G.op).obj \u2131.val).map fV'.op = x (fV' \u226b fV) hV' ** convert hx fV' (\ud835\udfd9 _) hV hV' (by rw [Category.id_comp]) ** case h.e'_3.h C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV W : StructuredArrow (op V) G.op V' : C fV' : G.obj V' \u27f6 V hV' : S.arrows (fV' \u226b fV) e_1\u271d : (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')) \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') ** simp only [op_id, FunctorToTypes.map_id_apply] ** C D : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : Category.{v, u} D A : Type w inst\u271d\u00b9 : Category.{max u v, w} A inst\u271d : HasLimits A J : GrothendieckTopology C K : GrothendieckTopology D G : C \u2964 D hu : CoverLifting J K G \u2131 : Sheaf J A X : A U : D S : Sieve U hS : S \u2208 GrothendieckTopology.sieves K U x : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows hx : Compatible x V : D fV : V \u27f6 U hV : S.arrows fV W : StructuredArrow (op V) G.op V' : C fV' : G.obj V' \u27f6 V hV' : S.arrows (fV' \u226b fV) \u22a2 fV' \u226b fV = \ud835\udfd9 (G.obj V') \u226b fV' \u226b fV ** rw [Category.id_comp] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_zero_eq_aeval_zero ** R : Type u S : Type v T : Type w A : Type z A' : Type u_1 B' : Type u_2 a b : R n : \u2115 inst\u271d\u2076 : CommSemiring A' inst\u271d\u2075 : Semiring B' inst\u271d\u2074 : CommSemiring R p\u271d q : R[X] inst\u271d\u00b3 : Semiring A inst\u271d\u00b2 : Algebra R A B : Type u_3 inst\u271d\u00b9 : Semiring B inst\u271d : Algebra R B x : A p : R[X] \u22a2 coeff p 0 = \u2191(aeval 0) p ** simp [coeff_zero_eq_eval_zero] ** Qed", + "informal": "" + }, + { + "formal": "nonempty_sections_of_finite_cofiltered_system.init ** J : Type u inst\u271d\u00b9 : SmallCategory J inst\u271d : IsCofilteredOrEmpty J F : J \u2964 Type u hf : \u2200 (j : J), Finite (F.obj j) hne : \u2200 (j : J), _root_.Nonempty (F.obj j) \u22a2 Set.Nonempty (Functor.sections F) ** let F' : J \u2964 TopCat := F \u22d9 TopCat.discrete ** J : Type u inst\u271d\u00b9 : SmallCategory J inst\u271d : IsCofilteredOrEmpty J F : J \u2964 Type u hf : \u2200 (j : J), Finite (F.obj j) hne : \u2200 (j : J), _root_.Nonempty (F.obj j) F' : J \u2964 TopCat := F \u22d9 TopCat.discrete \u22a2 Set.Nonempty (Functor.sections F) ** haveI : \u2200 j, DiscreteTopology (F'.obj j) := fun _ => \u27e8rfl\u27e9 ** J : Type u inst\u271d\u00b9 : SmallCategory J inst\u271d : IsCofilteredOrEmpty J F : J \u2964 Type u hf : \u2200 (j : J), Finite (F.obj j) hne : \u2200 (j : J), _root_.Nonempty (F.obj j) F' : J \u2964 TopCat := F \u22d9 TopCat.discrete this : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j) \u22a2 Set.Nonempty (Functor.sections F) ** haveI : \u2200 j, Finite (F'.obj j) := hf ** J : Type u inst\u271d\u00b9 : SmallCategory J inst\u271d : IsCofilteredOrEmpty J F : J \u2964 Type u hf : \u2200 (j : J), Finite (F.obj j) hne : \u2200 (j : J), _root_.Nonempty (F.obj j) F' : J \u2964 TopCat := F \u22d9 TopCat.discrete this\u271d : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j) this : \u2200 (j : J), Finite \u2191(F'.obj j) \u22a2 Set.Nonempty (Functor.sections F) ** haveI : \u2200 j, Nonempty (F'.obj j) := hne ** J : Type u inst\u271d\u00b9 : SmallCategory J inst\u271d : IsCofilteredOrEmpty J F : J \u2964 Type u hf : \u2200 (j : J), Finite (F.obj j) hne : \u2200 (j : J), _root_.Nonempty (F.obj j) F' : J \u2964 TopCat := F \u22d9 TopCat.discrete this\u271d\u00b9 : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j) this\u271d : \u2200 (j : J), Finite \u2191(F'.obj j) this : \u2200 (j : J), _root_.Nonempty \u2191(F'.obj j) \u22a2 Set.Nonempty (Functor.sections F) ** obtain \u27e8\u27e8u, hu\u27e9\u27e9 := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} F' ** case intro.mk J : Type u inst\u271d\u00b9 : SmallCategory J inst\u271d : IsCofilteredOrEmpty J F : J \u2964 Type u hf : \u2200 (j : J), Finite (F.obj j) hne : \u2200 (j : J), _root_.Nonempty (F.obj j) F' : J \u2964 TopCat := F \u22d9 TopCat.discrete this\u271d\u00b9 : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j) this\u271d : \u2200 (j : J), Finite \u2191(F'.obj j) this : \u2200 (j : J), _root_.Nonempty \u2191(F'.obj j) u : (j : J) \u2192 \u2191(F'.obj j) hu : u \u2208 {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F'.map f) (u i) = u j} \u22a2 Set.Nonempty (Functor.sections F) ** exact \u27e8u, hu\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03b2 : Type u_5 h\u03b1 : IsEmpty \u03b1 f : \u03b1 \u2192\u209b \u03b2 \u22a2 SimpleFunc.range f = \u2205 ** rw [\u2190 Finset.not_nonempty_iff_eq_empty] ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03b2 : Type u_5 h\u03b1 : IsEmpty \u03b1 f : \u03b1 \u2192\u209b \u03b2 \u22a2 \u00acFinset.Nonempty (SimpleFunc.range f) ** by_contra h ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03b2 : Type u_5 h\u03b1 : IsEmpty \u03b1 f : \u03b1 \u2192\u209b \u03b2 h : Finset.Nonempty (SimpleFunc.range f) \u22a2 False ** obtain \u27e8y, hy_mem\u27e9 := h ** case intro \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03b2 : Type u_5 h\u03b1 : IsEmpty \u03b1 f : \u03b1 \u2192\u209b \u03b2 y : \u03b2 hy_mem : y \u2208 SimpleFunc.range f \u22a2 False ** rw [SimpleFunc.mem_range, Set.mem_range] at hy_mem ** case intro \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03b2 : Type u_5 h\u03b1 : IsEmpty \u03b1 f : \u03b1 \u2192\u209b \u03b2 y : \u03b2 hy_mem : \u2203 y_1, \u2191f y_1 = y \u22a2 False ** obtain \u27e8x, hxy\u27e9 := hy_mem ** case intro.intro \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03b2 : Type u_5 h\u03b1 : IsEmpty \u03b1 f : \u03b1 \u2192\u209b \u03b2 y : \u03b2 x : \u03b1 hxy : \u2191f x = y \u22a2 False ** rw [isEmpty_iff] at h\u03b1 ** case intro.intro \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03b2 : Type u_5 h\u03b1 : \u03b1 \u2192 False f : \u03b1 \u2192\u209b \u03b2 y : \u03b2 x : \u03b1 hxy : \u2191f x = y \u22a2 False ** exact h\u03b1 x ** Qed", + "informal": "" + }, + { + "formal": "Nat.div_left_inj ** m n k a b d : \u2115 hda : d \u2223 a hdb : d \u2223 b \u22a2 a / d = b / d \u2194 a = b ** refine \u27e8fun h => ?_, congr_arg fun n => n / d\u27e9 ** m n k a b d : \u2115 hda : d \u2223 a hdb : d \u2223 b h : a / d = b / d \u22a2 a = b ** rw [\u2190 Nat.mul_div_cancel' hda, \u2190 Nat.mul_div_cancel' hdb, h] ** Qed", + "informal": "" + }, + { + "formal": "ZMod.nat_coe_zmod_eq_iff ** p n : \u2115 z : ZMod p inst\u271d : NeZero p \u22a2 \u2191n = z \u2194 \u2203 k, n = val z + p * k ** constructor ** case mp p n : \u2115 z : ZMod p inst\u271d : NeZero p \u22a2 \u2191n = z \u2192 \u2203 k, n = val z + p * k ** rintro rfl ** case mp p n : \u2115 inst\u271d : NeZero p \u22a2 \u2203 k, n = val \u2191n + p * k ** refine' \u27e8n / p, _\u27e9 ** case mp p n : \u2115 inst\u271d : NeZero p \u22a2 n = val \u2191n + p * (n / p) ** rw [val_nat_cast, Nat.mod_add_div] ** case mpr p n : \u2115 z : ZMod p inst\u271d : NeZero p \u22a2 (\u2203 k, n = val z + p * k) \u2192 \u2191n = z ** rintro \u27e8k, rfl\u27e9 ** case mpr.intro p : \u2115 z : ZMod p inst\u271d : NeZero p k : \u2115 \u22a2 \u2191(val z + p * k) = z ** rw [Nat.cast_add, nat_cast_zmod_val, Nat.cast_mul, nat_cast_self, zero_mul,\n add_zero] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.reverse_natDegree_le ** R : Type u_1 inst\u271d : Semiring R f\u271d f : R[X] \u22a2 natDegree (reverse f) \u2264 natDegree f ** rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero] ** R : Type u_1 inst\u271d : Semiring R f\u271d f : R[X] \u22a2 \u2200 (m : \u2115), \u2191(natDegree f) < \u2191m \u2192 coeff (reverse f) m = 0 ** intro n hn ** R : Type u_1 inst\u271d : Semiring R f\u271d f : R[X] n : \u2115 hn : \u2191(natDegree f) < \u2191n \u22a2 coeff (reverse f) n = 0 ** rw [Nat.cast_lt] at hn ** R : Type u_1 inst\u271d : Semiring R f\u271d f : R[X] n : \u2115 hn : natDegree f < n \u22a2 coeff (reverse f) n = 0 ** rw [coeff_reverse, revAt, Function.Embedding.coeFn_mk, if_neg (not_le_of_gt hn),\n coeff_eq_zero_of_natDegree_lt hn] ** Qed", + "informal": "" + }, + { + "formal": "StrictMonoOn.exists_slope_lt_deriv ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) \u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a ** by_cases h : \u2200 w \u2208 Ioo x y, deriv f w \u2260 0 ** case pos E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) h : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0 \u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a ** apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h ** case neg E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) h : \u00ac\u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0 \u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a ** push_neg at h ** case neg E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) h : \u2203 w, w \u2208 Ioo x y \u2227 deriv f w = 0 \u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a ** rcases h with \u27e8w, \u27e8hxw, hwy\u27e9, hw\u27e9 ** case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w b : \u211d hb : (f y - f w) / (y - w) < deriv f b hwb : w < b hby : b < y \u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a ** refine' \u27e8b, \u27e8hxw.trans hwb, hby\u27e9, _\u27e9 ** case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w b : \u211d hb : (f y - f w) / (y - w) < deriv f b hwb : w < b hby : b < y \u22a2 (f y - f x) / (y - x) < deriv f b ** simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb \u22a2 ** case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d hxa : x < a haw : a < w b : \u211d hwb : w < b hby : b < y ha : f w - f x < deriv f a * (w - x) hb : f y - f w < deriv f b * (y - w) this : deriv f a * (w - x) < deriv f b * (w - x) \u22a2 f y - f x < deriv f b * (y - x) ** linarith ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y \u22a2 \u2203 a, a \u2208 Ioo x w \u2227 (f w - f x) / (w - x) < deriv f a ** apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _ ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y \u22a2 ContinuousOn (fun w => f w) (Icc x w) ** exact hf.mono (Icc_subset_Icc le_rfl hwy.le) ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y \u22a2 StrictMonoOn (deriv fun w => f w) (Ioo x w) ** exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le) ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y \u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo x w \u2192 deriv (fun w => f w) w_1 \u2260 0 ** intro z hz ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y z : \u211d hz : z \u2208 Ioo x w \u22a2 deriv (fun w => f w) z \u2260 0 ** rw [\u2190 hw] ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y z : \u211d hz : z \u2208 Ioo x w \u22a2 deriv (fun w => f w) z \u2260 deriv f w ** apply ne_of_lt ** case h E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y z : \u211d hz : z \u2208 Ioo x w \u22a2 deriv (fun w => f w) z < deriv f w ** exact hf'_mono \u27e8hz.1, hz.2.trans hwy\u27e9 \u27e8hxw, hwy\u27e9 hz.2 ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w \u22a2 \u2203 b, b \u2208 Ioo w y \u2227 (f y - f w) / (y - w) < deriv f b ** apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _ ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w \u22a2 ContinuousOn (fun {y} => f y) (Icc w y) ** refine' hf.mono (Icc_subset_Icc hxw.le le_rfl) ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w \u22a2 StrictMonoOn (deriv fun {y} => f y) (Ioo w y) ** exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl) ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w \u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo w y \u2192 deriv (fun {y} => f y) w_1 \u2260 0 ** intro z hz ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w z : \u211d hz : z \u2208 Ioo w y \u22a2 deriv (fun {y} => f y) z \u2260 0 ** rw [\u2190 hw] ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w z : \u211d hz : z \u2208 Ioo w y \u22a2 deriv (fun {y} => f y) z \u2260 deriv f w ** apply ne_of_gt ** case h E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d ha : (f w - f x) / (w - x) < deriv f a hxa : x < a haw : a < w z : \u211d hz : z \u2208 Ioo w y \u22a2 deriv f w < deriv (fun {y} => f y) z ** exact hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hz.1, hz.2\u27e9 hz.1 ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d hxa : x < a haw : a < w b : \u211d hwb : w < b hby : b < y ha : f w - f x < deriv f a * (w - x) hb : f y - f w < deriv f b * (y - w) \u22a2 deriv f a * (w - x) < deriv f b * (w - x) ** apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _ ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d hxa : x < a haw : a < w b : \u211d hwb : w < b hby : b < y ha : f w - f x < deriv f a * (w - x) hb : f y - f w < deriv f b * (y - w) \u22a2 deriv f a < deriv f b ** exact hf'_mono \u27e8hxa, haw.trans hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 (haw.trans hwb) ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d hxa : x < a haw : a < w b : \u211d hwb : w < b hby : b < y ha : f w - f x < deriv f a * (w - x) hb : f y - f w < deriv f b * (y - w) \u22a2 0 \u2264 deriv f b ** rw [\u2190 hw] ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E F : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F x y : \u211d f : \u211d \u2192 \u211d hf : ContinuousOn f (Icc x y) hxy : x < y hf'_mono : StrictMonoOn (deriv f) (Ioo x y) w : \u211d hw : deriv f w = 0 hxw : x < w hwy : w < y a : \u211d hxa : x < a haw : a < w b : \u211d hwb : w < b hby : b < y ha : f w - f x < deriv f a * (w - x) hb : f y - f w < deriv f b * (y - w) \u22a2 deriv f w \u2264 deriv f b ** exact (hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 hwb).le ** Qed", + "informal": "" + }, + { + "formal": "List.mem_range' ** m s step : Nat \u22a2 m \u2208 range' s 0 step \u2194 \u2203 i, i < 0 \u2227 m = s + step * i ** simp [range', Nat.not_lt_zero] ** m s step n : Nat \u22a2 m \u2208 range' s (n + 1) step \u2194 \u2203 i, i < n + 1 \u2227 m = s + step * i ** 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] ** m s step n : Nat h : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n \u22a2 m \u2208 range' s (n + 1) step \u2194 \u2203 i, i < n + 1 \u2227 m = s + step * i ** simp [range', mem_range', Nat.lt_succ, h] ** m s step n : Nat h : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < 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 ** simp only [\u2190 exists_and_right, and_assoc] ** m s step n : Nat h : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < 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 ** rw [exists_comm] ** m s step n : Nat h : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < 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 ** simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm] ** m s step n i : Nat \u22a2 i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n ** cases i <;> simp [Nat.succ_le] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Arrow.iso_w ** T : Type u inst\u271d : Category.{v, u} T f g : Arrow T e : f \u2245 g \u22a2 g.hom = e.inv.left \u226b f.hom \u226b e.hom.right ** have eq := Arrow.hom.congr_right e.inv_hom_id ** T : Type u inst\u271d : Category.{v, u} T f g : Arrow T e : f \u2245 g eq : (e.inv \u226b e.hom).right = (\ud835\udfd9 g).right \u22a2 g.hom = e.inv.left \u226b f.hom \u226b e.hom.right ** rw [Arrow.comp_right, Arrow.id_right] at eq ** T : Type u inst\u271d : Category.{v, u} T f g : Arrow T e : f \u2245 g eq : e.inv.right \u226b e.hom.right = \ud835\udfd9 g.right \u22a2 g.hom = e.inv.left \u226b f.hom \u226b e.hom.right ** erw [Arrow.w_assoc, eq, Category.comp_id] ** Qed", + "informal": "" + }, + { + "formal": "Finset.centerMass_eq_of_sum_1 ** R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hw : \u2211 i in t, w i = 1 \u22a2 centerMass t w z = \u2211 i in t, w i \u2022 z i ** simp only [Finset.centerMass, hw, inv_one, one_smul] ** Qed", + "informal": "" + }, + { + "formal": "Set.bounded_le_inter_lt ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop s t : Set \u03b1 inst\u271d : LinearOrder \u03b1 a : \u03b1 \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 ** simp_rw [\u2190 not_le, bounded_le_inter_not_le] ** Qed", + "informal": "" + }, + { + "formal": "sInf_inv ** \u03b1 : Type u_1 inst\u271d\u00b3 : CompleteLattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 s\u271d t s : Set \u03b1 \u22a2 sInf s\u207b\u00b9 = (sSup s)\u207b\u00b9 ** rw [\u2190 image_inv, sInf_image] ** \u03b1 : Type u_1 inst\u271d\u00b3 : CompleteLattice \u03b1 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 inst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1 s\u271d t s : Set \u03b1 \u22a2 \u2a05 a \u2208 s, a\u207b\u00b9 = (sSup s)\u207b\u00b9 ** exact ((OrderIso.inv \u03b1).map_sSup _).symm ** Qed", + "informal": "" + }, + { + "formal": "BoxIntegral.IntegrationParams.toFilteriUnion_congr ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 I\u271d J : Box \u03b9 c c\u2081 c\u2082 : \u211d\u22650 r r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) \u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\u271d l\u271d l\u2081 l\u2082 : IntegrationParams I : Box \u03b9 l : IntegrationParams \u03c0\u2081 \u03c0\u2082 : Prepartition I h : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082 \u22a2 toFilteriUnion l I \u03c0\u2081 = toFilteriUnion l I \u03c0\u2082 ** simp only [toFilteriUnion, toFilterDistortioniUnion, h] ** Qed", + "informal": "" + }, + { + "formal": "Orientation.norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V hd2 : Fact (finrank \u211d V = 2) o : Orientation \u211d V (Fin 2) x y : V h : oangle o x y = \u2191(\u03c0 / 2) \u22a2 \u2016y\u2016 / Real.Angle.cos (oangle o y (y - x)) = \u2016y - x\u2016 ** 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] ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V hd2 : Fact (finrank \u211d V = 2) o : Orientation \u211d V (Fin 2) x y : V h : oangle o x y = \u2191(\u03c0 / 2) hs : Real.Angle.sign (oangle o y (y - x)) = 1 \u22a2 \u2016y\u2016 / Real.Angle.cos (oangle o y (y - x)) = \u2016y - x\u2016 ** 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))] ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V hd2 : Fact (finrank \u211d V = 2) o : Orientation \u211d V (Fin 2) x y : V h : oangle o x y = \u2191(\u03c0 / 2) \u22a2 Real.Angle.sign (oangle o y (y - x)) = 1 ** rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two] ** Qed", + "informal": "" + }, + { + "formal": "Subtype.preirreducibleSpace ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 \u03c0 : \u03b9 \u2192 Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t s : Set \u03b1 h : IsPreirreducible s \u22a2 IsPreirreducible univ ** rintro _ _ \u27e8u, hu, rfl\u27e9 \u27e8v, hv, rfl\u27e9 \u27e8\u27e8x, hxs\u27e9, -, hxu\u27e9 \u27e8\u27e8y, hys\u27e9, -, hyv\u27e9 ** case intro.intro.intro.intro.intro.mk.intro.intro.mk.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 \u03c0 : \u03b9 \u2192 Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t s : Set \u03b1 h : IsPreirreducible s u : Set \u03b1 hu : IsOpen u v : Set \u03b1 hv : IsOpen v x : \u03b1 hxs : x \u2208 s hxu : { val := x, property := hxs } \u2208 val \u207b\u00b9' u y : \u03b1 hys : y \u2208 s hyv : { val := y, property := hys } \u2208 val \u207b\u00b9' v \u22a2 Set.Nonempty (univ \u2229 (val \u207b\u00b9' u \u2229 val \u207b\u00b9' v)) ** rcases h u v hu hv \u27e8x, hxs, hxu\u27e9 \u27e8y, hys, hyv\u27e9 with \u27e8z, hzs, \u27e8hzu, hzv\u27e9\u27e9 ** case intro.intro.intro.intro.intro.mk.intro.intro.mk.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 \u03c0 : \u03b9 \u2192 Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 s\u271d t s : Set \u03b1 h : IsPreirreducible s u : Set \u03b1 hu : IsOpen u v : Set \u03b1 hv : IsOpen v x : \u03b1 hxs : x \u2208 s hxu : { val := x, property := hxs } \u2208 val \u207b\u00b9' u y : \u03b1 hys : y \u2208 s hyv : { val := y, property := hys } \u2208 val \u207b\u00b9' v z : \u03b1 hzs : z \u2208 s hzu : z \u2208 u hzv : z \u2208 v \u22a2 Set.Nonempty (univ \u2229 (val \u207b\u00b9' u \u2229 val \u207b\u00b9' v)) ** exact \u27e8\u27e8z, hzs\u27e9, \u27e8Set.mem_univ _, \u27e8hzu, hzv\u27e9\u27e9\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Subobject.mk_eq_of_comm ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C X\u271d Y Z : C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D B A : C X : Subobject B f : A \u27f6 B inst\u271d : Mono f i : A \u2245 underlying.obj X w : i.hom \u226b arrow X = f \u22a2 i.symm.hom \u226b f = arrow X ** rw [Iso.symm_hom, Iso.inv_comp_eq, w] ** Qed", + "informal": "" + }, + { + "formal": "PNat.Coprime.coprime_dvd_left ** m k n : \u2115+ \u22a2 m \u2223 k \u2192 Coprime k n \u2192 Coprime m n ** rw [dvd_iff] ** m k n : \u2115+ \u22a2 \u2191m \u2223 \u2191k \u2192 Coprime k n \u2192 Coprime m n ** repeat' rw [\u2190 coprime_coe] ** m k n : \u2115+ \u22a2 \u2191m \u2223 \u2191k \u2192 Nat.Coprime \u2191k \u2191n \u2192 Nat.Coprime \u2191m \u2191n ** apply Nat.Coprime.coprime_dvd_left ** m k n : \u2115+ \u22a2 \u2191m \u2223 \u2191k \u2192 Nat.Coprime \u2191k \u2191n \u2192 Coprime m n ** rw [\u2190 coprime_coe] ** Qed", + "informal": "" + }, + { + "formal": "norm_sup_sub_sup_le_add_norm ** \u03b1 : Type u_1 inst\u271d : NormedLatticeAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 \u2016a \u2294 b - c \u2294 d\u2016 \u2264 \u2016a - c\u2016 + \u2016b - d\u2016 ** rw [\u2190 norm_abs_eq_norm (a - c), \u2190 norm_abs_eq_norm (b - d)] ** \u03b1 : Type u_1 inst\u271d : NormedLatticeAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 \u2016a \u2294 b - c \u2294 d\u2016 \u2264 \u2016|a - c|\u2016 + \u2016|b - d|\u2016 ** refine' le_trans (solid _) (norm_add_le |a - c| |b - d|) ** \u03b1 : Type u_1 inst\u271d : NormedLatticeAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 |a \u2294 b - c \u2294 d| \u2264 ||a - c| + |b - d|| ** rw [abs_of_nonneg (|a - c| + |b - d|) (add_nonneg (abs_nonneg (a - c)) (abs_nonneg (b - d)))] ** \u03b1 : Type u_1 inst\u271d : NormedLatticeAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 |a \u2294 b - c \u2294 d| = |a \u2294 b - c \u2294 b + (c \u2294 b - c \u2294 d)| ** rw [sub_add_sub_cancel] ** \u03b1 : Type u_1 inst\u271d : NormedLatticeAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 |a \u2294 b - c \u2294 b| + |c \u2294 b - c \u2294 d| \u2264 |a - c| + |b - d| ** apply add_le_add ** case h\u2081 \u03b1 : Type u_1 inst\u271d : NormedLatticeAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 |a \u2294 b - c \u2294 b| \u2264 |a - c| ** exact abs_sup_sub_sup_le_abs _ _ _ ** case h\u2082 \u03b1 : Type u_1 inst\u271d : NormedLatticeAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 |c \u2294 b - c \u2294 d| \u2264 |b - d| ** rw [@sup_comm _ _ c, @sup_comm _ _ c] ** case h\u2082 \u03b1 : Type u_1 inst\u271d : NormedLatticeAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 |b \u2294 c - d \u2294 c| \u2264 |b - d| ** exact abs_sup_sub_sup_le_abs _ _ _ ** Qed", + "informal": "" + }, + { + "formal": "Basis.flag_strictMono ** R : Type u_1 M : Type u_2 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R M n : \u2115 inst\u271d : Nontrivial R b : Basis (Fin n) R M x\u271d : Fin n \u22a2 flag b (Fin.castSucc x\u271d) < flag b (Fin.succ x\u271d) ** simp [flag_succ] ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.derivFamily_isNormal ** \u03b9 : Type u f\u271d : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} f : \u03b9 \u2192 Ordinal.{max u u_1} \u2192 Ordinal.{max u u_1} o : Ordinal.{max u_1 u} \u22a2 derivFamily f o < derivFamily f (succ o) ** rw [derivFamily_succ, \u2190 succ_le_iff] ** \u03b9 : Type u f\u271d : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} f : \u03b9 \u2192 Ordinal.{max u u_1} \u2192 Ordinal.{max u u_1} o : Ordinal.{max u_1 u} \u22a2 succ (derivFamily f o) \u2264 nfpFamily f (succ (derivFamily f o)) ** apply le_nfpFamily ** \u03b9 : Type u f\u271d : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} f : \u03b9 \u2192 Ordinal.{max u u_1} \u2192 Ordinal.{max u u_1} o : Ordinal.{max u_1 u} l : IsLimit o a : Ordinal.{max u_1 u} \u22a2 derivFamily f o \u2264 a \u2194 \u2200 (b : Ordinal.{max u_1 u}), b < o \u2192 derivFamily f b \u2264 a ** rw [derivFamily_limit _ l, bsup_le_iff] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.VectorMeasure.of_diff ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M\u271d : Type u_3 inst\u271d\u2075 : AddCommMonoid M\u271d inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : T2Space M\u271d v\u271d : VectorMeasure \u03b1 M\u271d f : \u2115 \u2192 Set \u03b1 M : Type u_4 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : T2Space M v : VectorMeasure \u03b1 M A B : Set \u03b1 hA : MeasurableSet A hB : MeasurableSet B h : A \u2286 B \u22a2 \u2191v (B \\ A) = \u2191v B - \u2191v A ** rw [\u2190 of_add_of_diff hA hB h, add_sub_cancel'] ** Qed", + "informal": "" + }, + { + "formal": "Valuation.map_sub_le ** K : Type u_1 F : Type u_2 R : Type u_3 inst\u271d\u00b3 : DivisionRing K \u0393\u2080 : Type u_4 \u0393'\u2080 : Type u_5 \u0393''\u2080 : Type u_6 inst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080 inst\u271d\u00b9 : Ring R inst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080 v : Valuation R \u0393\u2080 x\u271d y\u271d z x y : R g : (fun x => \u0393\u2080) x hx : \u2191v x \u2264 g hy : \u2191v y \u2264 g \u22a2 \u2191v (x - y) \u2264 g ** rw [sub_eq_add_neg] ** K : Type u_1 F : Type u_2 R : Type u_3 inst\u271d\u00b3 : DivisionRing K \u0393\u2080 : Type u_4 \u0393'\u2080 : Type u_5 \u0393''\u2080 : Type u_6 inst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080 inst\u271d\u00b9 : Ring R inst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080 v : Valuation R \u0393\u2080 x\u271d y\u271d z x y : R g : (fun x => \u0393\u2080) x hx : \u2191v x \u2264 g hy : \u2191v y \u2264 g \u22a2 \u2191v (x + -y) \u2264 g ** exact v.map_add_le hx (le_trans (le_of_eq (v.map_neg y)) hy) ** Qed", + "informal": "" + }, + { + "formal": "IsLocalization.exists_smul_mem_of_mem_adjoin ** R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s \u22a2 \u2203 m, m \u2022 x \u2208 A ** let g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' ** R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' \u22a2 \u2203 m, m \u2022 x \u2208 A ** let y := IsLocalization.commonDenomOfFinset M s ** R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s \u22a2 \u2203 m, m \u2022 x \u2208 A ** have hx\u2081 : (y : S) \u2022 (s : Set S') = g '' _ :=\n (IsLocalization.finsetIntegerMultiple_image _ s).symm ** R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) \u22a2 \u2203 m, m \u2022 x \u2208 A ** 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)) ** case intro R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) n : \u2115 hn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A \u22a2 \u2203 m, m \u2022 x \u2208 A ** obtain \u27e8x', hx', hx''\u27e9 := hn n (le_of_eq rfl) ** case intro.intro.intro R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) n : \u2115 hn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A x' : S hx' : x' \u2208 \u2191A.toSubsemiring hx'' : \u2191\u2191g x' = \u2191y ^ n \u2022 \u2191(algebraMap S S') x \u22a2 \u2203 m, m \u2022 x \u2208 A ** rw [Algebra.smul_def, \u2190 _root_.map_mul] at hx'' ** case intro.intro.intro R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) n : \u2115 hn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A x' : S hx' : x' \u2208 \u2191A.toSubsemiring hx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x) \u22a2 \u2203 m, m \u2022 x \u2208 A ** obtain \u27e8a, ha\u2082\u27e9 := (IsLocalization.eq_iff_exists M S').mp hx'' ** case intro.intro.intro.intro R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) n : \u2115 hn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A x' : S hx' : x' \u2208 \u2191A.toSubsemiring hx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x) a : { x // x \u2208 M } ha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x) \u22a2 \u2203 m, m \u2022 x \u2208 A ** use a * y ^ n ** case h R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) n : \u2115 hn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A x' : S hx' : x' \u2208 \u2191A.toSubsemiring hx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x) a : { x // x \u2208 M } ha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x) \u22a2 (a * y ^ n) \u2022 x \u2208 A ** convert A.mul_mem hx' (hA\u2082 a.prop) using 1 ** case h.e'_4 R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) n : \u2115 hn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A x' : S hx' : x' \u2208 \u2191A.toSubsemiring hx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x) a : { x // x \u2208 M } ha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x) \u22a2 (a * y ^ n) \u2022 x = x' * \u2191a ** rw [Submonoid.smul_def, smul_eq_mul, Submonoid.coe_mul, SubmonoidClass.coe_pow, mul_assoc, \u2190 ha\u2082,\n mul_comm] ** R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) \u22a2 \u2191y \u2022 \u2191s \u2286 \u2191(Subalgebra.map g A) ** rw [hx\u2081] ** R S : Type u inst\u271d\u2079 : CommRing R inst\u271d\u2078 : CommRing S M\u271d : Submonoid R N : Submonoid S R' S' : Type u inst\u271d\u2077 : CommRing R' inst\u271d\u2076 : CommRing S' f : R \u2192+* S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra S S' inst\u271d\u00b3 : Algebra R S inst\u271d\u00b2 : Algebra R S' inst\u271d\u00b9 : IsScalarTower R S S' M : Submonoid S inst\u271d : IsLocalization M S' x : S s : Finset S' A : Subalgebra R S hA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A hA\u2082 : M \u2264 A.toSubmonoid hx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S' y : { x // x \u2208 M } := commonDenomOfFinset M s hx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s) \u22a2 \u2191g '' \u2191(finsetIntegerMultiple M s) \u2286 \u2191(Subalgebra.map g A) ** exact Set.image_subset _ hA\u2081 ** Qed", + "informal": "" + }, + { + "formal": "Sylow.subsingleton_of_normal ** G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P \u22a2 Subsingleton (Sylow p G) ** apply Subsingleton.intro ** case allEq G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P \u22a2 \u2200 (a b : Sylow p G), a = b ** intro Q R ** case allEq G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P Q R : Sylow p G \u22a2 Q = R ** obtain \u27e8x, h1\u27e9 := exists_smul_eq G P Q ** case allEq.intro G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P Q R : Sylow p G x : G h1 : x \u2022 P = Q \u22a2 Q = R ** obtain \u27e8x, h2\u27e9 := exists_smul_eq G P R ** case allEq.intro.intro G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P Q R : Sylow p G x\u271d : G h1 : x\u271d \u2022 P = Q x : G h2 : x \u2022 P = R \u22a2 Q = R ** rw [Sylow.smul_eq_of_normal] at h1 h2 ** case allEq.intro.intro G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P Q R : Sylow p G x\u271d : G h1 : P = Q x : G h2 : P = R \u22a2 Q = R ** rw [\u2190 h1, \u2190 h2] ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Language.age_directLimit ** L : Language K : Set (Bundled (Structure L)) M : Type w inst\u271d\u2076 : Structure L M N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) \u22a2 age L (DirectLimit G f) = \u22c3 i, age L (G i) ** ext M ** case h L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) \u22a2 M \u2208 age L (DirectLimit G f) \u2194 M \u2208 \u22c3 i, age L (G i) ** simp only [mem_iUnion] ** case h L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) \u22a2 M \u2208 age L (DirectLimit G f) \u2194 \u2203 i, M \u2208 age L (G i) ** constructor ** case h.mp L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) \u22a2 M \u2208 age L (DirectLimit G f) \u2192 \u2203 i, M \u2208 age L (G i) ** rintro \u27e8Mfg, \u27e8e\u27e9\u27e9 ** case h.mp.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f \u22a2 \u2203 i, M \u2208 age L (G i) ** obtain \u27e8s, hs\u27e9 := Mfg.range e.toHom ** case h.mp.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) \u22a2 \u2203 i, M \u2208 age L (G i) ** let out := @Quotient.out _ (DirectLimit.setoid G f) ** case h.mp.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out \u22a2 \u2203 i, M \u2208 age L (G i) ** obtain \u27e8i, hi\u27e9 := Finset.exists_le (s.image (Sigma.fst \u2218 out)) ** case h.mp.intro.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out i : \u03b9 hi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i \u22a2 \u2203 i, M \u2208 age L (G i) ** have e' := (DirectLimit.of L \u03b9 G f i).equivRange.symm.toEmbedding ** case h.mp.intro.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out i : \u03b9 hi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i e' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i \u22a2 \u2203 i, M \u2208 age L (G i) ** refine' \u27e8i, Mfg, \u27e8e'.comp ((Substructure.inclusion _).comp e.equivRange.toEmbedding)\u27e9\u27e9 ** case h.mp.intro.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out i : \u03b9 hi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i e' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i \u22a2 Hom.range (Embedding.toHom e) \u2264 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) ** rw [\u2190 hs, closure_le] ** case h.mp.intro.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out i : \u03b9 hi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i e' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i \u22a2 \u2191s \u2286 \u2191(Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i))) ** intro x hx ** case h.mp.intro.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out i : \u03b9 hi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i e' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i x : DirectLimit G f hx : x \u2208 \u2191s \u22a2 x \u2208 \u2191(Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i))) ** refine' \u27e8f (out x).1 i (hi (out x).1 (Finset.mem_image_of_mem _ hx)) (out x).2, _\u27e9 ** case h.mp.intro.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out i : \u03b9 hi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i e' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i x : DirectLimit G f hx : x \u2208 \u2191s \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 ** 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] ** L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out i : \u03b9 hi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i e' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i x : DirectLimit G f hx : x \u2208 \u2191s \u22a2 (Structure.Sigma.mk f i (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd)).fst \u2264 i L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] DirectLimit G f s : Finset (DirectLimit G f) hs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e) out : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out i : \u03b9 hi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i e' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i x : DirectLimit G f hx : x \u2208 \u2191s \u22a2 (Structure.Sigma.mk f i (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd)).fst \u2264 i ** rfl ** case h.mpr L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) \u22a2 (\u2203 i, M \u2208 age L (G i)) \u2192 M \u2208 age L (DirectLimit G f) ** rintro \u27e8i, Mfg, \u27e8e\u27e9\u27e9 ** case h.mpr.intro.intro.intro L : Language K : Set (Bundled (Structure L)) M\u271d : Type w inst\u271d\u2076 : Structure L M\u271d N : Type w inst\u271d\u2075 : Structure L N \u03b9 : Type w inst\u271d\u2074 : Preorder \u03b9 inst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1 inst\u271d\u00b2 : Nonempty \u03b9 G : \u03b9 \u2192 Type (max w w') inst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i) f : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j inst\u271d : DirectedSystem G fun i j h => \u2191(f i j h) M : Bundled (Structure L) i : \u03b9 Mfg : Structure.FG L \u2191M e : \u2191M \u21aa[L] G i \u22a2 M \u2208 age L (DirectLimit G f) ** exact \u27e8Mfg, \u27e8Embedding.comp (DirectLimit.of L \u03b9 G f i) e\u27e9\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Bitvec.toNat_ofNat ** k n : \u2115 \u22a2 Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k ** induction' k with k ih generalizing n ** case zero n\u271d n : \u2115 \u22a2 Bitvec.toNat (Bitvec.ofNat zero n) = n % 2 ^ zero ** simp [Nat.mod_one] ** case zero n\u271d n : \u2115 \u22a2 Bitvec.toNat (Bitvec.ofNat 0 n) = 0 ** rfl ** case succ n\u271d k : \u2115 ih : \u2200 {n : \u2115}, Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k n : \u2115 \u22a2 Bitvec.toNat (Bitvec.ofNat (succ k) n) = n % 2 ^ succ k ** rw [ofNat_succ, toNat_append, ih, bits_toNat_decide, mod_pow_succ, Nat.mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.IsFundamentalDomain.set_integral_eq ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 E hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x \u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc ** by_cases hfs : IntegrableOn f s \u03bc ** case pos G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 E hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x hfs : IntegrableOn f s \u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc ** have hft : IntegrableOn f t \u03bc := by rwa [ht.integrableOn_iff hs hf] ** case pos G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 E hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x hfs : IntegrableOn f s hft : IntegrableOn f t \u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc ** 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 ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 E hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x hfs : IntegrableOn f s \u22a2 IntegrableOn f t ** rwa [ht.integrableOn_iff hs hf] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 E hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x hfs : IntegrableOn f s hft : IntegrableOn f t \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 ** simp only [hf, inter_comm] ** case neg G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 E hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x hfs : \u00acIntegrableOn f s \u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc ** rw [integral_undef hfs, integral_undef] ** case neg G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 E hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x hfs : \u00acIntegrableOn f s \u22a2 \u00acIntegrable fun x => f x ** rwa [hs.integrableOn_iff ht hf] at hfs ** Qed", + "informal": "" + }, + { + "formal": "Units.neg_divp ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w R : Type x inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : HasDistribNeg \u03b1 a\u271d b a : \u03b1 u : \u03b1\u02e3 \u22a2 -(a /\u209a u) = -a /\u209a u ** simp only [divp, neg_mul] ** Qed", + "informal": "" + }, + { + "formal": "AffineMap.linear_eq_zero_iff_exists_const ** k : Type u_1 V1 : Type u_2 P1 : Type u_3 V2 : Type u_4 P2 : Type u_5 V3 : Type u_6 P3 : Type u_7 V4 : Type u_8 P4 : Type u_9 inst\u271d\u00b9\u00b2 : Ring k inst\u271d\u00b9\u00b9 : AddCommGroup V1 inst\u271d\u00b9\u2070 : Module k V1 inst\u271d\u2079 : AffineSpace V1 P1 inst\u271d\u2078 : AddCommGroup V2 inst\u271d\u2077 : Module k V2 inst\u271d\u2076 : AffineSpace V2 P2 inst\u271d\u2075 : AddCommGroup V3 inst\u271d\u2074 : Module k V3 inst\u271d\u00b3 : AffineSpace V3 P3 inst\u271d\u00b2 : AddCommGroup V4 inst\u271d\u00b9 : Module k V4 inst\u271d : AffineSpace V4 P4 f : P1 \u2192\u1d43[k] P2 \u22a2 f.linear = 0 \u2194 \u2203 q, f = const k P1 q ** refine' \u27e8fun h => _, fun h => _\u27e9 ** case refine'_1 k : Type u_1 V1 : Type u_2 P1 : Type u_3 V2 : Type u_4 P2 : Type u_5 V3 : Type u_6 P3 : Type u_7 V4 : Type u_8 P4 : Type u_9 inst\u271d\u00b9\u00b2 : Ring k inst\u271d\u00b9\u00b9 : AddCommGroup V1 inst\u271d\u00b9\u2070 : Module k V1 inst\u271d\u2079 : AffineSpace V1 P1 inst\u271d\u2078 : AddCommGroup V2 inst\u271d\u2077 : Module k V2 inst\u271d\u2076 : AffineSpace V2 P2 inst\u271d\u2075 : AddCommGroup V3 inst\u271d\u2074 : Module k V3 inst\u271d\u00b3 : AffineSpace V3 P3 inst\u271d\u00b2 : AddCommGroup V4 inst\u271d\u00b9 : Module k V4 inst\u271d : AffineSpace V4 P4 f : P1 \u2192\u1d43[k] P2 h : f.linear = 0 \u22a2 \u2203 q, f = const k P1 q ** use f (Classical.arbitrary P1) ** case h k : Type u_1 V1 : Type u_2 P1 : Type u_3 V2 : Type u_4 P2 : Type u_5 V3 : Type u_6 P3 : Type u_7 V4 : Type u_8 P4 : Type u_9 inst\u271d\u00b9\u00b2 : Ring k inst\u271d\u00b9\u00b9 : AddCommGroup V1 inst\u271d\u00b9\u2070 : Module k V1 inst\u271d\u2079 : AffineSpace V1 P1 inst\u271d\u2078 : AddCommGroup V2 inst\u271d\u2077 : Module k V2 inst\u271d\u2076 : AffineSpace V2 P2 inst\u271d\u2075 : AddCommGroup V3 inst\u271d\u2074 : Module k V3 inst\u271d\u00b3 : AffineSpace V3 P3 inst\u271d\u00b2 : AddCommGroup V4 inst\u271d\u00b9 : Module k V4 inst\u271d : AffineSpace V4 P4 f : P1 \u2192\u1d43[k] P2 h : f.linear = 0 \u22a2 f = const k P1 (\u2191f (Classical.arbitrary P1)) ** ext ** case h.h k : Type u_1 V1 : Type u_2 P1 : Type u_3 V2 : Type u_4 P2 : Type u_5 V3 : Type u_6 P3 : Type u_7 V4 : Type u_8 P4 : Type u_9 inst\u271d\u00b9\u00b2 : Ring k inst\u271d\u00b9\u00b9 : AddCommGroup V1 inst\u271d\u00b9\u2070 : Module k V1 inst\u271d\u2079 : AffineSpace V1 P1 inst\u271d\u2078 : AddCommGroup V2 inst\u271d\u2077 : Module k V2 inst\u271d\u2076 : AffineSpace V2 P2 inst\u271d\u2075 : AddCommGroup V3 inst\u271d\u2074 : Module k V3 inst\u271d\u00b3 : AffineSpace V3 P3 inst\u271d\u00b2 : AddCommGroup V4 inst\u271d\u00b9 : Module k V4 inst\u271d : AffineSpace V4 P4 f : P1 \u2192\u1d43[k] P2 h : f.linear = 0 p\u271d : P1 \u22a2 \u2191f p\u271d = \u2191(const k P1 (\u2191f (Classical.arbitrary P1))) p\u271d ** rw [coe_const, Function.const_apply, \u2190 @vsub_eq_zero_iff_eq V2, \u2190 f.linearMap_vsub, h,\n LinearMap.zero_apply] ** case refine'_2 k : Type u_1 V1 : Type u_2 P1 : Type u_3 V2 : Type u_4 P2 : Type u_5 V3 : Type u_6 P3 : Type u_7 V4 : Type u_8 P4 : Type u_9 inst\u271d\u00b9\u00b2 : Ring k inst\u271d\u00b9\u00b9 : AddCommGroup V1 inst\u271d\u00b9\u2070 : Module k V1 inst\u271d\u2079 : AffineSpace V1 P1 inst\u271d\u2078 : AddCommGroup V2 inst\u271d\u2077 : Module k V2 inst\u271d\u2076 : AffineSpace V2 P2 inst\u271d\u2075 : AddCommGroup V3 inst\u271d\u2074 : Module k V3 inst\u271d\u00b3 : AffineSpace V3 P3 inst\u271d\u00b2 : AddCommGroup V4 inst\u271d\u00b9 : Module k V4 inst\u271d : AffineSpace V4 P4 f : P1 \u2192\u1d43[k] P2 h : \u2203 q, f = const k P1 q \u22a2 f.linear = 0 ** rcases h with \u27e8q, rfl\u27e9 ** case refine'_2.intro k : Type u_1 V1 : Type u_2 P1 : Type u_3 V2 : Type u_4 P2 : Type u_5 V3 : Type u_6 P3 : Type u_7 V4 : Type u_8 P4 : Type u_9 inst\u271d\u00b9\u00b2 : Ring k inst\u271d\u00b9\u00b9 : AddCommGroup V1 inst\u271d\u00b9\u2070 : Module k V1 inst\u271d\u2079 : AffineSpace V1 P1 inst\u271d\u2078 : AddCommGroup V2 inst\u271d\u2077 : Module k V2 inst\u271d\u2076 : AffineSpace V2 P2 inst\u271d\u2075 : AddCommGroup V3 inst\u271d\u2074 : Module k V3 inst\u271d\u00b3 : AffineSpace V3 P3 inst\u271d\u00b2 : AddCommGroup V4 inst\u271d\u00b9 : Module k V4 inst\u271d : AffineSpace V4 P4 q : P2 \u22a2 (const k P1 q).linear = 0 ** exact const_linear k P1 q ** Qed", + "informal": "" + }, + { + "formal": "rightCoset_eq_iff ** \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 \u22a2 \u2191s *r x = \u2191s *r y \u2194 y * x\u207b\u00b9 \u2208 s ** rw [Set.ext_iff] ** \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 \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 ** simp_rw [mem_rightCoset_iff, SetLike.mem_coe] ** \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 \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 ** constructor ** case mp \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 \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 ** intro h ** case mp \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 h : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s \u22a2 y * x\u207b\u00b9 \u2208 s ** apply (h y).mpr ** case mp \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 h : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s \u22a2 y * y\u207b\u00b9 \u2208 s ** rw [mul_right_inv] ** case mp \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 h : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s \u22a2 1 \u2208 s ** exact s.one_mem ** case mpr \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 \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 ** intro h z ** case mpr \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 h : y * x\u207b\u00b9 \u2208 s z : \u03b1 \u22a2 z * x\u207b\u00b9 \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s ** rw [\u2190 inv_mul_cancel_left y x\u207b\u00b9] ** case mpr \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 h : y * x\u207b\u00b9 \u2208 s z : \u03b1 \u22a2 z * (y\u207b\u00b9 * (y * x\u207b\u00b9)) \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s ** rw [\u2190 mul_assoc] ** case mpr \u03b1 : Type u_1 inst\u271d : Group \u03b1 s : Subgroup \u03b1 x y : \u03b1 h : y * x\u207b\u00b9 \u2208 s z : \u03b1 \u22a2 z * y\u207b\u00b9 * (y * x\u207b\u00b9) \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s ** exact s.mul_mem_cancel_right h ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.pullbackZeroZeroIso_hom_fst ** C : Type u_1 inst\u271d\u00b3 : Category.{u_2, u_1} C inst\u271d\u00b2 : HasZeroObject C inst\u271d\u00b9 : HasZeroMorphisms C X Y : C inst\u271d : HasBinaryProduct X Y \u22a2 (pullbackZeroZeroIso X Y).hom \u226b prod.fst = pullback.fst ** simp [\u2190 Iso.eq_inv_comp] ** Qed", + "informal": "" + }, + { + "formal": "Rat.neg_divInt ** n d : Int \u22a2 -(n /. d) = -n /. d ** rcases Int.eq_nat_or_neg d with \u27e8_, rfl | rfl\u27e9 <;> simp [divInt_neg', neg_mkRat] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.isOpenImmersionCat_comp_of_sourceAffineLocally ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 P (Scheme.\u0393.map (f \u226b g).op) ** 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] ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 P (Scheme.\u0393.map ((IsOpenImmersion.isoOfRangeEq (Scheme.ofRestrict Y ?m.145001) f ?m.145008).hom \u226b f \u226b g).op) P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 TopCat P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 OpenEmbedding \u2191?m.145000 P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 TopCat P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 OpenEmbedding \u2191?m.145000 P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 TopCat P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 OpenEmbedding \u2191?m.145000 P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base ** convert h\u2082 \u27e8_, rangeIsAffineOpenOfOpenImmersion f\u27e9 using 3 ** case h.e'_5.h.h.e'_8.h.e'_5 P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 (IsOpenImmersion.isoOfRangeEq (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion \u2191{ val := Scheme.Hom.opensRange f, property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) }))) f ?m.145008).hom \u226b f \u226b g = Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion \u2191{ val := Scheme.Hom.opensRange f, property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) })) \u226b g ** rw [IsOpenImmersion.isoOfRangeEq_hom_fac_assoc] ** case h.e'_5.h.h.e'_8.h.e'_5.e P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop hP : RingHom.PropertyIsLocal P h\u2081 : RingHom.RespectsIso P X Y Z : Scheme inst\u271d\u00b2 : IsAffine X inst\u271d\u00b9 : IsAffine Z f : X \u27f6 Y inst\u271d : IsOpenImmersion f g : Y \u27f6 Z h\u2082 : sourceAffineLocally P g \u22a2 Set.range \u2191(Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion \u2191{ val := Scheme.Hom.opensRange f, property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) }))).val.base = Set.range \u2191f.val.base ** exact Subtype.range_coe ** Qed", + "informal": "" + }, + { + "formal": "Nat.pos_of_floor_pos ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedSemiring \u03b1 inst\u271d : FloorSemiring \u03b1 a : \u03b1 n : \u2115 h : 0 < \u230aa\u230b\u208a ha : a \u2264 0 \u22a2 0 < 0 ** rwa [floor_of_nonpos ha] at h ** Qed", + "informal": "" + }, + { + "formal": "Behrend.le_sqrt_log ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 log (2 / (1 - 2 / rexp 1)) * (69 / 50) \u2264 Real.sqrt (log \u2191N) ** 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 _ ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 log 8 * (69 / 50) \u2264 Real.sqrt (log \u2191N) ** have l8 : log 8 = (3 : \u2115) * log 2 := by\n rw [\u2190 log_rpow zero_lt_two, rpow_nat_cast]\n norm_num ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 log 8 * (69 / 50) \u2264 Real.sqrt (log \u2191N) ** rw [l8] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 \u21913 * log 2 * (69 / 50) \u2264 Real.sqrt (log \u2191N) ** apply le_sqrt_of_sq_le (le_trans _ this) ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 (\u21913 * log 2 * (69 / 50)) ^ 2 \u2264 \u219112 * log 2 ** rw [mul_right_comm, mul_pow, sq (log 2), \u2190 mul_assoc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 (\u21913 * (69 / 50)) ^ 2 * log 2 * log 2 \u2264 \u219112 * log 2 ** apply mul_le_mul_of_nonneg_right _ (log_nonneg one_le_two) ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 (\u21913 * (69 / 50)) ^ 2 * log 2 \u2264 \u219112 ** rw [\u2190 le_div_iff'] ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 0 < (\u21913 * (69 / 50)) ^ 2 ** exact sq_pos_of_ne_zero _ (by norm_num1) ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N \u22a2 \u219112 * log 2 \u2264 log \u2191N ** rw [\u2190 log_rpow zero_lt_two, log_le_log, rpow_nat_cast] ** case h\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N \u22a2 0 < \u2191N ** rw [cast_pos] ** case h\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N \u22a2 0 < N ** exact hN.trans_lt' (by norm_num1) ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N \u22a2 2 ^ 12 \u2264 \u2191N ** norm_num1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N \u22a2 4096 \u2264 \u2191N ** exact_mod_cast hN ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N \u22a2 0 < 2 ^ \u219112 ** exact rpow_pos_of_pos zero_lt_two _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N \u22a2 0 < 4096 ** norm_num1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 0 < 8 ** norm_num1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 0 \u2264 69 / 50 ** norm_num1 ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 0 < 2 / (1 - 2 / rexp 1) ** refine' div_pos zero_lt_two _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 0 < 1 - 2 / rexp 1 ** rw [sub_pos, div_lt_one (exp_pos _)] ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 2 < rexp 1 ** exact exp_one_gt_d9.trans_le' (by norm_num1) ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 2 \u2264 2.7182818283 ** norm_num1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 log 8 = \u21913 * log 2 ** rw [\u2190 log_rpow zero_lt_two, rpow_nat_cast] ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N \u22a2 log 8 = log (2 ^ 3) ** norm_num ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 log 2 \u2264 \u219112 / (\u21913 * (69 / 50)) ^ 2 ** exact log_two_lt_d9.le.trans (by norm_num1) ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 0.6931471808 \u2264 \u219112 / (\u21913 * (69 / 50)) ^ 2 ** norm_num1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N this : \u219112 * log 2 \u2264 log \u2191N l8 : log 8 = \u21913 * log 2 \u22a2 \u21913 * (69 / 50) \u2260 0 ** norm_num1 ** Qed", + "informal": "" + }, + { + "formal": "Multiset.prod_pair ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : CommMonoid \u03b1 s t : Multiset \u03b1 a\u271d : \u03b1 m : Multiset \u03b9 f g : \u03b9 \u2192 \u03b1 a b : \u03b1 \u22a2 prod {a, b} = a * b ** rw [insert_eq_cons, prod_cons, prod_singleton] ** Qed", + "informal": "" + }, + { + "formal": "MeasurableEmbedding.integrableOn_map_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : MeasurableSpace \u03b2 e : \u03b1 \u2192 \u03b2 he : MeasurableEmbedding e f : \u03b2 \u2192 E \u03bc : Measure \u03b1 s : Set \u03b2 \u22a2 IntegrableOn f s \u2194 IntegrableOn (f \u2218 e) (e \u207b\u00b9' s) ** simp only [IntegrableOn, he.restrict_map, he.integrable_map_iff] ** Qed", + "informal": "" + }, + { + "formal": "Trivialization.coe_linearMapAt_of_mem ** R : Type u_1 B : Type u_2 F : Type u_3 E : B \u2192 Type u_4 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : TopologicalSpace F inst\u271d\u2076 : TopologicalSpace B inst\u271d\u2075 : TopologicalSpace (TotalSpace F E) e\u271d : Trivialization F TotalSpace.proj x : TotalSpace F E b\u271d : B y : E b\u271d inst\u271d\u2074 : AddCommMonoid F inst\u271d\u00b3 : Module R F inst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x) inst\u271d\u00b9 : (x : B) \u2192 Module R (E x) e : Trivialization F TotalSpace.proj inst\u271d : Trivialization.IsLinear R e b : B hb : b \u2208 e.baseSet \u22a2 \u2191(Trivialization.linearMapAt R e b) = fun y => (\u2191e { proj := b, snd := y }).2 ** simp_rw [coe_linearMapAt, if_pos hb] ** Qed", + "informal": "" + }, + { + "formal": "invOf_mul_self_assoc' ** \u03b1 : Type u inst\u271d : Monoid \u03b1 a b : \u03b1 x\u271d : Invertible a \u22a2 \u215fa * (a * b) = b ** rw [\u2190 mul_assoc, invOf_mul_self, one_mul] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.NatTrans.leftDerived_eq ** C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 (leftDerived \u03b1 n).app X = (Functor.leftDerivedObjIso F n P).hom \u226b (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) \u226b (Functor.leftDerivedObjIso G n P).inv ** symm ** C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 (Functor.leftDerivedObjIso F n P).hom \u226b (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) \u226b (Functor.leftDerivedObjIso G n P).inv = (leftDerived \u03b1 n).app X ** dsimp [NatTrans.leftDerived, Functor.leftDerivedObjIso] ** C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b \ud835\udfd9 ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).obj ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).obj ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex)))) \u226b homology.map (_ : HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n = 0) (_ : HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n = 0) (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n) (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n) (_ : (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right = (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b \ud835\udfd9 (HomologicalComplex.homology ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n) \u226b (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) = (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)) ** simp only [Category.comp_id, Category.id_comp] ** C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b homology.map (_ : HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n = 0) (_ : HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n = 0) (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n) (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n) (_ : (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right = (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) = (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)) ** rw [\u2190 homologyFunctor_map, HomotopyCategory.homologyFunctor_map_factors] ** C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex)) \u226b (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) = (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)) ** simp only [\u2190 Functor.map_comp] ** C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) = (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)) ** congr 1 ** case e_a C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv) = (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as) ** apply HomotopyCategory.eq_of_homotopy ** case e_a.h C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 Homotopy ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv) ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as) ** simp only [NatTrans.mapHomologicalComplex_naturality_assoc, \u2190 Functor.map_comp] ** case e_a.h C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 Homotopy ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as \u226b (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as) ** apply Homotopy.compLeftId ** case e_a.h.h C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 Homotopy ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as)) ** refine' (Functor.mapHomotopy _ (HomotopyEquiv.homotopyHomInvId _) ).trans _ ** case e_a.h.h C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 Homotopy ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as)) (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as)) ** apply Homotopy.ofEq ** case e_a.h.h.h C : Type u inst\u271d\u00b9\u00b3 : Category.{v, u} C D : Type u_1 inst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D inst\u271d\u00b9\u00b9 : Preadditive C inst\u271d\u00b9\u2070 : HasZeroObject C inst\u271d\u2079 : HasEqualizers C inst\u271d\u2078 : HasImages C inst\u271d\u2077 : HasProjectiveResolutions C inst\u271d\u2076 : Preadditive D inst\u271d\u2075 : HasEqualizers D inst\u271d\u2074 : HasCokernels D inst\u271d\u00b3 : HasImages D inst\u271d\u00b2 : HasImageMaps D F G : C \u2964 D inst\u271d\u00b9 : Functor.Additive F inst\u271d : Functor.Additive G \u03b1 : F \u27f6 G n : \u2115 X : C P : ProjectiveResolution X \u22a2 (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as) = \ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as) ** simp only [Functor.map_id] ** Qed", + "informal": "" + }, + { + "formal": "EReal.coe_toReal_le ** x : EReal h : x \u2260 \u22a5 \u22a2 \u2191(toReal x) \u2264 x ** by_cases h' : x = \u22a4 ** case pos x : EReal h : x \u2260 \u22a5 h' : x = \u22a4 \u22a2 \u2191(toReal x) \u2264 x ** simp only [h', le_top] ** case neg x : EReal h : x \u2260 \u22a5 h' : \u00acx = \u22a4 \u22a2 \u2191(toReal x) \u2264 x ** simp only [le_refl, coe_toReal h' h] ** Qed", + "informal": "" + }, + { + "formal": "Commute.div_sub_div ** \u03b1 : Type u_1 \u03b2 : Type u_2 K : Type u_3 inst\u271d : DivisionRing K a b c d : K hbc : Commute b c hbd : Commute b d hb : b \u2260 0 hd : d \u2260 0 \u22a2 a / b - c / d = (a * d - b * c) / (b * d) ** simpa only [mul_neg, neg_div, \u2190 sub_eq_add_neg] using hbc.neg_right.div_add_div hbd hb hd ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Mem\u2112p.snorm_indicator_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \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 ** have h\u2112p := hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p \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 ** obtain \u27e8\u27e8f', hf', heq\u27e9, _\u27e9 := hf ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \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 ** 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 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u03b4 : \u211d h\u03b4pos : 0 < \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 \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 ** refine' \u27e8\u03b4, h\u03b4pos, fun s hs h\u03bcs => _\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u03b4 : \u211d h\u03b4pos : 0 < \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 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** convert h\u03b4 s hs h\u03bcs using 1 ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u03b4 : \u211d h\u03b4pos : 0 < \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 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 snorm (Set.indicator s f) p \u03bc = snorm (Set.indicator s f') p \u03bc ** rw [snorm_indicator_eq_snorm_restrict hs, snorm_indicator_eq_snorm_restrict hs] ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u03b4 : \u211d h\u03b4pos : 0 < \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 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 snorm f p (Measure.restrict \u03bc s) = snorm f' p (Measure.restrict \u03bc s) ** refine' snorm_congr_ae heq.restrict ** Qed", + "informal": "" + }, + { + "formal": "LieSubmodule.lieIdeal_oper_eq_linear_span ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \u22a2 \u2191\u2045I, N\u2046 = Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** apply le_antisymm ** case a R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \u22a2 \u2191\u2045I, N\u2046 \u2264 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** let s := { m : M | \u2203 (x : \u21a5I) (n : \u21a5N), \u2045(x : L), (n : M)\u2046 = m } ** case a R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} aux : \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s \u22a2 \u2191\u2045I, N\u2046 \u2264 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** change _ \u2264 ({ Submodule.span R s with lie_mem := fun hm' => aux _ _ hm' } : LieSubmodule R L M) ** case a R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} aux : \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s \u22a2 \u2045I, N\u2046 \u2264 let src := Submodule.span R s; { toSubmodule := { toAddSubmonoid := src.toAddSubmonoid, smul_mem' := (_ : \u2200 (c : R) {x : M}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) }, lie_mem := (_ : \u2200 {x : L} {m : M}, m \u2208 { toAddSubmonoid := src.toAddSubmonoid, smul_mem' := (_ : \u2200 (c : R) {x : M}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192 \u2045x, m\u2046 \u2208 Submodule.span R s) } ** rw [lieIdeal_oper_eq_span, lieSpan_le] ** case a R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} aux : \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s \u22a2 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2286 \u2191(let src := Submodule.span R s; { toSubmodule := { toAddSubmonoid := src.toAddSubmonoid, smul_mem' := (_ : \u2200 (c : R) {x : M}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) }, lie_mem := (_ : \u2200 {x : L} {m : M}, m \u2208 { toAddSubmonoid := src.toAddSubmonoid, smul_mem' := (_ : \u2200 (c : R) {x : M}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192 \u2045x, m\u2046 \u2208 Submodule.span R s) }) ** exact Submodule.subset_span ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u22a2 \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s ** intro y m' hm' ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s \u22a2 \u2045y, m'\u2046 \u2208 Submodule.span R s ** refine Submodule.span_induction (R := R) (M := M) (s := s)\n (p := fun m' \u21a6 \u2045y, m'\u2046 \u2208 Submodule.span R s) hm' ?_ ?_ ?_ ?_ ** case refine_1 R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s \u22a2 \u2200 (x : M), x \u2208 s \u2192 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) x ** rintro m'' \u27e8x, n, hm''\u27e9 ** case refine_1.intro.intro R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s m'' : M x : { x // x \u2208 I } n : { x // x \u2208 N } hm'' : \u2045\u2191x, \u2191n\u2046 = m'' \u22a2 \u2045y, m''\u2046 \u2208 Submodule.span R s ** rw [\u2190 hm'', leibniz_lie] ** case refine_1.intro.intro R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s m'' : M x : { x // x \u2208 I } n : { x // x \u2208 N } hm'' : \u2045\u2191x, \u2191n\u2046 = m'' \u22a2 \u2045\u2045y, \u2191x\u2046, \u2191n\u2046 + \u2045\u2191x, \u2045y, \u2191n\u2046\u2046 \u2208 Submodule.span R s ** refine Submodule.add_mem _ ?_ ?_ <;> apply Submodule.subset_span ** case refine_1.intro.intro.refine_1.a R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s m'' : M x : { x // x \u2208 I } n : { x // x \u2208 N } hm'' : \u2045\u2191x, \u2191n\u2046 = m'' \u22a2 \u2045\u2045y, \u2191x\u2046, \u2191n\u2046 \u2208 s ** use \u27e8\u2045y, \u2191x\u2046, I.lie_mem x.property\u27e9, n ** case refine_1.intro.intro.refine_2.a R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s m'' : M x : { x // x \u2208 I } n : { x // x \u2208 N } hm'' : \u2045\u2191x, \u2191n\u2046 = m'' \u22a2 \u2045\u2191x, \u2045y, \u2191n\u2046\u2046 \u2208 s ** use x, \u27e8\u2045y, \u2191n\u2046, N.lie_mem n.property\u27e9 ** case refine_2 R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s \u22a2 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) 0 ** simp only [lie_zero, Submodule.zero_mem] ** case refine_3 R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s \u22a2 \u2200 (x y_1 : M), (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) x \u2192 (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) ** intro m\u2081 m\u2082 hm\u2081 hm\u2082 ** case refine_3 R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s m\u2081 m\u2082 : M hm\u2081 : \u2045y, m\u2081\u2046 \u2208 Submodule.span R s hm\u2082 : \u2045y, m\u2082\u2046 \u2208 Submodule.span R s \u22a2 \u2045y, m\u2081 + m\u2082\u2046 \u2208 Submodule.span R s ** rw [lie_add] ** case refine_3 R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s m\u2081 m\u2082 : M hm\u2081 : \u2045y, m\u2081\u2046 \u2208 Submodule.span R s hm\u2082 : \u2045y, m\u2082\u2046 \u2208 Submodule.span R s \u22a2 \u2045y, m\u2081\u2046 + \u2045y, m\u2082\u2046 \u2208 Submodule.span R s ** exact Submodule.add_mem _ hm\u2081 hm\u2082 ** case refine_4 R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s \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) ** intro t m'' hm'' ** case refine_4 R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s t : R m'' : M hm'' : \u2045y, m''\u2046 \u2208 Submodule.span R s \u22a2 \u2045y, t \u2022 m''\u2046 \u2208 Submodule.span R s ** rw [lie_smul] ** case refine_4 R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 s : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} y : L m' : M hm' : m' \u2208 Submodule.span R s t : R m'' : M hm'' : \u2045y, m''\u2046 \u2208 Submodule.span R s \u22a2 t \u2022 \u2045y, m''\u2046 \u2208 Submodule.span R s ** exact Submodule.smul_mem _ t hm'' ** case a R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \u22a2 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2264 \u2191\u2045I, N\u2046 ** rw [lieIdeal_oper_eq_span] ** case a R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \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}) ** apply submodule_span_le_lieSpan ** Qed", + "informal": "" + }, + { + "formal": "Submodule.map_inl ** R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid M\u2082 inst\u271d\u00b9 : Module R M inst\u271d : Module R M\u2082 p : Submodule R M q : Submodule R M\u2082 \u22a2 map (inl R M M\u2082) p = prod p \u22a5 ** ext \u27e8x, y\u27e9 ** case h.mk R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid M\u2082 inst\u271d\u00b9 : Module R M inst\u271d : Module R M\u2082 p : Submodule R M q : Submodule R M\u2082 x : M y : M\u2082 \u22a2 (x, y) \u2208 map (inl R M M\u2082) p \u2194 (x, y) \u2208 prod p \u22a5 ** simp only [and_left_comm, eq_comm, mem_map, Prod.mk.inj_iff, inl_apply, mem_bot, exists_eq_left',\n mem_prod] ** Qed", + "informal": "" + }, + { + "formal": "List.prod_eq_zero_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u00b2 : MonoidWithZero M\u2080 inst\u271d\u00b9 : Nontrivial M\u2080 inst\u271d : NoZeroDivisors M\u2080 L : List M\u2080 \u22a2 prod L = 0 \u2194 0 \u2208 L ** induction' L with a L ihL ** case nil \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u00b2 : MonoidWithZero M\u2080 inst\u271d\u00b9 : Nontrivial M\u2080 inst\u271d : NoZeroDivisors M\u2080 \u22a2 prod [] = 0 \u2194 0 \u2208 [] ** simp ** case cons \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u00b2 : MonoidWithZero M\u2080 inst\u271d\u00b9 : Nontrivial M\u2080 inst\u271d : NoZeroDivisors M\u2080 a : M\u2080 L : List M\u2080 ihL : prod L = 0 \u2194 0 \u2208 L \u22a2 prod (a :: L) = 0 \u2194 0 \u2208 a :: L ** rw [prod_cons, mul_eq_zero, ihL, mem_cons, eq_comm] ** Qed", + "informal": "" + }, + { + "formal": "contDiffGroupoid_prod ** m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : e \u2208 contDiffGroupoid \u22a4 I he' : e' \u2208 contDiffGroupoid \u22a4 I' \u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I') ** cases' he with he he_symm ** case intro m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he' : e' \u2208 contDiffGroupoid \u22a4 I' he : Pregroupoid.property { property := fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I), comp := (_ : \u2200 {f g : H \u2192 H} {u v : Set H}, ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192 IsOpen u \u2192 IsOpen v \u2192 IsOpen (u \u2229 f \u207b\u00b9' v) \u2192 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)) (g \u2218 f) (u \u2229 f \u207b\u00b9' v)), id_mem := (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)), locality := (_ : \u2200 {f : H \u2192 H} {u : Set H}, IsOpen u \u2192 (\u2200 (x : H), x \u2208 u \u2192 \u2203 v, IsOpen v \u2227 x \u2208 v \u2227 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)) f (u \u2229 v)) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)), congr := (_ : \u2200 {f g : H \u2192 H} {u : Set H}, IsOpen u \u2192 (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) } (\u2191e) e.source he_symm : Pregroupoid.property { property := fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I), comp := (_ : \u2200 {f g : H \u2192 H} {u v : Set H}, ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192 IsOpen u \u2192 IsOpen v \u2192 IsOpen (u \u2229 f \u207b\u00b9' v) \u2192 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)) (g \u2218 f) (u \u2229 f \u207b\u00b9' v)), id_mem := (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)), locality := (_ : \u2200 {f : H \u2192 H} {u : Set H}, IsOpen u \u2192 (\u2200 (x : H), x \u2208 u \u2192 \u2203 v, IsOpen v \u2227 x \u2208 v \u2227 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)) f (u \u2229 v)) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)), congr := (_ : \u2200 {f g : H \u2192 H} {u : Set H}, IsOpen u \u2192 (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) } (\u2191(LocalHomeomorph.symm e)) e.target \u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I') ** cases' he' with he' he'_symm ** case intro.intro m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : Pregroupoid.property { property := fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I), comp := (_ : \u2200 {f g : H \u2192 H} {u v : Set H}, ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192 IsOpen u \u2192 IsOpen v \u2192 IsOpen (u \u2229 f \u207b\u00b9' v) \u2192 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)) (g \u2218 f) (u \u2229 f \u207b\u00b9' v)), id_mem := (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)), locality := (_ : \u2200 {f : H \u2192 H} {u : Set H}, IsOpen u \u2192 (\u2200 (x : H), x \u2208 u \u2192 \u2203 v, IsOpen v \u2227 x \u2208 v \u2227 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)) f (u \u2229 v)) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)), congr := (_ : \u2200 {f g : H \u2192 H} {u : Set H}, IsOpen u \u2192 (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) } (\u2191e) e.source he_symm : Pregroupoid.property { property := fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I), comp := (_ : \u2200 {f g : H \u2192 H} {u v : Set H}, ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192 IsOpen u \u2192 IsOpen v \u2192 IsOpen (u \u2229 f \u207b\u00b9' v) \u2192 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)) (g \u2218 f) (u \u2229 f \u207b\u00b9' v)), id_mem := (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)), locality := (_ : \u2200 {f : H \u2192 H} {u : Set H}, IsOpen u \u2192 (\u2200 (x : H), x \u2208 u \u2192 \u2203 v, IsOpen v \u2227 x \u2208 v \u2227 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)) f (u \u2229 v)) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)), congr := (_ : \u2200 {f g : H \u2192 H} {u : Set H}, IsOpen u \u2192 (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) } (\u2191(LocalHomeomorph.symm e)) e.target he' : Pregroupoid.property { property := fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'), comp := (_ : \u2200 {f g : H' \u2192 H'} {u v : Set H'}, ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' v \u2229 range \u2191I') \u2192 IsOpen u \u2192 IsOpen v \u2192 IsOpen (u \u2229 f \u207b\u00b9' v) \u2192 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I')) (g \u2218 f) (u \u2229 f \u207b\u00b9' v)), id_mem := (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 id \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' univ \u2229 range \u2191I')), locality := (_ : \u2200 {f : H' \u2192 H'} {u : Set H'}, IsOpen u \u2192 (\u2200 (x : H'), x \u2208 u \u2192 \u2203 v, IsOpen v \u2227 x \u2208 v \u2227 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I')) f (u \u2229 v)) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')), congr := (_ : \u2200 {f g : H' \u2192 H'} {u : Set H'}, IsOpen u \u2192 (\u2200 (x : H'), x \u2208 u \u2192 g x = f x) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')) } (\u2191e') e'.source he'_symm : Pregroupoid.property { property := fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'), comp := (_ : \u2200 {f g : H' \u2192 H'} {u v : Set H'}, ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' v \u2229 range \u2191I') \u2192 IsOpen u \u2192 IsOpen v \u2192 IsOpen (u \u2229 f \u207b\u00b9' v) \u2192 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I')) (g \u2218 f) (u \u2229 f \u207b\u00b9' v)), id_mem := (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 id \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' univ \u2229 range \u2191I')), locality := (_ : \u2200 {f : H' \u2192 H'} {u : Set H'}, IsOpen u \u2192 (\u2200 (x : H'), x \u2208 u \u2192 \u2203 v, IsOpen v \u2227 x \u2208 v \u2227 (fun f s => ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I')) f (u \u2229 v)) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')), congr := (_ : \u2200 {f g : H' \u2192 H'} {u : Set H'}, IsOpen u \u2192 (\u2200 (x : H'), x \u2208 u \u2192 g x = f x) \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192 ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')) } (\u2191(LocalHomeomorph.symm e')) e'.target \u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I') ** simp only at he he_symm he' he'_symm ** case intro.intro m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') \u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I') ** constructor <;> simp only [LocalEquiv.prod_source, LocalHomeomorph.prod_toLocalEquiv] ** case intro.intro.left m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') \u22a2 ContDiffOn \ud835\udd5c \u22a4 (\u2191(ModelWithCorners.prod I I') \u2218 \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I'))) (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229 range \u2191(ModelWithCorners.prod I I')) ** have h3 := ContDiffOn.prod_map he he' ** case intro.intro.left m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') h3 : ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I'))) ((\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) \u00d7\u02e2 (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')) \u22a2 ContDiffOn \ud835\udd5c \u22a4 (\u2191(ModelWithCorners.prod I I') \u2218 \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I'))) (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229 range \u2191(ModelWithCorners.prod I I')) ** rw [\u2190 I.image_eq, \u2190 I'.image_eq, Set.prod_image_image_eq] at h3 ** case intro.intro.left m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') h3 : ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I'))) ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.source \u00d7\u02e2 e'.source) \u22a2 ContDiffOn \ud835\udd5c \u22a4 (\u2191(ModelWithCorners.prod I I') \u2218 \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I'))) (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229 range \u2191(ModelWithCorners.prod I I')) ** rw [\u2190 (I.prod I').image_eq] ** case intro.intro.left m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') h3 : ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I'))) ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.source \u00d7\u02e2 e'.source) \u22a2 ContDiffOn \ud835\udd5c \u22a4 (\u2191(ModelWithCorners.prod I I') \u2218 \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I'))) (\u2191(ModelWithCorners.prod I I') '' e.source \u00d7\u02e2 e'.source) ** exact h3 ** case intro.intro.right m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') \u22a2 ContDiffOn \ud835\udd5c \u22a4 (\u2191(ModelWithCorners.prod I I') \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I'))) (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229 range \u2191(ModelWithCorners.prod I I')) ** have h3 := ContDiffOn.prod_map he_symm he'_symm ** case intro.intro.right m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') h3 : ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))) ((\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) \u00d7\u02e2 (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')) \u22a2 ContDiffOn \ud835\udd5c \u22a4 (\u2191(ModelWithCorners.prod I I') \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I'))) (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229 range \u2191(ModelWithCorners.prod I I')) ** rw [\u2190 I.image_eq, \u2190 I'.image_eq, Set.prod_image_image_eq] at h3 ** case intro.intro.right m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') h3 : ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))) ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.target \u00d7\u02e2 e'.target) \u22a2 ContDiffOn \ud835\udd5c \u22a4 (\u2191(ModelWithCorners.prod I I') \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I'))) (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229 range \u2191(ModelWithCorners.prod I I')) ** rw [\u2190 (I.prod I').image_eq] ** case intro.intro.right m n : \u2115\u221e \ud835\udd5c : Type u_1 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u2074 : TopologicalSpace H I\u271d : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3 : TopologicalSpace M E' : Type u_5 H' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \ud835\udd5c E' inst\u271d : TopologicalSpace H' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' e : LocalHomeomorph H H e' : LocalHomeomorph H' H' he : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) he_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) he' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I') he'_symm : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I') h3 : ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))) ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.target \u00d7\u02e2 e'.target) \u22a2 ContDiffOn \ud835\udd5c \u22a4 (\u2191(ModelWithCorners.prod I I') \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I'))) (\u2191(ModelWithCorners.prod I I') '' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target) ** exact h3 ** Qed", + "informal": "" + }, + { + "formal": "TruncatedWittVector.ext_iff ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 x y : TruncatedWittVector p n R h : x = y i : Fin n \u22a2 coeff i x = coeff i y ** rw [h] ** Qed", + "informal": "" + }, + { + "formal": "Orientation.rightAngleRotationAux\u2081_rightAngleRotationAux\u2081 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x : E \u22a2 \u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x) = -x ** apply ext_inner_left \u211d ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x : E \u22a2 \u2200 (v : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)), inner v (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner v (-x) ** intro y ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x : E y : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x) \u22a2 inner y (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner y (-x) ** have : \u27eao.rightAngleRotationAux\u2081 y, o.rightAngleRotationAux\u2081 x\u27eb = \u27eay, x\u27eb :=\n LinearIsometry.inner_map_map o.rightAngleRotationAux\u2082 y x ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x : E y : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x) this : inner (\u2191(rightAngleRotationAux\u2081 o) y) (\u2191(rightAngleRotationAux\u2081 o) x) = inner y x \u22a2 inner y (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner y (-x) ** rw [o.inner_rightAngleRotationAux\u2081_right, \u2190 o.inner_rightAngleRotationAux\u2081_left, this,\n inner_neg_right] ** Qed", + "informal": "" + }, + { + "formal": "Basis.equiv_trans ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : Module R M inst\u271d\u00b3 : AddCommMonoid M' inst\u271d\u00b2 : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x : M b' : Basis \u03b9' R M' e\u271d : \u03b9 \u2243 \u03b9' inst\u271d\u00b9 : AddCommMonoid M'' inst\u271d : Module R M'' \u03b9'' : Type u_10 b'' : Basis \u03b9'' R M'' e : \u03b9 \u2243 \u03b9' e' : \u03b9' \u2243 \u03b9'' i : \u03b9 \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) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Nat.Partrec'.tail ** n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : Partrec' f v : Vector \u2115 (succ n) \u22a2 (mOfFn fun i => (\u2191fun v => Vector.get v (Fin.succ i)) v) >>= f = f (Vector.tail v) ** simp ** n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : Partrec' f v : Vector \u2115 (succ n) \u22a2 f (ofFn fun i => Vector.get v (Fin.succ i)) = f (Vector.tail v) ** rw [\u2190 ofFn_get v.tail] ** n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : Partrec' f v : Vector \u2115 (succ n) \u22a2 f (ofFn fun i => Vector.get v (Fin.succ i)) = f (ofFn (Vector.get (Vector.tail v))) ** congr ** case e_a.e_a n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : Partrec' f v : Vector \u2115 (succ n) \u22a2 (fun i => Vector.get v (Fin.succ i)) = Vector.get (Vector.tail v) ** funext i ** case e_a.e_a.h n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : Partrec' f v : Vector \u2115 (succ n) i : Fin n \u22a2 Vector.get v (Fin.succ i) = Vector.get (Vector.tail v) i ** simp ** Qed", + "informal": "" + }, + { + "formal": "LieAlgebra.center_eq_bot_of_semisimple ** R : Type u L : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : LieRing L inst\u271d : LieAlgebra R L h : IsSemisimple R L \u22a2 center R L = \u22a5 ** rw [isSemisimple_iff_no_abelian_ideals] at h ** R : Type u L : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : LieRing L inst\u271d : LieAlgebra R L h : \u2200 (I : LieIdeal R L), IsLieAbelian { x // x \u2208 \u2191I } \u2192 I = \u22a5 \u22a2 center R L = \u22a5 ** apply h ** case a R : Type u L : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : LieRing L inst\u271d : LieAlgebra R L h : \u2200 (I : LieIdeal R L), IsLieAbelian { x // x \u2208 \u2191I } \u2192 I = \u22a5 \u22a2 IsLieAbelian { x // x \u2208 \u2191(center R L) } ** infer_instance ** Qed", + "informal": "" + }, + { + "formal": "Set.relatively_discrete_of_finite ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : EMetricSpace \u03b1 x y z : \u03b1 s t : Set \u03b1 C : \u211d\u22650\u221e sC : Set \u211d\u22650\u221e inst\u271d : Finite \u2191s \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 ** rw [\u2190 einfsep_pos] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : EMetricSpace \u03b1 x y z : \u03b1 s t : Set \u03b1 C : \u211d\u22650\u221e sC : Set \u211d\u22650\u221e inst\u271d : Finite \u2191s \u22a2 0 < einfsep s ** exact einfsep_pos_of_finite ** Qed", + "informal": "" + }, + { + "formal": "Multiset.fold_bind ** \u03b1 : Type u_1 \u03b2 : Type u_2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 hc : IsCommutative \u03b1 op ha : IsAssociative \u03b1 op \u03b9 : Type u_3 s : Multiset \u03b9 t : \u03b9 \u2192 Multiset \u03b1 b : \u03b9 \u2192 \u03b1 b\u2080 : \u03b1 \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) ** induction' s using Multiset.induction_on with a ha ih ** case empty \u03b1 : Type u_1 \u03b2 : Type u_2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 hc : IsCommutative \u03b1 op ha : IsAssociative \u03b1 op \u03b9 : Type u_3 t : \u03b9 \u2192 Multiset \u03b1 b : \u03b9 \u2192 \u03b1 b\u2080 : \u03b1 \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) ** rw [zero_bind, map_zero, map_zero, fold_zero] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 hc : IsCommutative \u03b1 op ha\u271d : IsAssociative \u03b1 op \u03b9 : Type u_3 t : \u03b9 \u2192 Multiset \u03b1 b : \u03b9 \u2192 \u03b1 b\u2080 : \u03b1 a : \u03b9 ha : Multiset \u03b9 ih : 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) \u22a2 fold op (fold op b\u2080 (map b (a ::\u2098 ha))) (bind (a ::\u2098 ha) t) = fold op b\u2080 (map (fun i => fold op (b i) (t i)) (a ::\u2098 ha)) ** rw [cons_bind, map_cons, map_cons, fold_cons_left, fold_cons_left, fold_add, ih] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Presieve.extend_agrees ** C : Type u\u2081 inst\u271d : Category.{v\u2081, u\u2081} C P Q U : C\u1d52\u1d56 \u2964 Type w X Y : C S : Sieve X R : Presieve X J J\u2082 : GrothendieckTopology C x : FamilyOfElements P R t : FamilyOfElements.Compatible x f : Y \u27f6 X hf : R f \u22a2 FamilyOfElements.sieveExtend x f (_ : f \u2208 (generate R).arrows) = x f hf ** have h := (le_generate R Y hf).choose_spec ** C : Type u\u2081 inst\u271d : Category.{v\u2081, u\u2081} C P Q U : C\u1d52\u1d56 \u2964 Type w X Y : C S : Sieve X R : Presieve X J J\u2082 : GrothendieckTopology C x : FamilyOfElements P R t : FamilyOfElements.Compatible x f : Y \u27f6 X hf : R f h : \u2203 h g, R g \u2227 h \u226b g = f \u22a2 FamilyOfElements.sieveExtend x f (_ : f \u2208 (generate R).arrows) = x f hf ** unfold FamilyOfElements.sieveExtend ** C : Type u\u2081 inst\u271d : Category.{v\u2081, u\u2081} C P Q U : C\u1d52\u1d56 \u2964 Type w X Y : C S : Sieve X R : Presieve X J J\u2082 : GrothendieckTopology C x : FamilyOfElements P R t : FamilyOfElements.Compatible x f : Y \u27f6 X hf : R f h : \u2203 h g, R g \u2227 h \u226b g = f \u22a2 P.map (Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f)).op (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f)) (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f)))) = x f hf ** rw [t h.choose (\ud835\udfd9 _) _ hf _] ** C : Type u\u2081 inst\u271d : Category.{v\u2081, u\u2081} C P Q U : C\u1d52\u1d56 \u2964 Type w X Y : C S : Sieve X R : Presieve X J J\u2082 : GrothendieckTopology C x : FamilyOfElements P R t : FamilyOfElements.Compatible x f : Y \u27f6 X hf : R f h : \u2203 h g, R g \u2227 h \u226b g = f \u22a2 P.map (\ud835\udfd9 Y).op (x f hf) = x f hf ** simp ** C : Type u\u2081 inst\u271d : Category.{v\u2081, u\u2081} C P Q U : C\u1d52\u1d56 \u2964 Type w X Y : C S : Sieve X R : Presieve X J J\u2082 : GrothendieckTopology C x : FamilyOfElements P R t : FamilyOfElements.Compatible x f : Y \u27f6 X hf : R f h : \u2203 h g, R g \u2227 h \u226b g = f \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 ** rw [id_comp] ** C : Type u\u2081 inst\u271d : Category.{v\u2081, u\u2081} C P Q U : C\u1d52\u1d56 \u2964 Type w X Y : C S : Sieve X R : Presieve X J J\u2082 : GrothendieckTopology C x : FamilyOfElements P R t : FamilyOfElements.Compatible x f : Y \u27f6 X hf : R f h : \u2203 h g, R g \u2227 h \u226b g = f \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 ** exact h.choose_spec.choose_spec.2 ** Qed", + "informal": "" + }, + { + "formal": "NormedSpace.isVonNBounded_iff ** \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E \u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 Bornology.IsBounded s ** rw [Metric.isBounded_iff_subset_closedBall (0 : E)] ** \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E \u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 \u2203 r, s \u2286 Metric.closedBall 0 r ** constructor ** case mp \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E \u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2192 \u2203 r, s \u2286 Metric.closedBall 0 r ** intro h ** case mp \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E h : Bornology.IsVonNBounded \ud835\udd5c s \u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r ** rcases h (Metric.ball_mem_nhds 0 zero_lt_one) with \u27e8\u03c1, h\u03c1, h\u03c1ball\u27e9 ** case mp.intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E h : Bornology.IsVonNBounded \ud835\udd5c s \u03c1 : \u211d h\u03c1 : 0 < \u03c1 h\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1 \u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r ** rcases NormedField.exists_lt_norm \ud835\udd5c \u03c1 with \u27e8a, ha\u27e9 ** case mp.intro.intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E h : Bornology.IsVonNBounded \ud835\udd5c s \u03c1 : \u211d h\u03c1 : 0 < \u03c1 h\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1 a : \ud835\udd5c ha : \u03c1 < \u2016a\u2016 \u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r ** specialize h\u03c1ball a ha.le ** case mp.intro.intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E h : Bornology.IsVonNBounded \ud835\udd5c s \u03c1 : \u211d h\u03c1 : 0 < \u03c1 a : \ud835\udd5c ha : \u03c1 < \u2016a\u2016 h\u03c1ball : s \u2286 a \u2022 Metric.ball 0 1 \u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r ** 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 ** case mp.intro.intro.intro \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E h : Bornology.IsVonNBounded \ud835\udd5c s \u03c1 : \u211d h\u03c1 : 0 < \u03c1 a : \ud835\udd5c ha : \u03c1 < \u2016a\u2016 h\u03c1ball : s \u2286 Metric.ball 0 \u2016a\u2016 \u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r ** exact \u27e8\u2016a\u2016, h\u03c1ball.trans Metric.ball_subset_closedBall\u27e9 ** case mpr \ud835\udd5c : Type u_1 \ud835\udd5c' : Type u_2 E : Type u_3 E' : Type u_4 F : Type u_5 \u03b9 : Type u_6 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : SeminormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E s : Set E \u22a2 (\u2203 r, s \u2286 Metric.closedBall 0 r) \u2192 Bornology.IsVonNBounded \ud835\udd5c s ** exact fun \u27e8C, hC\u27e9 => (isVonNBounded_closedBall \ud835\udd5c E C).subset hC ** Qed", + "informal": "" + }, + { + "formal": "List.get?_inj ** i : Nat \u03b1\u271d : Type u_1 xs : List \u03b1\u271d j : Nat h\u2080 : i < length xs h\u2081 : Nodup xs h\u2082 : get? xs i = get? xs j \u22a2 i = j ** 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 ** case nil \u03b1\u271d : Type u_1 i j : Nat h\u2080 : i < length [] h\u2081 : Nodup [] h\u2082 : get? [] i = get? [] j \u22a2 i = j ** cases h\u2080 ** case cons \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j : Nat h\u2080 : i < length (x :: xs) h\u2081 : Nodup (x :: xs) h\u2082 : get? (x :: xs) i = get? (x :: xs) j \u22a2 i = j ** 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 => ?_ ** case cons.refine_1 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i\u271d j : Nat h\u2081 : Nodup (x :: xs) i : Nat h\u2080 : i + 1 < length (x :: xs) h\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) 0 \u22a2 i + 1 = 0 case cons.refine_2 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d : Nat h\u2081 : Nodup (x :: xs) j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : get? (x :: xs) 0 = get? (x :: xs) (j + 1) \u22a2 0 = j + 1 ** 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?] ** case cons.refine_1.cons.refine_1 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i\u271d j i : Nat h\u2080 : i + 1 < length (x :: xs) h\u2082 : get? xs i = some x h' : Pairwise (fun x x_1 => x \u2260 x_1) xs h : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' \u22a2 \u2203 n, get? xs n = some x case cons.refine_2.cons.refine_1 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : some x = get? xs j h' : Pairwise (fun x x_1 => x \u2260 x_1) xs h : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' \u22a2 \u2203 n, get? xs n = some x ** exact \u27e8_, h\u2082\u27e9 ** case cons.refine_2.cons.refine_1 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : some x = get? xs j h' : Pairwise (fun x x_1 => x \u2260 x_1) xs h : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' \u22a2 \u2203 n, get? xs n = some x ** exact \u27e8_ , h\u2082.symm\u27e9 ** \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j : Nat h\u2081 : Nodup (x :: xs) h\u2080 : 0 < length (x :: xs) h\u2082 : get? (x :: xs) 0 = get? (x :: xs) 0 \u22a2 0 = 0 ** rfl ** \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i\u271d j\u271d : Nat h\u2081 : Nodup (x :: xs) i j : Nat h\u2080 : i + 1 < length (x :: xs) h\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) (j + 1) \u22a2 i + 1 = j + 1 ** simp ** \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i\u271d j\u271d : Nat h\u2081 : Nodup (x :: xs) i j : Nat h\u2080 : i + 1 < length (x :: xs) h\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) (j + 1) \u22a2 i = j ** cases h\u2081 with\n| cons ha h\u2081 => exact ih (Nat.lt_of_succ_lt_succ h\u2080) h\u2081 h\u2082 ** case cons \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i\u271d j\u271d i j : Nat h\u2080 : i + 1 < length (x :: xs) h\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) (j + 1) h\u2081 : Pairwise (fun x x_1 => x \u2260 x_1) xs ha : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' \u22a2 i = j ** exact ih (Nat.lt_of_succ_lt_succ h\u2080) h\u2081 h\u2082 ** case cons.refine_2 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d : Nat h\u2081 : Nodup (x :: xs) j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : get? (x :: xs) 0 = get? (x :: xs) (j + 1) \u22a2 0 = j + 1 ** simp at h\u2082 ** case cons.refine_2 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d : Nat h\u2081 : Nodup (x :: xs) j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : some x = get? xs j \u22a2 0 = j + 1 ** cases h\u2081 ** case cons.refine_2.cons \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : some x = get? xs j a\u271d\u00b9 : Pairwise (fun x x_1 => x \u2260 x_1) xs a\u271d : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' \u22a2 0 = j + 1 ** rename_i h' h ** case cons.refine_2.cons \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : some x = get? xs j h' : Pairwise (fun x x_1 => x \u2260 x_1) xs h : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' \u22a2 0 = j + 1 ** have := h x ?_ rfl ** case cons.refine_2.cons.refine_2 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : some x = get? xs j h' : Pairwise (fun x x_1 => x \u2260 x_1) xs h : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' this : False \u22a2 0 = j + 1 case cons.refine_2.cons.refine_1 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : some x = get? xs j h' : Pairwise (fun x x_1 => x \u2260 x_1) xs h : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' \u22a2 x \u2208 xs ** cases this ** case cons.refine_2.cons.refine_1 \u03b1\u271d : Type u_1 x : \u03b1\u271d xs : List \u03b1\u271d ih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j i j\u271d j : Nat h\u2080 : 0 < length (x :: xs) h\u2082 : some x = get? xs j h' : Pairwise (fun x x_1 => x \u2260 x_1) xs h : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a' \u22a2 x \u2208 xs ** rw [mem_iff_get?] ** Qed", + "informal": "" + }, + { + "formal": "DedekindDomain.ProdAdicCompletions.IsFiniteAdele.zero ** R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v : HeightOneSpectrum R \u22a2 IsFiniteAdele 0 ** rw [IsFiniteAdele, Filter.eventually_cofinite] ** R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v : HeightOneSpectrum R \u22a2 Set.Finite {x | \u00acOfNat.ofNat 0 x \u2208 adicCompletionIntegers K x} ** 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' _ ** R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v : HeightOneSpectrum R h_empty : {v | \u00ac0 \u2208 adicCompletionIntegers K v} = \u2205 \u22a2 Set.Finite {x | \u00acOfNat.ofNat 0 x \u2208 adicCompletionIntegers K x} ** convert finite_empty ** R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v : HeightOneSpectrum R \u22a2 {v | \u00ac0 \u2208 adicCompletionIntegers K v} = \u2205 ** ext v ** case h R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v\u271d v : HeightOneSpectrum R \u22a2 v \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v} \u2194 v \u2208 \u2205 ** rw [mem_empty_iff_false, iff_false_iff] ** case h R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v\u271d v : HeightOneSpectrum R \u22a2 \u00acv \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v} ** intro hv ** case h R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v\u271d v : HeightOneSpectrum R hv : v \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v} \u22a2 False ** rw [mem_setOf] at hv ** case h R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v\u271d v : HeightOneSpectrum R hv : \u00ac0 \u2208 adicCompletionIntegers K v \u22a2 False ** apply hv ** case h R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v\u271d v : HeightOneSpectrum R hv : \u00ac0 \u2208 adicCompletionIntegers K v \u22a2 0 \u2208 adicCompletionIntegers K v ** rw [mem_adicCompletionIntegers] ** case h R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v\u271d v : HeightOneSpectrum R hv : \u00ac0 \u2208 adicCompletionIntegers K v \u22a2 \u2191Valued.v 0 \u2264 1 ** have h_zero : (Valued.v (0 : v.adicCompletion K) : WithZero (Multiplicative \u2124)) = 0 :=\n Valued.v.map_zero' ** case h R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v\u271d v : HeightOneSpectrum R hv : \u00ac0 \u2208 adicCompletionIntegers K v h_zero : \u2191Valued.v 0 = 0 \u22a2 \u2191Valued.v 0 \u2264 1 ** rw [h_zero] ** case h R : Type u_1 K : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : IsDedekindDomain R inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K v\u271d v : HeightOneSpectrum R hv : \u00ac0 \u2208 adicCompletionIntegers K v h_zero : \u2191Valued.v 0 = 0 \u22a2 0 \u2264 1 ** exact zero_le_one' _ ** Qed", + "informal": "" + }, + { + "formal": "inv_le_inv ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedSemifield \u03b1 a b c d e : \u03b1 m n : \u2124 ha : 0 < a hb : 0 < b \u22a2 a\u207b\u00b9 \u2264 b\u207b\u00b9 \u2194 b \u2264 a ** rw [\u2190 one_div, div_le_iff ha, \u2190 div_eq_inv_mul, le_div_iff hb, one_mul] ** Qed", + "informal": "" + }, + { + "formal": "div_self' ** \u03b1 : Type u_1 \u03b2 : Type u_2 G : Type u_3 inst\u271d : Group G a\u271d b c d a : G \u22a2 a / a = 1 ** rw [div_eq_mul_inv, mul_right_inv a] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.snorm_indicator_const\u2080 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G hs : NullMeasurableSet s hp : p \u2260 0 hp_top : p \u2260 \u22a4 hp_pos : 0 < ENNReal.toReal p \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator s (fun x => c) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) = (\u222b\u207b (x : \u03b1), Set.indicator s (fun x => \u2191\u2016c\u2016\u208a ^ ENNReal.toReal p) x \u2202\u03bc) ^ (1 / ENNReal.toReal p) ** congr 2 ** case e_a.e_f \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G hs : NullMeasurableSet s hp : p \u2260 0 hp_top : p \u2260 \u22a4 hp_pos : 0 < ENNReal.toReal p \u22a2 (fun x => \u2191\u2016Set.indicator s (fun x => c) x\u2016\u208a ^ ENNReal.toReal p) = fun x => Set.indicator s (fun x => \u2191\u2016c\u2016\u208a ^ ENNReal.toReal p) x ** refine (Set.comp_indicator_const c (fun x : G \u21a6 (\u2016x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) ?_) ** case e_a.e_f \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G hs : NullMeasurableSet s hp : p \u2260 0 hp_top : p \u2260 \u22a4 hp_pos : 0 < ENNReal.toReal p \u22a2 (fun x => \u2191\u2016x\u2016\u208a ^ ENNReal.toReal p) 0 = 0 ** simp [hp_pos] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G hs : NullMeasurableSet s hp : p \u2260 0 hp_top : p \u2260 \u22a4 hp_pos : 0 < ENNReal.toReal p \u22a2 (\u222b\u207b (x : \u03b1), Set.indicator s (fun x => \u2191\u2016c\u2016\u208a ^ ENNReal.toReal p) x \u2202\u03bc) ^ (1 / ENNReal.toReal p) = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** 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] ** case hz \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G hs : NullMeasurableSet s hp : p \u2260 0 hp_top : p \u2260 \u22a4 hp_pos : 0 < ENNReal.toReal p \u22a2 0 \u2264 1 / ENNReal.toReal p ** positivity ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.NewtonIdentities.disjUnion_filter_pairs_eq_pairs ** \u03c3 : Type u_1 inst\u271d\u00b2 : Fintype \u03c3 inst\u271d\u00b9 : DecidableEq \u03c3 R : Type u_2 inst\u271d : CommRing R k : \u2115 \u22a2 disjUnion (filter (fun t => card t.1 < k) (MvPolynomial.NewtonIdentities.pairs \u03c3 k)) (filter (fun t => card t.1 = k) (MvPolynomial.NewtonIdentities.pairs \u03c3 k)) (_ : Disjoint (filter (fun t => card t.1 < k) (MvPolynomial.NewtonIdentities.pairs \u03c3 k)) (filter (fun t => card t.1 = k) (MvPolynomial.NewtonIdentities.pairs \u03c3 k))) = MvPolynomial.NewtonIdentities.pairs \u03c3 k ** simp only [disjUnion_eq_union, Finset.ext_iff, pairs, filter_filter, mem_filter] ** \u03c3 : Type u_1 inst\u271d\u00b2 : Fintype \u03c3 inst\u271d\u00b9 : DecidableEq \u03c3 R : Type u_2 inst\u271d : CommRing R k : \u2115 \u22a2 \u2200 (a : Finset \u03c3 \u00d7 \u03c3), a \u2208 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 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 a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1) ** intro a ** \u03c3 : Type u_1 inst\u271d\u00b2 : Fintype \u03c3 inst\u271d\u00b9 : DecidableEq \u03c3 R : Type u_2 inst\u271d : CommRing R k : \u2115 a : Finset \u03c3 \u00d7 \u03c3 \u22a2 a \u2208 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 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 a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1) ** rw [\u2190 filter_or, mem_filter] ** \u03c3 : Type u_1 inst\u271d\u00b2 : Fintype \u03c3 inst\u271d\u00b9 : DecidableEq \u03c3 R : Type u_2 inst\u271d : CommRing R k : \u2115 a : Finset \u03c3 \u00d7 \u03c3 \u22a2 a \u2208 univ \u2227 ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228 (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) \u2194 a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1) ** refine' \u27e8fun ha \u21a6 by tauto, fun ha \u21a6 _\u27e9 ** \u03c3 : Type u_1 inst\u271d\u00b2 : Fintype \u03c3 inst\u271d\u00b9 : DecidableEq \u03c3 R : Type u_2 inst\u271d : CommRing R k : \u2115 a : Finset \u03c3 \u00d7 \u03c3 ha : a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1) \u22a2 a \u2208 univ \u2227 ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228 (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) ** have hacard := le_iff_lt_or_eq.mp ha.2.1 ** \u03c3 : Type u_1 inst\u271d\u00b2 : Fintype \u03c3 inst\u271d\u00b9 : DecidableEq \u03c3 R : Type u_2 inst\u271d : CommRing R k : \u2115 a : Finset \u03c3 \u00d7 \u03c3 ha : a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1) hacard : card a.1 < k \u2228 card a.1 = k \u22a2 a \u2208 univ \u2227 ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228 (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) ** tauto ** \u03c3 : Type u_1 inst\u271d\u00b2 : Fintype \u03c3 inst\u271d\u00b9 : DecidableEq \u03c3 R : Type u_2 inst\u271d : CommRing R k : \u2115 a : Finset \u03c3 \u00d7 \u03c3 ha : a \u2208 univ \u2227 ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228 (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) \u22a2 a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1) ** tauto ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Walk.darts_dropUntil_subset ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' inst\u271d : DecidableEq V u v w : V p : Walk G v w h : u \u2208 support p x : Dart G hx : x \u2208 darts (dropUntil p u h) \u22a2 x \u2208 darts p ** rw [\u2190 take_spec p h, darts_append, List.mem_append] ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' inst\u271d : DecidableEq V u v w : V p : Walk G v w h : u \u2208 support p x : Dart G hx : x \u2208 darts (dropUntil p u h) \u22a2 x \u2208 darts (takeUntil p u h) \u2228 x \u2208 darts (dropUntil p u h) ** exact Or.inr hx ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.terminal.comp_from ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : HasTerminal C P Q : C f : P \u27f6 Q \u22a2 f \u226b from Q = from P ** aesop ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.sup_eq_lsub_or_sup_succ_eq_lsub ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u22a2 sup f = lsub f \u2228 succ (sup f) = lsub f ** cases' eq_or_lt_of_le (sup_le_lsub.{_, v} f) with h h ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} h : sup f = lsub f \u22a2 sup f = lsub f \u2228 succ (sup f) = lsub f ** exact Or.inl h ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} h : sup f < lsub f \u22a2 sup f = lsub f \u2228 succ (sup f) = lsub f ** exact Or.inr ((succ_le_of_lt h).antisymm (lsub_le_sup_succ f)) ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.isLocallySurjective_iff_whisker_forget ** C : Type u inst\u271d\u00b2 : Category.{v, u} C J : GrothendieckTopology C A : Type u' inst\u271d\u00b9 : Category.{v', u'} A inst\u271d : ConcreteCategory A F G : C\u1d52\u1d56 \u2964 A f : F \u27f6 G \u22a2 IsLocallySurjective J f \u2194 IsLocallySurjective J (whiskerRight f (forget A)) ** simp only [isLocallySurjective_iff_imagePresheaf_sheafify_eq_top] ** C : Type u inst\u271d\u00b2 : Category.{v, u} C J : GrothendieckTopology C A : Type u' inst\u271d\u00b9 : Category.{v', u'} A inst\u271d : ConcreteCategory A F G : C\u1d52\u1d56 \u2964 A f : F \u27f6 G \u22a2 Subpresheaf.sheafify J (imagePresheaf (whiskerRight f (forget A))) = \u22a4 \u2194 Subpresheaf.sheafify J (imagePresheaf (whiskerRight (whiskerRight f (forget A)) (forget (Type w')))) = \u22a4 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "orthogonalProjection_norm_le ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : InnerProductSpace \u211d F K : Submodule \ud835\udd5c E inst\u271d : HasOrthogonalProjection K \u22a2 0 \u2264 1 ** norm_num ** Qed", + "informal": "" + }, + { + "formal": "Nat.choose_two_right ** n : \u2115 \u22a2 choose n 2 = n * (n - 1) / 2 ** induction' n with n ih ** case zero \u22a2 choose zero 2 = zero * (zero - 1) / 2 ** simp ** case succ n : \u2115 ih : choose n 2 = n * (n - 1) / 2 \u22a2 choose (succ n) 2 = succ n * (succ n - 1) / 2 ** rw [triangle_succ n, choose, ih] ** case succ n : \u2115 ih : choose n 2 = n * (n - 1) / 2 \u22a2 choose n 1 + n * (n - 1) / 2 = n * (n - 1) / 2 + n ** simp [add_comm] ** Qed", + "informal": "" + }, + { + "formal": "BilinForm.comp_id_id ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B\u271d : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' B : BilinForm R M \u22a2 comp B LinearMap.id LinearMap.id = B ** ext ** case H R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B\u271d : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' B : BilinForm R M x\u271d y\u271d : M \u22a2 bilin (comp B LinearMap.id LinearMap.id) x\u271d y\u271d = bilin B x\u271d y\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "integral_sin_sq_mul_cos ** a b : \u211d n : \u2115 \u22a2 \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3 ** have := @integral_sin_pow_mul_cos_pow_odd a b 2 0 ** a b : \u211d n : \u2115 this : \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 \u22a2 \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3 ** norm_num at this ** a b : \u211d n : \u2115 this : \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3 \u22a2 \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3 ** exact this ** Qed", + "informal": "" + }, + { + "formal": "countable_cover_nhdsWithin_of_sigma_compact ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 s\u271d t : Set \u03b1 inst\u271d : SigmaCompactSpace \u03b1 f : \u03b1 \u2192 Set \u03b1 s : Set \u03b1 hs : IsClosed s hf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 \ud835\udcdd[s] x \u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, f x ** simp only [nhdsWithin, mem_inf_principal] at hf ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 s\u271d t : Set \u03b1 inst\u271d : SigmaCompactSpace \u03b1 f : \u03b1 \u2192 Set \u03b1 s : Set \u03b1 hs : IsClosed s hf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x \u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, f x ** 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 ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 s\u271d t\u271d : Set \u03b1 inst\u271d : SigmaCompactSpace \u03b1 f : \u03b1 \u2192 Set \u03b1 s : Set \u03b1 hs : IsClosed s hf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x t : \u2115 \u2192 Finset \u03b1 ht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s hsub : \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} \u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, f x ** 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 ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 s\u271d t\u271d : Set \u03b1 inst\u271d : SigmaCompactSpace \u03b1 f : \u03b1 \u2192 Set \u03b1 s : Set \u03b1 hs : IsClosed s hf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x t : \u2115 \u2192 Finset \u03b1 ht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s hsub : \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} x : \u03b1 hx : x \u2208 s \u22a2 \u2203 i j, x \u2208 f i ** rcases exists_mem_compactCovering x with \u27e8n, hn\u27e9 ** case intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 s\u271d t\u271d : Set \u03b1 inst\u271d : SigmaCompactSpace \u03b1 f : \u03b1 \u2192 Set \u03b1 s : Set \u03b1 hs : IsClosed s hf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x t : \u2115 \u2192 Finset \u03b1 ht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s hsub : \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} x : \u03b1 hx : x \u2208 s n : \u2115 hn : x \u2208 compactCovering \u03b1 n \u22a2 \u2203 i j, x \u2208 f i ** 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 ** case intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Type u_1 \u03c0 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 s\u271d t\u271d : Set \u03b1 inst\u271d : SigmaCompactSpace \u03b1 f : \u03b1 \u2192 Set \u03b1 s : Set \u03b1 hs : IsClosed s hf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x t : \u2115 \u2192 Finset \u03b1 ht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s hsub : \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} x : \u03b1 hx : x \u2208 s n : \u2115 hn : x \u2208 compactCovering \u03b1 n y : \u03b1 hyt : y \u2208 t n hyf : x \u2208 s \u2192 x \u2208 f y \u22a2 \u2203 i j, x \u2208 f i ** exact \u27e8y, mem_iUnion.2 \u27e8n, hyt\u27e9, hyf hx\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "measurableSet_region_between_oc ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s \u22a2 MeasurableSet {p | p.1 \u2208 s \u2227 p.2 \u2208 Ioc (f p.1) (g p.1)} ** dsimp only [regionBetween, Ioc, mem_setOf_eq, setOf_and] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s \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}}) ** refine'\n MeasurableSet.inter _\n ((measurableSet_lt (hf.rst.immp measurable_fst) measurable_snd).inter\n (measurableSet_le measurable_snd (hg.comp measurable_fst))) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s \u22a2 MeasurableSet {a | a.1 \u2208 s} ** exact measurable_fst hs ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.exists_countable_dense_no_zero_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e \u22a2 \u2203 s, Set.Countable s \u2227 Dense s \u2227 \u00ac0 \u2208 s \u2227 \u00ac\u22a4 \u2208 s ** 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 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s\u271d s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s hs : (\u2200 (x : \u211d\u22650\u221e), IsBot x \u2192 \u00acx \u2208 s) \u2227 \u2200 (x : \u211d\u22650\u221e), IsTop x \u2192 \u00acx \u2208 s \u22a2 \u2203 s, Set.Countable s \u2227 Dense s \u2227 \u00ac0 \u2208 s \u2227 \u00ac\u22a4 \u2208 s ** exact \u27e8s, s_count, s_dense, fun h => hs.1 0 (by simp) h, fun h => hs.2 \u221e (by simp) h\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s\u271d s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s hs : (\u2200 (x : \u211d\u22650\u221e), IsBot x \u2192 \u00acx \u2208 s) \u2227 \u2200 (x : \u211d\u22650\u221e), IsTop x \u2192 \u00acx \u2208 s h : 0 \u2208 s \u22a2 IsBot 0 ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s\u271d s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s hs : (\u2200 (x : \u211d\u22650\u221e), IsBot x \u2192 \u00acx \u2208 s) \u2227 \u2200 (x : \u211d\u22650\u221e), IsTop x \u2192 \u00acx \u2208 s h : \u22a4 \u2208 s \u22a2 IsTop \u22a4 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Fin.prod_univ_five ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : CommMonoid \u03b2 f : Fin 5 \u2192 \u03b2 \u22a2 \u220f i : Fin 5, f i = f 0 * f 1 * f 2 * f 3 * f 4 ** rw [prod_univ_castSucc, prod_univ_four] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : CommMonoid \u03b2 f : Fin 5 \u2192 \u03b2 \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 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Group.toDivInvMonoid_injective ** G\u271d : Type u_1 G : Type u_2 \u22a2 Injective (@toDivInvMonoid G) ** rintro \u27e8\u27e9 \u27e8\u27e9 \u27e8\u27e9 ** case mk.mk.refl G\u271d : Type u_1 G : Type u_2 toDivInvMonoid\u271d : DivInvMonoid G mul_left_inv\u271d\u00b9 mul_left_inv\u271d : \u2200 (a : G), a\u207b\u00b9 * a = 1 \u22a2 mk mul_left_inv\u271d\u00b9 = mk mul_left_inv\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Sylow.characteristic_of_normal ** G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P \u22a2 Characteristic \u2191P ** haveI := Sylow.subsingleton_of_normal P h ** G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P this : Subsingleton (Sylow p G) \u22a2 Characteristic \u2191P ** rw [characteristic_iff_map_eq] ** G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P this : Subsingleton (Sylow p G) \u22a2 \u2200 (\u03d5 : G \u2243* G), Subgroup.map (MulEquiv.toMonoidHom \u03d5) \u2191P = \u2191P ** intro \u03a6 ** G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P this : Subsingleton (Sylow p G) \u03a6 : G \u2243* G \u22a2 Subgroup.map (MulEquiv.toMonoidHom \u03a6) \u2191P = \u2191P ** show (\u03a6 \u2022 P).toSubgroup = P.toSubgroup ** G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P this : Subsingleton (Sylow p G) \u03a6 : G \u2243* G \u22a2 \u2191(\u03a6 \u2022 P) = \u2191P ** congr ** case e_self G : Type u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b2 : Group G p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : Finite (Sylow p G) P : Sylow p G h : Normal \u2191P this : Subsingleton (Sylow p G) \u03a6 : G \u2243* G \u22a2 \u03a6 \u2022 P = P ** simp ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.MonoidalOfChosenFiniteProducts.triangle ** C : Type u inst\u271d : Category.{v, u} C X\u271d Y\u271d : C \ud835\udcaf : LimitCone (Functor.empty C) \u212c : (X Y : C) \u2192 LimitCone (pair X Y) X Y : C \u22a2 (BinaryFan.associatorOfLimitCone \u212c X \ud835\udcaf.cone.pt Y).hom \u226b tensorHom \u212c (\ud835\udfd9 X) (BinaryFan.leftUnitor \ud835\udcaf.isLimit (\u212c \ud835\udcaf.cone.pt Y).isLimit).hom = tensorHom \u212c (BinaryFan.rightUnitor \ud835\udcaf.isLimit (\u212c X \ud835\udcaf.cone.pt).isLimit).hom (\ud835\udfd9 Y) ** dsimp [tensorHom] ** C : Type u inst\u271d : Category.{v, u} C X\u271d Y\u271d : C \ud835\udcaf : LimitCone (Functor.empty C) \u212c : (X Y : C) \u2192 LimitCone (pair X Y) X Y : C \u22a2 (BinaryFan.associatorOfLimitCone \u212c X \ud835\udcaf.cone.pt Y).hom \u226b IsLimit.lift (\u212c X Y).isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b \ud835\udfd9 X) (BinaryFan.snd (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b BinaryFan.snd (\u212c \ud835\udcaf.cone.pt Y).cone)) = IsLimit.lift (\u212c X Y).isLimit (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) (BinaryFan.snd (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b \ud835\udfd9 Y)) ** apply IsLimit.hom_ext (\u212c _ _).isLimit ** C : Type u inst\u271d : Category.{v, u} C X\u271d Y\u271d : C \ud835\udcaf : LimitCone (Functor.empty C) \u212c : (X Y : C) \u2192 LimitCone (pair X Y) X Y : C \u22a2 \u2200 (j : Discrete WalkingPair), ((BinaryFan.associatorOfLimitCone \u212c X \ud835\udcaf.cone.pt Y).hom \u226b IsLimit.lift (\u212c X Y).isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b \ud835\udfd9 X) (BinaryFan.snd (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b BinaryFan.snd (\u212c \ud835\udcaf.cone.pt Y).cone))) \u226b (\u212c X Y).cone.\u03c0.app j = IsLimit.lift (\u212c X Y).isLimit (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) (BinaryFan.snd (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b \ud835\udfd9 Y)) \u226b (\u212c X Y).cone.\u03c0.app j ** rintro \u27e8\u27e8\u27e9\u27e9 <;> simp ** Qed", + "informal": "" + }, + { + "formal": "Nat.maxPowDiv.base_mul_eq_succ ** p n : \u2115 hp : 1 < p hn : 0 < n \u22a2 maxPowDiv p (p * n) = maxPowDiv p n + 1 ** have : 0 < p := lt_trans (b := 1) (by simp) hp ** p n : \u2115 hp : 1 < p hn : 0 < n this : 0 < p \u22a2 maxPowDiv p (p * n) = maxPowDiv p n + 1 ** dsimp [maxPowDiv] ** p n : \u2115 hp : 1 < p hn : 0 < n this : 0 < p \u22a2 go 0 p (p * n) = go 0 p n + 1 ** rw [maxPowDiv.go_eq, if_pos, mul_div_right _ this] ** p n : \u2115 hp : 1 < p hn : 0 < n \u22a2 0 < 1 ** simp ** p n : \u2115 hp : 1 < p hn : 0 < n this : 0 < p \u22a2 go (0 + 1) p n = go 0 p n + 1 ** apply go_succ ** case hc p n : \u2115 hp : 1 < p hn : 0 < n this : 0 < p \u22a2 1 < p \u2227 0 < p * n \u2227 p * n % p = 0 ** refine \u27e8hp, ?_, by simp\u27e9 ** case hc p n : \u2115 hp : 1 < p hn : 0 < n this : 0 < p \u22a2 0 < p * n ** apply Nat.mul_pos this hn ** p n : \u2115 hp : 1 < p hn : 0 < n this : 0 < p \u22a2 p * n % p = 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "disjoint_sSup_iff ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : Frame \u03b1 s\u271d t : Set \u03b1 a b : \u03b1 s : Set \u03b1 \u22a2 Disjoint a (sSup s) \u2194 \u2200 (b : \u03b1), b \u2208 s \u2192 Disjoint a b ** simpa only [disjoint_comm] using @sSup_disjoint_iff ** Qed", + "informal": "" + }, + { + "formal": "ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K \u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0 ** obtain \u27e8z, hzK, infi\u27e9 := exists_norm_eq_iInf_of_complete_convex ne hc.isComplete K.convex b ** case intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 \u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0 ** have hinner := (norm_eq_iInf_iff_real_inner_le_zero K.convex hzK).1 infi ** case intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 \u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0 ** use z - b ** case h \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 \u22a2 (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)) \u2227 inner (z - b) b < 0 ** constructor ** case h.left \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 \u22a2 \u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b) ** rintro x hxK ** case h.left \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 x : H hxK : x \u2208 K \u22a2 0 \u2264 inner x (z - b) ** specialize hinner _ (K.add_mem hxK hzK) ** case h.left \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 x : H hxK : x \u2208 K hinner : inner (b - z) (x + z - z) \u2264 0 \u22a2 0 \u2264 inner x (z - b) ** 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 ** case h.right \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 \u22a2 inner (z - b) b < 0 ** have hinner\u2080 := hinner 0 (K.pointed_of_nonempty_of_isClosed ne hc) ** case h.right \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 hinner\u2080 : inner (b - z) (0 - z) \u2264 0 \u22a2 inner (z - b) b < 0 ** rw [zero_sub, inner_neg_right, Right.neg_nonpos_iff] at hinner\u2080 ** case h.right \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 hinner\u2080 : 0 \u2264 inner (b - z) z \u22a2 inner (z - b) b < 0 ** have hbz : b - z \u2260 0 := by\n rw [sub_ne_zero]\n contrapose! hzK\n rwa [\u2190 hzK] ** case h.right \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 hinner\u2080 : 0 \u2264 inner (b - z) z hbz : b - z \u2260 0 \u22a2 inner (z - b) b < 0 ** 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] ** case h.right \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 hinner\u2080 : 0 \u2264 inner (b - z) z hbz : b - z \u2260 0 \u22a2 0 < inner (b - z) b ** 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] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 hinner\u2080 : 0 \u2264 inner (b - z) z \u22a2 b - z \u2260 0 ** rw [sub_ne_zero] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 hinner\u2080 : 0 \u2264 inner (b - z) z \u22a2 b \u2260 z ** contrapose! hzK ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 hinner\u2080 : 0 \u2264 inner (b - z) z hzK : b = z \u22a2 \u00acz \u2208 \u2191K ** rwa [\u2190 hzK] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup H inst\u271d\u00b9 : InnerProductSpace \u211d H s t : Set H inst\u271d : CompleteSpace H K : ConvexCone \u211d H ne : Set.Nonempty \u2191K hc : IsClosed \u2191K b : H disj : \u00acb \u2208 K z : H hzK : z \u2208 \u2191K infi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016 hinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0 hinner\u2080 : 0 \u2264 inner (b - z) z hbz : b - z \u2260 0 \u22a2 inner (b - z) (b - z + z) = inner (b - z) b ** rw [sub_add_cancel] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Iso.cancel_iso_hom_right ** C : Type u inst\u271d : Category.{v, u} C X\u271d Y\u271d Z\u271d X Y Z : C f f' : X \u27f6 Y g : Y \u2245 Z \u22a2 f \u226b g.hom = f' \u226b g.hom \u2194 f = f' ** simp only [cancel_mono] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.isUnit_fromBlocks_iff_of_invertible\u2081\u2081 ** l : Type u_1 m : Type u_2 n : Type u_3 \u03b1 : Type u_4 inst\u271d\u2077 : Fintype l inst\u271d\u2076 : Fintype m inst\u271d\u2075 : Fintype n inst\u271d\u2074 : DecidableEq l inst\u271d\u00b3 : DecidableEq m inst\u271d\u00b2 : DecidableEq n inst\u271d\u00b9 : CommRing \u03b1 A : Matrix m m \u03b1 B : Matrix m n \u03b1 C : Matrix n m \u03b1 D : Matrix n n \u03b1 inst\u271d : Invertible A \u22a2 IsUnit (fromBlocks A B C D) \u2194 IsUnit (D - C * \u215fA * B) ** simp only [\u2190 nonempty_invertible_iff_isUnit,\n (invertibleEquivFromBlocks\u2081\u2081Invertible A B C D).nonempty_congr] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.sum_comm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9' : Type u_8 \u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1 \u22a2 (sum fun n => sum (\u03bc n)) = sum fun m => sum fun n => \u03bc n m ** ext1 s hs ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03b9' : Type u_8 \u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(sum fun n => sum (\u03bc n)) s = \u2191\u2191(sum fun m => sum fun n => \u03bc n m) s ** simp_rw [sum_apply _ hs] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03b9' : Type u_8 \u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s \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 ** rw [ENNReal.tsum_comm] ** Qed", + "informal": "" + }, + { + "formal": "Equiv.embeddingCongr_apply_trans ** \u03b1\u2081 : Sort u_1 \u03b2\u2081 : Sort u_2 \u03b3\u2081 : Sort u_3 \u03b1\u2082 : Sort u_4 \u03b2\u2082 : Sort u_5 \u03b3\u2082 : Sort u_6 ea : \u03b1\u2081 \u2243 \u03b1\u2082 eb : \u03b2\u2081 \u2243 \u03b2\u2082 ec : \u03b3\u2081 \u2243 \u03b3\u2082 f : \u03b1\u2081 \u21aa \u03b2\u2081 g : \u03b2\u2081 \u21aa \u03b3\u2081 \u22a2 \u2191(embeddingCongr ea ec) (Embedding.trans f g) = Embedding.trans (\u2191(embeddingCongr ea eb) f) (\u2191(embeddingCongr eb ec) g) ** ext ** case h \u03b1\u2081 : Sort u_1 \u03b2\u2081 : Sort u_2 \u03b3\u2081 : Sort u_3 \u03b1\u2082 : Sort u_4 \u03b2\u2082 : Sort u_5 \u03b3\u2082 : Sort u_6 ea : \u03b1\u2081 \u2243 \u03b1\u2082 eb : \u03b2\u2081 \u2243 \u03b2\u2082 ec : \u03b3\u2081 \u2243 \u03b3\u2082 f : \u03b1\u2081 \u21aa \u03b2\u2081 g : \u03b2\u2081 \u21aa \u03b3\u2081 x\u271d : \u03b1\u2082 \u22a2 \u2191(\u2191(embeddingCongr ea ec) (Embedding.trans f g)) x\u271d = \u2191(Embedding.trans (\u2191(embeddingCongr ea eb) f) (\u2191(embeddingCongr eb ec) g)) x\u271d ** simp ** Qed", + "informal": "" + }, + { + "formal": "MulOpposite.semiconjBy_unop ** \u03b1 : Type u inst\u271d : Mul \u03b1 a x y : \u03b1\u1d50\u1d52\u1d56 \u22a2 SemiconjBy (unop a) (unop y) (unop x) \u2194 SemiconjBy a x y ** conv_rhs => rw [\u2190 op_unop a, \u2190 op_unop x, \u2190 op_unop y, semiconjBy_op] ** Qed", + "informal": "" + }, + { + "formal": "continuousAt_extChartAt' ** \ud835\udd5c : Type u_1 E : Type u_2 M : Type u_3 H : Type u_4 E' : Type u_5 M' : Type u_6 H' : Type u_7 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : TopologicalSpace H inst\u271d\u2076 : TopologicalSpace M f f' : LocalHomeomorph M H I : ModelWithCorners \ud835\udd5c E H inst\u271d\u2075 : NormedAddCommGroup E' inst\u271d\u2074 : NormedSpace \ud835\udd5c E' inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' I' : ModelWithCorners \ud835\udd5c E' H' x : M s t : Set M inst\u271d\u00b9 : ChartedSpace H M inst\u271d : ChartedSpace H' M' x' : M h : x' \u2208 (extChartAt I x).source \u22a2 x' \u2208 (chartAt H x).toLocalEquiv.source ** rwa [\u2190 extChartAt_source I] ** Qed", + "informal": "" + }, + { + "formal": "Set.preimage_inl_image_inr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b9 \u2192 \u03b1 s\u271d t : Set \u03b1 s : Set \u03b2 \u22a2 Sum.inl \u207b\u00b9' (Sum.inr '' s) = \u2205 ** ext ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b9 \u2192 \u03b1 s\u271d t : Set \u03b1 s : Set \u03b2 x\u271d : \u03b1 \u22a2 x\u271d \u2208 Sum.inl \u207b\u00b9' (Sum.inr '' s) \u2194 x\u271d \u2208 \u2205 ** simp ** Qed", + "informal": "" + }, + { + "formal": "dite_ne_left_iff ** \u03b1 : Sort u_2 \u03b2 : Sort ?u.32959 \u03c3 : \u03b1 \u2192 Sort u_1 f : \u03b1 \u2192 \u03b2 P Q : Prop inst\u271d\u00b9 : Decidable P inst\u271d : Decidable Q a b c : \u03b1 A : P \u2192 \u03b1 B : \u00acP \u2192 \u03b1 \u22a2 dite P (fun x => a) B \u2260 a \u2194 \u2203 h, a \u2260 B h ** rw [Ne.def, dite_eq_left_iff, not_forall] ** \u03b1 : Sort u_2 \u03b2 : Sort ?u.32959 \u03c3 : \u03b1 \u2192 Sort u_1 f : \u03b1 \u2192 \u03b2 P Q : Prop inst\u271d\u00b9 : Decidable P inst\u271d : Decidable Q a b c : \u03b1 A : P \u2192 \u03b1 B : \u00acP \u2192 \u03b1 \u22a2 (\u2203 x, \u00acB x = a) \u2194 \u2203 h, a \u2260 B h ** exact exists_congr fun h \u21a6 by rw [ne_comm] ** \u03b1 : Sort u_2 \u03b2 : Sort ?u.32959 \u03c3 : \u03b1 \u2192 Sort u_1 f : \u03b1 \u2192 \u03b2 P Q : Prop inst\u271d\u00b9 : Decidable P inst\u271d : Decidable Q a b c : \u03b1 A : P \u2192 \u03b1 B : \u00acP \u2192 \u03b1 h : \u00acP \u22a2 \u00acB h = a \u2194 a \u2260 B h ** rw [ne_comm] ** Qed", + "informal": "" + }, + { + "formal": "intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedSpace \u211d E f : \u211d \u2192 E c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 a b z : \u211d u v ua ub va vb : \u03b9 \u2192 \u211d inst\u271d\u00b9 : FTCFilter a la la' inst\u271d : FTCFilter b lb lb' hab : IntervalIntegrable f volume a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae volume) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae volume) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb \u22a2 (fun t => ((\u222b (x : \u211d) in va t..vb t, f x) - \u222b (x : \u211d) in ua t..ub t, f x) - ((vb t - ub t) \u2022 cb - (va t - ua t) \u2022 ca)) =o[lt] fun t => \u2016va t - ua t\u2016 + \u2016vb t - ub t\u2016 ** simpa [integral_const]\n using measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hab hmeas_a hmeas_b\n ha_lim hb_lim hua hva hub hvb ** Qed", + "informal": "" + }, + { + "formal": "DirectSum.IsInternal.subordinateOrthonormalBasis_subordinate ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E' inst\u271d\u2077 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : InnerProductSpace \u211d F' inst\u271d\u00b2 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E n : \u2115 hn : finrank \ud835\udd5c E = n inst\u271d : DecidableEq \u03b9 V : \u03b9 \u2192 Submodule \ud835\udd5c E hV : IsInternal V a : Fin n hV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i) \u22a2 \u2191(subordinateOrthonormalBasis hn hV hV') a \u2208 V (subordinateOrthonormalBasisIndex hn hV a hV') ** simpa only [DirectSum.IsInternal.subordinateOrthonormalBasis, OrthonormalBasis.coe_reindex,\n DirectSum.IsInternal.subordinateOrthonormalBasisIndex] using\n hV.collectedOrthonormalBasis_mem hV' (fun i => stdOrthonormalBasis \ud835\udd5c (V i))\n ((hV.sigmaOrthonormalBasisIndexEquiv hn hV').symm a) ** Qed", + "informal": "" + }, + { + "formal": "RingQuot.ringQuot_ext' ** R : Type uR inst\u271d\u2076 : Semiring R S : Type uS inst\u271d\u2075 : CommSemiring S T : Type uT A : Type uA inst\u271d\u2074 : Semiring A inst\u271d\u00b3 : Algebra S A r : R \u2192 R \u2192 Prop inst\u271d\u00b2 : Semiring T B : Type u\u2084 inst\u271d\u00b9 : Semiring B inst\u271d : Algebra S B s : A \u2192 A \u2192 Prop f g : RingQuot s \u2192\u2090[S] B w : AlgHom.comp f (mkAlgHom S s) = AlgHom.comp g (mkAlgHom S s) \u22a2 f = g ** ext x ** case H R : Type uR inst\u271d\u2076 : Semiring R S : Type uS inst\u271d\u2075 : CommSemiring S T : Type uT A : Type uA inst\u271d\u2074 : Semiring A inst\u271d\u00b3 : Algebra S A r : R \u2192 R \u2192 Prop inst\u271d\u00b2 : Semiring T B : Type u\u2084 inst\u271d\u00b9 : Semiring B inst\u271d : Algebra S B s : A \u2192 A \u2192 Prop f g : RingQuot s \u2192\u2090[S] B w : AlgHom.comp f (mkAlgHom S s) = AlgHom.comp g (mkAlgHom S s) x : RingQuot s \u22a2 \u2191f x = \u2191g x ** rcases mkAlgHom_surjective S s x with \u27e8x, rfl\u27e9 ** case H.intro R : Type uR inst\u271d\u2076 : Semiring R S : Type uS inst\u271d\u2075 : CommSemiring S T : Type uT A : Type uA inst\u271d\u2074 : Semiring A inst\u271d\u00b3 : Algebra S A r : R \u2192 R \u2192 Prop inst\u271d\u00b2 : Semiring T B : Type u\u2084 inst\u271d\u00b9 : Semiring B inst\u271d : Algebra S B s : A \u2192 A \u2192 Prop f g : RingQuot s \u2192\u2090[S] B w : AlgHom.comp f (mkAlgHom S s) = AlgHom.comp g (mkAlgHom S s) x : A \u22a2 \u2191f (\u2191(mkAlgHom S s) x) = \u2191g (\u2191(mkAlgHom S s) x) ** exact AlgHom.congr_fun w x ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.map_\u03c0_preserves_coequalizer_inv ** C : Type u\u2081 inst\u271d\u2074 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b3 : Category.{v\u2082, u\u2082} D G : C \u2964 D X Y Z : C f g : X \u27f6 Y h : Y \u27f6 Z w : f \u226b h = g \u226b h inst\u271d\u00b2 : HasCoequalizer f g inst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g) inst\u271d : PreservesColimit (parallelPair f g) G \u22a2 G.map (coequalizer.\u03c0 f g) \u226b (PreservesCoequalizer.iso G f g).inv = coequalizer.\u03c0 (G.map f) (G.map g) ** rw [\u2190 \u03b9_comp_coequalizerComparison_assoc, \u2190 PreservesCoequalizer.iso_hom, Iso.hom_inv_id,\n comp_id] ** Qed", + "informal": "" + }, + { + "formal": "FreeGroup.Red.cons_cons_iff ** \u03b1 : Type u L L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool \u22a2 Red (p :: L\u2081) (p :: L\u2082) \u2192 Red L\u2081 L\u2082 ** generalize eq\u2081 : (p :: L\u2081 : List _) = LL\u2081 ** \u03b1 : Type u L L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool LL\u2081 : List (\u03b1 \u00d7 Bool) eq\u2081 : p :: L\u2081 = LL\u2081 \u22a2 Red LL\u2081 (p :: L\u2082) \u2192 Red L\u2081 L\u2082 ** generalize eq\u2082 : (p :: L\u2082 : List _) = LL\u2082 ** \u03b1 : Type u L L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool LL\u2081 : List (\u03b1 \u00d7 Bool) eq\u2081 : p :: L\u2081 = LL\u2081 LL\u2082 : List (\u03b1 \u00d7 Bool) eq\u2082 : p :: L\u2082 = LL\u2082 \u22a2 Red LL\u2081 LL\u2082 \u2192 Red L\u2081 L\u2082 ** intro h ** \u03b1 : Type u L L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool LL\u2081 : List (\u03b1 \u00d7 Bool) eq\u2081 : p :: L\u2081 = LL\u2081 LL\u2082 : List (\u03b1 \u00d7 Bool) eq\u2082 : p :: L\u2082 = LL\u2082 h : Red LL\u2081 LL\u2082 \u22a2 Red L\u2081 L\u2082 ** 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 ** case refl \u03b1 : Type u L L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool LL\u2081 : List (\u03b1 \u00d7 Bool) eq\u2081\u271d : p :: L\u2081\u271d = LL\u2081 LL\u2082 : List (\u03b1 \u00d7 Bool) eq\u2082\u271d : p :: L\u2082\u271d = LL\u2082 L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool) eq\u2081 : p :: L\u2081 = LL\u2082 eq\u2082 : p :: L\u2082 = LL\u2082 \u22a2 Red L\u2081 L\u2082 ** subst_vars ** case refl \u03b1 : Type u L L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool) eq\u2081 : p :: L\u2081 = p :: L\u2082\u271d eq\u2082 : p :: L\u2082 = p :: L\u2082\u271d \u22a2 Red L\u2081 L\u2082 ** cases eq\u2082 ** case refl.refl \u03b1 : Type u L L\u2081\u271d L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool L\u2081 : List (\u03b1 \u00d7 Bool) eq\u2081 : p :: L\u2081 = p :: L\u2082 \u22a2 Red L\u2081 L\u2082 ** cases eq\u2081 ** case refl.refl.refl \u03b1 : Type u L L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool \u22a2 Red L\u2082 L\u2082 ** constructor ** case head \u03b1 : Type u L L\u2081\u271d\u00b9 L\u2082\u271d\u00b9 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool LL\u2081 : List (\u03b1 \u00d7 Bool) eq\u2081\u271d : p :: L\u2081\u271d\u00b9 = LL\u2081 LL\u2082 : List (\u03b1 \u00d7 Bool) eq\u2082\u271d : p :: L\u2082\u271d\u00b9 = LL\u2082 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool) h\u2081\u2082 : Step L\u2081\u271d L\u2082\u271d h : ReflTransGen Step L\u2082\u271d LL\u2082 ih : \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 L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool) eq\u2081 : p :: L\u2081 = L\u2081\u271d eq\u2082 : p :: L\u2082 = LL\u2082 \u22a2 Red L\u2081 L\u2082 ** subst_vars ** case head \u03b1 : Type u L L\u2081\u271d L\u2082\u271d\u00b9 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool L\u2082\u271d L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool) h : ReflTransGen Step L\u2082\u271d (p :: L\u2082\u271d\u00b9) ih : \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 eq\u2082 : p :: L\u2082 = p :: L\u2082\u271d\u00b9 h\u2081\u2082 : Step (p :: L\u2081) L\u2082\u271d \u22a2 Red L\u2081 L\u2082 ** cases eq\u2082 ** case head.refl \u03b1 : Type u L L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool) p : \u03b1 \u00d7 Bool L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool) h : ReflTransGen Step L\u2082 (p :: L\u2082\u271d) ih : \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 h\u2081\u2082 : Step (p :: L\u2081) L\u2082 \u22a2 Red L\u2081 L\u2082\u271d ** cases' p with a b ** case head.refl.mk \u03b1 : Type u L L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool) a : \u03b1 b : Bool h : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d) ih : \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 h\u2081\u2082 : Step ((a, b) :: L\u2081) L\u2082 \u22a2 Red L\u2081 L\u2082\u271d ** rw [Step.cons_left_iff] at h\u2081\u2082 ** case head.refl.mk \u03b1 : Type u L L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool) a : \u03b1 b : Bool h : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d) ih : \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 h\u2081\u2082 : (\u2203 L, Step L\u2081 L \u2227 L\u2082 = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: L\u2082 \u22a2 Red L\u2081 L\u2082\u271d ** rcases h\u2081\u2082 with (\u27e8L, h\u2081\u2082, rfl\u27e9 | rfl) ** case head.refl.mk.inl.intro.intro \u03b1 : Type u L\u271d L\u2081\u271d L\u2082 L\u2083 L\u2084 L\u2081 : List (\u03b1 \u00d7 Bool) a : \u03b1 b : Bool L : List (\u03b1 \u00d7 Bool) h\u2081\u2082 : Step L\u2081 L h : ReflTransGen Step ((a, b) :: L) ((a, b) :: L\u2082) ih : \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 \u22a2 Red L\u2081 L\u2082 ** exact (ih rfl rfl).head h\u2081\u2082 ** case head.refl.mk.inr \u03b1 : Type u L L\u2081 L\u2082\u271d L\u2083 L\u2084 L\u2082 : List (\u03b1 \u00d7 Bool) a : \u03b1 b : Bool h : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d) ih : \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 \u22a2 Red ((a, !b) :: L\u2082) L\u2082\u271d ** exact (cons_cons h).tail Step.cons_not_rev ** Qed", + "informal": "" + }, + { + "formal": "ClosureOperator.closure_le_closed_iff_le ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03ba : \u03b9 \u2192 Sort u_3 inst\u271d\u00b9 inst\u271d : PartialOrder \u03b1 c : ClosureOperator \u03b1 x y : \u03b1 hy : closed c y \u22a2 \u2191c x \u2264 y \u2194 x \u2264 y ** rw [\u2190 c.closure_eq_self_of_mem_closed hy, \u2190 le_closure_iff] ** Qed", + "informal": "" + }, + { + "formal": "finite_of_fin_dim_affineIndependent ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : DivisionRing k inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module k V inst\u271d\u00b9 : AffineSpace V P inst\u271d : FiniteDimensional k V p : \u03b9 \u2192 P hi : AffineIndependent k p \u22a2 _root_.Finite \u03b9 ** nontriviality \u03b9 ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : DivisionRing k inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module k V inst\u271d\u00b9 : AffineSpace V P inst\u271d : FiniteDimensional k V p : \u03b9 \u2192 P hi : AffineIndependent k p \u271d : Nontrivial \u03b9 \u22a2 _root_.Finite \u03b9 ** inhabit \u03b9 ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : DivisionRing k inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module k V inst\u271d\u00b9 : AffineSpace V P inst\u271d : FiniteDimensional k V p : \u03b9 \u2192 P hi : AffineIndependent k p \u271d : Nontrivial \u03b9 inhabited_h : Inhabited \u03b9 \u22a2 _root_.Finite \u03b9 ** rw [affineIndependent_iff_linearIndependent_vsub k p default] at hi ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : DivisionRing k inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module k V inst\u271d\u00b9 : AffineSpace V P inst\u271d : FiniteDimensional k V p : \u03b9 \u2192 P \u271d : Nontrivial \u03b9 inhabited_h : Inhabited \u03b9 hi : LinearIndependent k fun i => p \u2191i -\u1d65 p default \u22a2 _root_.Finite \u03b9 ** letI : IsNoetherian k V := IsNoetherian.iff_fg.2 inferInstance ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : DivisionRing k inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module k V inst\u271d\u00b9 : AffineSpace V P inst\u271d : FiniteDimensional k V p : \u03b9 \u2192 P \u271d : Nontrivial \u03b9 inhabited_h : Inhabited \u03b9 hi : LinearIndependent k fun i => p \u2191i -\u1d65 p default this : IsNoetherian k V := IsNoetherian.iff_fg.mpr inferInstance \u22a2 _root_.Finite \u03b9 ** exact\n (Set.finite_singleton default).finite_of_compl (Set.finite_coe_iff.1 hi.finite_of_isNoetherian) ** Qed", + "informal": "" + }, + { + "formal": "Rel.interedges_biUnion_left ** \ud835\udd5c : Type u_1 \u03b9 : Type u_2 \u03ba : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b3 : LinearOrderedField \ud835\udd5c r : \u03b1 \u2192 \u03b2 \u2192 Prop inst\u271d\u00b2 : (a : \u03b1) \u2192 DecidablePred (r a) s\u271d s\u2081 s\u2082 : Finset \u03b1 t\u271d t\u2081 t\u2082 : Finset \u03b2 a : \u03b1 b : \u03b2 \u03b4 : \ud835\udd5c inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 s : Finset \u03b9 t : Finset \u03b2 f : \u03b9 \u2192 Finset \u03b1 \u22a2 interedges r (Finset.biUnion s f) t = Finset.biUnion s fun a => interedges r (f a) t ** ext ** case a \ud835\udd5c : Type u_1 \u03b9 : Type u_2 \u03ba : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b3 : LinearOrderedField \ud835\udd5c r : \u03b1 \u2192 \u03b2 \u2192 Prop inst\u271d\u00b2 : (a : \u03b1) \u2192 DecidablePred (r a) s\u271d s\u2081 s\u2082 : Finset \u03b1 t\u271d t\u2081 t\u2082 : Finset \u03b2 a : \u03b1 b : \u03b2 \u03b4 : \ud835\udd5c inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 s : Finset \u03b9 t : Finset \u03b2 f : \u03b9 \u2192 Finset \u03b1 a\u271d : \u03b1 \u00d7 \u03b2 \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 ** simp only [mem_biUnion, mem_interedges_iff, exists_and_right, \u2190 and_assoc] ** Qed", + "informal": "" + }, + { + "formal": "IsProjective.iff_projective ** R : Type u inst\u271d\u00b2 : Ring R P : Type (max u v) inst\u271d\u00b9 : AddCommGroup P inst\u271d : Module R P \u22a2 Module.Projective R P \u2194 Projective (of R P) ** refine' \u27e8fun h => _, fun h => _\u27e9 ** case refine'_1 R : Type u inst\u271d\u00b2 : Ring R P : Type (max u v) inst\u271d\u00b9 : AddCommGroup P inst\u271d : Module R P h : Module.Projective R P \u22a2 Projective (of R P) ** letI : Module.Projective R (ModuleCat.of R P) := h ** case refine'_1 R : Type u inst\u271d\u00b2 : Ring R P : Type (max u v) inst\u271d\u00b9 : AddCommGroup P inst\u271d : Module R P h : Module.Projective R P this : Module.Projective R \u2191(of R P) := h \u22a2 Projective (of R P) ** exact \u27e8fun E X epi => Module.projective_lifting_property _ _\n ((ModuleCat.epi_iff_surjective _).mp epi)\u27e9 ** case refine'_2 R : Type u inst\u271d\u00b2 : Ring R P : Type (max u v) inst\u271d\u00b9 : AddCommGroup P inst\u271d : Module R P h : Projective (of R P) \u22a2 Module.Projective R P ** refine' Module.Projective.of_lifting_property.{u,v} _ ** case refine'_2 R : Type u inst\u271d\u00b2 : Ring R P : Type (max u v) inst\u271d\u00b9 : AddCommGroup P inst\u271d : Module R P h : Projective (of R P) \u22a2 \u2200 {M : Type (max v u)} {N : Type (max u v)} [inst : AddCommGroup M] [inst_1 : AddCommGroup N] [inst_2 : Module R M] [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 ** intro E X mE mX sE sX f g s ** case refine'_2 R : Type u inst\u271d\u00b2 : Ring R P : Type (max u v) inst\u271d\u00b9 : AddCommGroup P inst\u271d : Module R P h : Projective (of R P) E : Type (max v u) X : Type (max u v) mE : AddCommGroup E mX : AddCommGroup X sE : Module R E sX : Module R X f : E \u2192\u2097[R] X g : P \u2192\u2097[R] X s : Function.Surjective \u2191f \u22a2 \u2203 h, comp f h = g ** haveI : Epi (\u219ff) := (ModuleCat.epi_iff_surjective (\u219ff)).mpr s ** case refine'_2 R : Type u inst\u271d\u00b2 : Ring R P : Type (max u v) inst\u271d\u00b9 : AddCommGroup P inst\u271d : Module R P h : Projective (of R P) E : Type (max v u) X : Type (max u v) mE : AddCommGroup E mX : AddCommGroup X sE : Module R E sX : Module R X f : E \u2192\u2097[R] X g : P \u2192\u2097[R] X s : Function.Surjective \u2191f this : Epi (\u219ff) \u22a2 \u2203 h, comp f h = g ** letI : Projective (ModuleCat.of R P) := h ** case refine'_2 R : Type u inst\u271d\u00b2 : Ring R P : Type (max u v) inst\u271d\u00b9 : AddCommGroup P inst\u271d : Module R P h : Projective (of R P) E : Type (max v u) X : Type (max u v) mE : AddCommGroup E mX : AddCommGroup X sE : Module R E sX : Module R X f : E \u2192\u2097[R] X g : P \u2192\u2097[R] X s : Function.Surjective \u2191f this\u271d : Epi (\u219ff) this : Projective (of R P) := h \u22a2 \u2203 h, comp f h = g ** exact \u27e8Projective.factorThru (\u219fg) (\u219ff), Projective.factorThru_comp (\u219fg) (\u219ff)\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.biproduct.lift_map ** J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g P : C k : (j : J) \u2192 P \u27f6 f j p : (j : J) \u2192 f j \u27f6 g j \u22a2 lift k \u226b map p = lift fun j => k j \u226b p j ** ext ** case w J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g P : C k : (j : J) \u2192 P \u27f6 f j p : (j : J) \u2192 f j \u27f6 g j j\u271d : J \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 ** simp ** Qed", + "informal": "" + }, + { + "formal": "TopologicalSpace.IsTopologicalBasis.sigma ** \u03b1 : Type u t : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) \u22a2 IsTopologicalBasis (\u22c3 i, (fun u => Sigma.mk i '' u) '' s i) ** apply isTopologicalBasis_of_open_of_nhds ** case h_open \u03b1 : Type u t : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) \u22a2 \u2200 (u : Set ((i : \u03b9) \u00d7 E i)), u \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2192 IsOpen u ** intro u hu ** case h_open \u03b1 : Type u t : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) u : Set ((i : \u03b9) \u00d7 E i) hu : u \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u22a2 IsOpen u ** 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 ** case h_open.intro.intro.intro \u03b1 : Type u t\u271d : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) i : \u03b9 t : Set (E i) ts : t \u2208 s i hu : Sigma.mk i '' t \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u22a2 IsOpen (Sigma.mk i '' t) ** exact isOpenMap_sigmaMk _ ((hs i).isOpen ts) ** \u03b1 : Type u t : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) u : Set ((i : \u03b9) \u00d7 E i) hu : u \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u22a2 \u2203 i t, t \u2208 s i \u2227 Sigma.mk i '' t = u ** simpa only [mem_iUnion, mem_image] using hu ** case h_nhds \u03b1 : Type u t : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) \u22a2 \u2200 (a : (i : \u03b9) \u00d7 E i) (u : Set ((i : \u03b9) \u00d7 E i)), 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 ** rintro \u27e8i, x\u27e9 u hxu u_open ** case h_nhds.mk \u03b1 : Type u t : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) i : \u03b9 x : E i u : Set ((i : \u03b9) \u00d7 E i) hxu : { fst := i, snd := x } \u2208 u u_open : IsOpen u \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 ** have hx : x \u2208 Sigma.mk i \u207b\u00b9' u := hxu ** case h_nhds.mk \u03b1 : Type u t : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) i : \u03b9 x : E i u : Set ((i : \u03b9) \u00d7 E i) hxu : { fst := i, snd := x } \u2208 u u_open : IsOpen u hx : x \u2208 Sigma.mk i \u207b\u00b9' u \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 ** 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) ** case h_nhds.mk.intro.intro.intro \u03b1 : Type u t : TopologicalSpace \u03b1 \u03b9 : Type u_1 E : \u03b9 \u2192 Type u_2 inst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i) s : (i : \u03b9) \u2192 Set (Set (E i)) hs : \u2200 (i : \u03b9), IsTopologicalBasis (s i) i : \u03b9 x : E i u : Set ((i : \u03b9) \u00d7 E i) hxu : { fst := i, snd := x } \u2208 u u_open : IsOpen u hx : x \u2208 Sigma.mk i \u207b\u00b9' u v : Set (E i) vs : v \u2208 s i xv : x \u2208 v hv : v \u2286 Sigma.mk i \u207b\u00b9' u \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 ** 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 ** Qed", + "informal": "" + }, + { + "formal": "integral_withDensity_eq_integral_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** by_cases hE : CompleteSpace E ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : \u00acCompleteSpace E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** by_cases hg : Integrable g (\u03bc.withDensity fun x => f x) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : \u00acIntegrable g \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** 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 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : \u00acCompleteSpace E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** simp [integral, hE] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : \u00acIntegrable g \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** rw [integral_undef hg, integral_undef] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : \u00acIntegrable g \u22a2 \u00acIntegrable fun a => f a \u2022 g a ** rwa [\u2190 integrable_withDensity_iff_integrable_smul f_meas] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u2192 (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) (indicator s fun x => c) ** intro c s s_meas hs ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 (\u222b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc ** rw [integral_indicator s_meas] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \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 ** simp_rw [\u2190 indicator_smul_apply, integral_indicator s_meas] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \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 ** simp only [s_meas, integral_const, Measure.restrict_apply', univ_inter, withDensity_apply] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \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 ** rw [lintegral_coe_eq_integral, ENNReal.toReal_ofReal, \u2190 integral_smul_const] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191(f x) \u2022 c \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc ** rfl ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 0 \u2264 \u222b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc ** exact integral_nonneg fun x => NNReal.coe_nonneg _ ** case pos.refine'_1.hfi \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 Integrable fun x => \u2191(f x) ** refine' \u27e8f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, _\u27e9 ** case pos.refine'_1.hfi \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 HasFiniteIntegral fun x => \u2191(f x) ** rw [withDensity_apply _ s_meas] at hs ** case pos.refine'_1.hfi \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4 \u22a2 HasFiniteIntegral fun x => \u2191(f x) ** rw [HasFiniteIntegral] ** case pos.refine'_1.hfi \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4 \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016\u2191(f a)\u2016\u208a \u2202\u03bc < \u22a4 ** convert hs with x ** case h.e'_3.h.e'_4.h.h.e'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4 x : \u03b1 \u22a2 \u2016\u2191(f x)\u2016\u208a = f x ** simp only [NNReal.nnnorm_eq] ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984, Disjoint (support f_1) (support g) \u2192 Integrable f_1 \u2192 Integrable g \u2192 (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 (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 (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) ** intro u u' _ u_int u'_int h h' ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \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 ** 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 ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \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 ** simp_rw [smul_add] ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \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 ** rw [integral_add u_int u'_int, h, h', integral_add] ** case pos.refine'_2.hf \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \u22a2 Integrable fun a => f a \u2022 u a ** exact (integrable_withDensity_iff_integrable_smul f_meas).1 u_int ** case pos.refine'_2.hg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \u22a2 Integrable fun a => f a \u2022 u' a ** exact (integrable_withDensity_iff_integrable_smul f_meas).1 u'_int ** case pos.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \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} ** 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 ** case pos.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) \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} ** 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 ** case pos.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) C2 : Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc \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} ** exact isClosed_eq C1 C2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) \u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc ** 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 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) this : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) \u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc ** convert this with u ** case h.e'_5.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) this : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u : { x // x \u2208 Lp E 1 } \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 ** simp only [Function.comp_apply, withDensitySMulLI_apply] ** case h.e'_5.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) this : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u : { x // x \u2208 Lp E 1 } \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 ** exact integral_congr_ae (mem\u21121_smul_of_L1_withDensity f_meas u).coeFn_toLp.symm ** case pos.refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984, f_1 =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g \u2192 Integrable f_1 \u2192 (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 (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 ** intro u v huv _ hu ** case pos.refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc \u22a2 (\u222b (a : \u03b1), v a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc ** rw [\u2190 integral_congr_ae huv, hu] ** case pos.refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc \u22a2 \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc ** apply integral_congr_ae ** case pos.refine'_4.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc \u22a2 (fun a => f a \u2022 u a) =\u1d50[\u03bc] fun a => f a \u2022 v a ** filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 huv] with x hx ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc x : \u03b1 hx : \u2191(f x) \u2260 0 \u2192 u x = v x \u22a2 f x \u2022 u x = f x \u2022 v x ** rcases eq_or_ne (f x) 0 with (h'x | h'x) ** case h.inl \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc x : \u03b1 hx : \u2191(f x) \u2260 0 \u2192 u x = v x h'x : f x = 0 \u22a2 f x \u2022 u x = f x \u2022 v x ** simp only [h'x, zero_smul] ** case h.inr \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc x : \u03b1 hx : \u2191(f x) \u2260 0 \u2192 u x = v x h'x : f x \u2260 0 \u22a2 f x \u2022 u x = f x \u2022 v x ** rw [hx _] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc x : \u03b1 hx : \u2191(f x) \u2260 0 \u2192 u x = v x h'x : f x \u2260 0 \u22a2 \u2191(f x) \u2260 0 ** simpa only [Ne.def, ENNReal.coe_eq_zero] using h'x ** Qed", + "informal": "" + }, + { + "formal": "Ultrafilter.compl_mem_iff_not_mem ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 f g : Ultrafilter \u03b1 s t : Set \u03b1 p q : \u03b1 \u2192 Prop \u22a2 s\u1d9c \u2208 f \u2194 \u00acs \u2208 f ** rw [\u2190 compl_not_mem_iff, compl_compl] ** Qed", + "informal": "" + }, + { + "formal": "acc_iff_cluster ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w a : \u03b1 s s\u2081 s\u2082 t : Set \u03b1 p p\u2081 p\u2082 : \u03b1 \u2192 Prop inst\u271d : TopologicalSpace \u03b1 x : \u03b1 F : Filter \u03b1 \u22a2 AccPt x F \u2194 ClusterPt x (\ud835\udcdf {x}\u1d9c \u2293 F) ** rw [AccPt, nhdsWithin, ClusterPt, inf_assoc] ** Qed", + "informal": "" + }, + { + "formal": "Set.Definable.preimage_comp ** M : Type w A : Set M L : Language inst\u271d : Structure L M \u03b1 : Type u\u2081 \u03b2 : Type u_1 B : Set M s\u271d : Set (\u03b1 \u2192 M) f : \u03b1 \u2192 \u03b2 s : Set (\u03b1 \u2192 M) h : Definable A L s \u22a2 Definable A L ((fun g => g \u2218 f) \u207b\u00b9' s) ** obtain \u27e8\u03c6, rfl\u27e9 := h ** case intro M : Type w A : Set M L : Language inst\u271d : Structure L M \u03b1 : Type u\u2081 \u03b2 : Type u_1 B : Set M s : Set (\u03b1 \u2192 M) f : \u03b1 \u2192 \u03b2 \u03c6 : Formula (L[[\u2191A]]) \u03b1 \u22a2 Definable A L ((fun g => g \u2218 f) \u207b\u00b9' setOf (Formula.Realize \u03c6)) ** refine' \u27e8\u03c6.relabel f, _\u27e9 ** case intro M : Type w A : Set M L : Language inst\u271d : Structure L M \u03b1 : Type u\u2081 \u03b2 : Type u_1 B : Set M s : Set (\u03b1 \u2192 M) f : \u03b1 \u2192 \u03b2 \u03c6 : Formula (L[[\u2191A]]) \u03b1 \u22a2 (fun g => g \u2218 f) \u207b\u00b9' setOf (Formula.Realize \u03c6) = setOf (Formula.Realize (Formula.relabel f \u03c6)) ** ext ** case intro.h M : Type w A : Set M L : Language inst\u271d : Structure L M \u03b1 : Type u\u2081 \u03b2 : Type u_1 B : Set M s : Set (\u03b1 \u2192 M) f : \u03b1 \u2192 \u03b2 \u03c6 : Formula (L[[\u2191A]]) \u03b1 x\u271d : \u03b2 \u2192 M \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)) ** simp only [Set.preimage_setOf_eq, mem_setOf_eq, Formula.realize_relabel] ** Qed", + "informal": "" + }, + { + "formal": "Finset.untrop_sum ** R : Type u_1 S : Type u_2 inst\u271d : ConditionallyCompleteLinearOrder R s : Finset S f : S \u2192 Tropical (WithTop R) \u22a2 untrop (\u2211 i in s, f i) = \u2a05 i, untrop (f \u2191i) ** simpa [\u2190 _root_.untrop_sum] using sum_attach.symm ** Qed", + "informal": "" + }, + { + "formal": "ascending_central_series_le_upper ** G : Type u_1 inst\u271d\u00b9 : Group G H\u271d : Subgroup G inst\u271d : Normal H\u271d H : \u2115 \u2192 Subgroup G hH : IsAscendingCentralSeries H n : \u2115 \u22a2 H (n + 1) \u2264 upperCentralSeries G (n + 1) ** intro x hx ** G : Type u_1 inst\u271d\u00b9 : Group G H\u271d : Subgroup G inst\u271d : Normal H\u271d H : \u2115 \u2192 Subgroup G hH : IsAscendingCentralSeries H n : \u2115 x : G hx : x \u2208 H (n + 1) \u22a2 x \u2208 upperCentralSeries G (n + 1) ** rw [mem_upperCentralSeries_succ_iff] ** G : Type u_1 inst\u271d\u00b9 : Group G H\u271d : Subgroup G inst\u271d : Normal H\u271d H : \u2115 \u2192 Subgroup G hH : IsAscendingCentralSeries H n : \u2115 x : G hx : x \u2208 H (n + 1) \u22a2 \u2200 (y : G), x * y * x\u207b\u00b9 * y\u207b\u00b9 \u2208 upperCentralSeries G n ** exact fun y => ascending_central_series_le_upper H hH n (hH.2 x n hx y) ** Qed", + "informal": "" + }, + { + "formal": "Fin.sort_univ ** n : \u2115 \u22a2 List.toFinset (Finset.sort (fun x y => x \u2264 y) Finset.univ) = List.toFinset (List.finRange n) ** simp ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.left_nonuniformWitnesses_subset ** \u03b1 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b9 : LinearOrderedField \ud835\udd5c G : SimpleGraph \u03b1 inst\u271d : DecidableRel G.Adj \u03b5 : \ud835\udd5c s t : Finset \u03b1 a b : \u03b1 h : \u00acIsUniform G \u03b5 s t \u22a2 (nonuniformWitnesses G \u03b5 s t).1 \u2286 s ** rw [nonuniformWitnesses, dif_pos h] ** \u03b1 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b9 : LinearOrderedField \ud835\udd5c G : SimpleGraph \u03b1 inst\u271d : DecidableRel G.Adj \u03b5 : \ud835\udd5c s t : Finset \u03b1 a b : \u03b1 h : \u00acIsUniform G \u03b5 s t \u22a2 (Exists.choose (_ : \u2203 s', s' \u2286 s \u2227 \u2203 t', t' \u2286 t \u2227 \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227 \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227 \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|), Exists.choose (_ : \u2203 t', t' \u2286 t \u2227 \u2191(card s) * \u03b5 \u2264 \u2191(card (Exists.choose (_ : \u2203 s', s' \u2286 s \u2227 \u2203 t', t' \u2286 t \u2227 \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227 \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227 \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|))) \u2227 \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227 \u03b5 \u2264 \u2191|edgeDensity G (Exists.choose (_ : \u2203 s', s' \u2286 s \u2227 \u2203 t', t' \u2286 t \u2227 \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227 \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227 \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|)) t' - edgeDensity G s t|)).1 \u2286 s ** exact (not_isUniform_iff.1 h).choose_spec.1 ** Qed", + "informal": "" + }, + { + "formal": "Complex.abs_of_nat ** n : \u2115 \u22a2 \u2191abs \u2191n = \u2191abs \u2191\u2191n ** rw [ofReal_nat_cast] ** Qed", + "informal": "" + }, + { + "formal": "BoxIntegral.Box.withBotCoe_inj ** \u03b9 : Type u_1 I\u271d J\u271d : Box \u03b9 x y : \u03b9 \u2192 \u211d I J : WithBot (Box \u03b9) \u22a2 \u2191I = \u2191J \u2194 I = J ** simp only [Subset.antisymm_iff, \u2190 le_antisymm_iff, withBotCoe_subset_iff] ** Qed", + "informal": "" + }, + { + "formal": "Ordnode.dual_node3L ** \u03b1 : Type u_1 l : Ordnode \u03b1 x : \u03b1 m : Ordnode \u03b1 y : \u03b1 r : Ordnode \u03b1 \u22a2 dual (node3L l x m y r) = node3R (dual r) y (dual m) x (dual l) ** simp [node3L, node3R, dual_node', add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Function.Periodic.div_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f g : \u03b1 \u2192 \u03b2 c c\u2081 c\u2082 x : \u03b1 inst\u271d : DivisionSemiring \u03b1 h : Periodic f c a : \u03b1 \u22a2 Periodic (fun x => f (x / a)) (c * a) ** simpa only [div_eq_mul_inv] using h.mul_const_inv a ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SimpleFunc.mem_restrict_range ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 K : Type u_5 inst\u271d : Zero \u03b2 r : \u03b2 s : Set \u03b1 f : \u03b1 \u2192\u209b \u03b2 hs : MeasurableSet s \u22a2 r \u2208 SimpleFunc.range (restrict f s) \u2194 r = 0 \u2227 s \u2260 univ \u2228 r \u2208 \u2191f '' s ** rw [\u2190 Finset.mem_coe, coe_range, coe_restrict _ hs, mem_range_indicator] ** Qed", + "informal": "" + }, + { + "formal": "frontier_Ioc ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : DenselyOrdered \u03b1 a\u271d b\u271d : \u03b1 s : Set \u03b1 inst\u271d : NoMaxOrder \u03b1 a b : \u03b1 h : a < b \u22a2 frontier (Ioc a b) = {a, b} ** rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le] ** Qed", + "informal": "" + }, + { + "formal": "derivWithin_congr ** \ud835\udd5c : Type u inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type w inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c hs : EqOn f\u2081 f s hx : f\u2081 x = f x \u22a2 derivWithin f\u2081 s x = derivWithin f s x ** unfold derivWithin ** \ud835\udd5c : Type u inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type w inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c hs : EqOn f\u2081 f s hx : f\u2081 x = f x \u22a2 \u2191(fderivWithin \ud835\udd5c f\u2081 s x) 1 = \u2191(fderivWithin \ud835\udd5c f s x) 1 ** rw [fderivWithin_congr hs hx] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.mul_iInf ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 \u03b9\u271d : Sort u_3 f\u271d g : \u03b9\u271d \u2192 \u211d\u22650\u221e \u03b9 : Sort u_4 inst\u271d : Nonempty \u03b9 f : \u03b9 \u2192 \u211d\u22650\u221e x : \u211d\u22650\u221e h : x \u2260 \u22a4 \u22a2 x * iInf f = \u2a05 i, x * f i ** simpa only [mul_comm] using iInf_mul h ** Qed", + "informal": "" + }, + { + "formal": "nndist_dist ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d : PseudoMetricSpace \u03b1 x y : \u03b1 \u22a2 nndist x y = Real.toNNReal (dist x y) ** rw [dist_nndist, Real.toNNReal_coe] ** Qed", + "informal": "" + }, + { + "formal": "invOf_two_add_invOf_two ** \u03b1 : Type u inst\u271d\u00b9 : NonAssocSemiring \u03b1 inst\u271d : Invertible 2 \u22a2 \u215f2 + \u215f2 = 1 ** rw [\u2190 two_mul, mul_invOf_self] ** Qed", + "informal": "" + }, + { + "formal": "PowerSeries.coeff_def ** R : Type u_1 inst\u271d : Semiring R s : Unit \u2192\u2080 \u2115 n : \u2115 h : \u2191s () = n \u22a2 coeff R n = MvPowerSeries.coeff R s ** erw [coeff, \u2190 h, \u2190 Finsupp.unique_single s] ** Qed", + "informal": "" + }, + { + "formal": "Metric.glueDist_swap ** X : Type u Y : Type v Z : Type w inst\u271d\u00b9 : MetricSpace X inst\u271d : MetricSpace Y \u03a6\u271d : Z \u2192 X \u03a8\u271d : Z \u2192 Y \u03b5\u271d : \u211d \u03a6 : Z \u2192 X \u03a8 : Z \u2192 Y \u03b5 : \u211d val\u271d\u00b9 : X val\u271d : Y \u22a2 glueDist \u03a8 \u03a6 \u03b5 (Sum.swap (Sum.inl val\u271d\u00b9)) (Sum.swap (Sum.inr val\u271d)) = glueDist \u03a6 \u03a8 \u03b5 (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d) ** simp only [glueDist, Sum.swap_inl, Sum.swap_inr, dist_comm, add_comm] ** X : Type u Y : Type v Z : Type w inst\u271d\u00b9 : MetricSpace X inst\u271d : MetricSpace Y \u03a6\u271d : Z \u2192 X \u03a8\u271d : Z \u2192 Y \u03b5\u271d : \u211d \u03a6 : Z \u2192 X \u03a8 : Z \u2192 Y \u03b5 : \u211d val\u271d\u00b9 : Y val\u271d : X \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) ** simp only [glueDist, Sum.swap_inl, Sum.swap_inr, dist_comm, add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Function.apply_update ** \u03b1\u271d : Sort u \u03b2\u271d : \u03b1\u271d \u2192 Sort v \u03b1' : Sort w inst\u271d\u00b2 : DecidableEq \u03b1\u271d inst\u271d\u00b9 : DecidableEq \u03b1' f\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a a : \u03b1\u271d b : \u03b2\u271d a \u03b9 : Sort u_1 inst\u271d : DecidableEq \u03b9 \u03b1 : \u03b9 \u2192 Sort u_2 \u03b2 : \u03b9 \u2192 Sort u_3 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i g : (i : \u03b9) \u2192 \u03b1 i i : \u03b9 v : \u03b1 i j : \u03b9 \u22a2 f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j ** by_cases h:j = i ** case pos \u03b1\u271d : Sort u \u03b2\u271d : \u03b1\u271d \u2192 Sort v \u03b1' : Sort w inst\u271d\u00b2 : DecidableEq \u03b1\u271d inst\u271d\u00b9 : DecidableEq \u03b1' f\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a a : \u03b1\u271d b : \u03b2\u271d a \u03b9 : Sort u_1 inst\u271d : DecidableEq \u03b9 \u03b1 : \u03b9 \u2192 Sort u_2 \u03b2 : \u03b9 \u2192 Sort u_3 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i g : (i : \u03b9) \u2192 \u03b1 i i : \u03b9 v : \u03b1 i j : \u03b9 h : j = i \u22a2 f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j ** subst j ** case pos \u03b1\u271d : Sort u \u03b2\u271d : \u03b1\u271d \u2192 Sort v \u03b1' : Sort w inst\u271d\u00b2 : DecidableEq \u03b1\u271d inst\u271d\u00b9 : DecidableEq \u03b1' f\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a a : \u03b1\u271d b : \u03b2\u271d a \u03b9 : Sort u_1 inst\u271d : DecidableEq \u03b9 \u03b1 : \u03b9 \u2192 Sort u_2 \u03b2 : \u03b9 \u2192 Sort u_3 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i g : (i : \u03b9) \u2192 \u03b1 i i : \u03b9 v : \u03b1 i \u22a2 f i (update g i v i) = update (fun k => f k (g k)) i (f i v) i ** simp ** case neg \u03b1\u271d : Sort u \u03b2\u271d : \u03b1\u271d \u2192 Sort v \u03b1' : Sort w inst\u271d\u00b2 : DecidableEq \u03b1\u271d inst\u271d\u00b9 : DecidableEq \u03b1' f\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a a : \u03b1\u271d b : \u03b2\u271d a \u03b9 : Sort u_1 inst\u271d : DecidableEq \u03b9 \u03b1 : \u03b9 \u2192 Sort u_2 \u03b2 : \u03b9 \u2192 Sort u_3 f : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i g : (i : \u03b9) \u2192 \u03b1 i i : \u03b9 v : \u03b1 i j : \u03b9 h : \u00acj = i \u22a2 f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j ** simp [h] ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.comapDomain_smul ** \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 \u03ba : Type u_1 inst\u271d\u00b2 : Monoid \u03b3 inst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i) inst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i) h : \u03ba \u2192 \u03b9 hh : Function.Injective h r : \u03b3 f : \u03a0\u2080 (i : \u03b9), \u03b2 i \u22a2 comapDomain h hh (r \u2022 f) = r \u2022 comapDomain h hh f ** ext ** case h \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 \u03ba : Type u_1 inst\u271d\u00b2 : Monoid \u03b3 inst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i) inst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i) h : \u03ba \u2192 \u03b9 hh : Function.Injective h r : \u03b3 f : \u03a0\u2080 (i : \u03b9), \u03b2 i i\u271d : \u03ba \u22a2 \u2191(comapDomain h hh (r \u2022 f)) i\u271d = \u2191(r \u2022 comapDomain h hh f) i\u271d ** rw [smul_apply, comapDomain_apply, smul_apply, comapDomain_apply] ** Qed", + "informal": "" + }, + { + "formal": "IsPGroup.card_modEq_card_fixedPoints ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) \u22a2 card \u03b1 \u2261 card \u2191(fixedPoints G \u03b1) [MOD p] ** 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 ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) \u22a2 card \u03b1 \u2261 card \u2191(fixedPoints G \u03b1) [MOD p] ** 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 ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) \u22a2 \u2211 a : Quotient (orbitRel G \u03b1), card { x // Quotient.mk'' x = a } \u2261 \u2211 _a : \u2191(fixedPoints G \u03b1), 1 [MOD p] ** rw [\u2190 ZMod.eq_iff_modEq_nat p, Nat.cast_sum, Nat.cast_sum] ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) \u22a2 \u2211 x : Quotient (orbitRel G \u03b1), \u2191(card { x_1 // Quotient.mk'' x_1 = x }) = \u2211 x : \u2191(fixedPoints G \u03b1), \u21911 ** 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 ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) key : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x) \u22a2 \u2211 x : Quotient (orbitRel G \u03b1), \u2191(card { x_1 // Quotient.mk'' x_1 = x }) = \u2211 x : \u2191(fixedPoints G \u03b1), \u21911 ** 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]) ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) key : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x) b\u271d : Quotient (orbitRel G \u03b1) b : \u03b1 x\u271d : Quotient.mk'' b \u2208 Finset.univ hb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0 \u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082 ** obtain \u27e8k, hk\u27e9 := hG.card_orbit b ** case intro p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) key : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x) b\u271d : Quotient (orbitRel G \u03b1) b : \u03b1 x\u271d : Quotient.mk'' b \u2208 Finset.univ hb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0 k : \u2115 hk : card \u2191(orbit G b) = p ^ k \u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082 ** 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])))) ** case intro p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) key : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x) b\u271d : Quotient (orbitRel G \u03b1) b : \u03b1 x\u271d : Quotient.mk'' b \u2208 Finset.univ hb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0 k : \u2115 hk : card \u2191(orbit G b) = p ^ k this : k = 0 \u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082 ** 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 ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) \u22a2 \u2211 _a : \u2191(fixedPoints G \u03b1), 1 = card \u2191(fixedPoints G \u03b1) ** simp ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) \u22a2 Finset.card Finset.univ = card \u2191(fixedPoints G \u03b1) ** rfl ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) x : \u03b1 \u22a2 card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x) ** simp only [Quotient.eq''] ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) x : \u03b1 \u22a2 card { y // Setoid.r y x } = card \u2191(orbit G x) ** congr ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) key : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x) a : \u2191(fixedPoints G \u03b1) ha : a \u2208 Finset.univ x\u271d : \u21911 \u2260 0 \u22a2 \u21911 = \u2191(card { x // Quotient.mk'' x = (fun a x x => Quotient.mk'' \u2191a) a ha x\u271d }) ** rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.2] ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) key : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x) b\u271d : Quotient (orbitRel G \u03b1) b : \u03b1 x\u271d : Quotient.mk'' b \u2208 Finset.univ hb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0 k : \u2115 hk : card \u2191(orbit G b) = p ^ k \u22a2 \u00acp ^ Nat.succ 0 \u2223 p ^ k ** rwa [pow_one, \u2190 hk, \u2190 Nat.modEq_zero_iff_dvd, \u2190 ZMod.eq_iff_modEq_nat, \u2190 key,\n Nat.cast_zero] ** p : \u2115 G : Type u_1 inst\u271d\u00b3 : Group G hG : IsPGroup p G hp : Fact (Nat.Prime p) \u03b1 : Type u_2 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u2191(fixedPoints G \u03b1) key : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x) b\u271d : Quotient (orbitRel G \u03b1) b : \u03b1 x\u271d : Quotient.mk'' b \u2208 Finset.univ hb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0 k : \u2115 hk : card \u2191(orbit G b) = p ^ k this : k = 0 \u22a2 card \u2191(orbit G b) = 1 ** rw [hk, this, pow_zero] ** Qed", + "informal": "" + }, + { + "formal": "CochainComplex.from_single\u2080_ext ** V : Type u inst\u271d\u00b2 : Category.{v, u} V inst\u271d\u00b9 : HasZeroMorphisms V inst\u271d : HasZeroObject V C : CochainComplex V \u2115 X : V f g : (single\u2080 V).obj X \u27f6 C h : Hom.f f 0 = Hom.f g 0 \u22a2 \u2191(fromSingle\u2080Equiv C X) f = \u2191(fromSingle\u2080Equiv C X) g ** ext ** case a V : Type u inst\u271d\u00b2 : Category.{v, u} V inst\u271d\u00b9 : HasZeroMorphisms V inst\u271d : HasZeroObject V C : CochainComplex V \u2115 X : V f g : (single\u2080 V).obj X \u27f6 C h : Hom.f f 0 = Hom.f g 0 \u22a2 \u2191(\u2191(fromSingle\u2080Equiv C X) f) = \u2191(\u2191(fromSingle\u2080Equiv C X) g) ** exact h ** Qed", + "informal": "" + }, + { + "formal": "IsPGroup.of_injective ** p : \u2115 G : Type u_1 inst\u271d\u00b9 : Group G hG : IsPGroup p G H : Type u_2 inst\u271d : Group H \u03d5 : H \u2192* G h\u03d5 : Function.Injective \u2191\u03d5 \u22a2 IsPGroup p H ** simp_rw [IsPGroup, \u2190 h\u03d5.eq_iff, \u03d5.map_pow, \u03d5.map_one] ** p : \u2115 G : Type u_1 inst\u271d\u00b9 : Group G hG : IsPGroup p G H : Type u_2 inst\u271d : Group H \u03d5 : H \u2192* G h\u03d5 : Function.Injective \u2191\u03d5 \u22a2 \u2200 (g : H), \u2203 k, \u2191\u03d5 g ^ p ^ k = 1 ** exact fun h => hG (\u03d5 h) ** Qed", + "informal": "" + }, + { + "formal": "Finset.empty_mem_ssubsets ** \u03b1 : Type u_1 s\u271d t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 h : Finset.Nonempty s \u22a2 \u2205 \u2208 ssubsets s ** rw [mem_ssubsets, ssubset_iff_subset_ne] ** \u03b1 : Type u_1 s\u271d t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 h : Finset.Nonempty s \u22a2 \u2205 \u2286 s \u2227 \u2205 \u2260 s ** exact \u27e8empty_subset s, h.ne_empty.symm\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** 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)) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** 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 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** refine' h.mono fun x hx => _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 hx : \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** cases' hx with l hx ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** 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)] ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) h_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 \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** rw [h_rw_sum] at hx ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_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 \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** 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 ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_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 hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** rw [hf_rw] ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_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 hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x \u22a2 \u2203 l, Tendsto (fun n => f n x - f 0 x + f 0 x) atTop (\ud835\udcdd l) ** exact \u27e8l + f 0 x, Tendsto.add_const _ hx\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** 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 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h1 : \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 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** 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) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h1 : \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 h2 : \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 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** 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 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h1 : \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 h2 : \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 h3 : (\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 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** 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 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h1 : \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 h2 : \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 h3 : (\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 h4 : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** 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)) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x \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) ** refine' h_summable.mono fun x hx => _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x x : \u03b1 hx : Summable fun i => f (i + 1) x - f i x \u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) ** let hx_sum := hx.hasSum.tendsto_sum_nat ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x x : \u03b1 hx : Summable fun i => f (i + 1) x - f i x hx_sum : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd (\u2211' (b : \u2115), (f (b + 1) x - f b x))) := HasSum.tendsto_sum_nat (Summable.hasSum hx) \u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) ** exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) \u22a2 (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x ** ext1 n ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) n : \u2115 \u22a2 \u2211 i in Finset.range n, (f (i + 1) x - f i x) = f n x - f 0 x ** 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 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) n : \u2115 \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 ** rw [Finset.sum_range_sub (fun m => f m x)] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_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 \u22a2 (fun n => f n x) = fun n => f n x - f 0 x + f 0 x ** ext1 n ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \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) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_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 : \u2115 \u22a2 f n x = f n x - f 0 x + f 0 x ** abel ** Qed", + "informal": "" + }, + { + "formal": "Real.logb_le_iff_le_rpow ** b x y : \u211d hb : 1 < b hx : 0 < x \u22a2 logb b x \u2264 y \u2194 x \u2264 b ^ y ** rw [\u2190 rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hx] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousMap.idealOfSet_isMaximal_iff ** X : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : CompactSpace X inst\u271d : T2Space X s : Opens X \u22a2 Ideal.IsMaximal (idealOfSet \ud835\udd5c \u2191s) \u2194 IsCoatom s ** rw [Ideal.isMaximal_def] ** X : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : CompactSpace X inst\u271d : T2Space X s : Opens X \u22a2 IsCoatom (idealOfSet \ud835\udd5c \u2191s) \u2194 IsCoatom s ** refine' (idealOpensGI X \ud835\udd5c).isCoatom_iff (fun I hI => _) s ** X : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : CompactSpace X inst\u271d : T2Space X s : Opens X I : Ideal C(X, \ud835\udd5c) hI : IsCoatom I \u22a2 idealOfSet \ud835\udd5c \u2191(opensOfIdeal I) = I ** rw [\u2190 Ideal.isMaximal_def] at hI ** X : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : CompactSpace X inst\u271d : T2Space X s : Opens X I : Ideal C(X, \ud835\udd5c) hI : Ideal.IsMaximal I \u22a2 idealOfSet \ud835\udd5c \u2191(opensOfIdeal I) = I ** exact idealOfSet_ofIdeal_isClosed inferInstance ** Qed", + "informal": "" + }, + { + "formal": "List.count_eq_of_nodup ** \u03b1 : Type u \u03b2 : Type v l\u271d l\u2081 l\u2082 : List \u03b1 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d b : \u03b1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l : List \u03b1 d : Nodup l \u22a2 count a l = if a \u2208 l then 1 else 0 ** split_ifs with h ** case pos \u03b1 : Type u \u03b2 : Type v l\u271d l\u2081 l\u2082 : List \u03b1 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d b : \u03b1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l : List \u03b1 d : Nodup l h : a \u2208 l \u22a2 count a l = 1 ** exact count_eq_one_of_mem d h ** case neg \u03b1 : Type u \u03b2 : Type v l\u271d l\u2081 l\u2082 : List \u03b1 r : \u03b1 \u2192 \u03b1 \u2192 Prop a\u271d b : \u03b1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l : List \u03b1 d : Nodup l h : \u00aca \u2208 l \u22a2 count a l = 0 ** exact count_eq_zero_of_not_mem h ** Qed", + "informal": "" + }, + { + "formal": "Real.cosh_eq ** x\u271d y x : \u211d \u22a2 cosh x * 2 = rexp x + rexp (-x) ** 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] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.forall_eq_iff_forall_eq ** R : Type u a b : R m n : \u2115 inst\u271d : Semiring R p q : R[X] \u22a2 (\u2200 (f g : R[X]), f = g) \u2194 \u2200 (a b : R), a = b ** simpa only [\u2190 subsingleton_iff] using subsingleton_iff_subsingleton ** Qed", + "informal": "" + }, + { + "formal": "GroupAlgebra.mul_average_right ** k : Type u_1 G : Type u_2 inst\u271d\u00b3 : CommSemiring k inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G inst\u271d : Invertible \u2191(Fintype.card G) g : G \u22a2 (average k G * fun\u2080 | g => 1) = average k G ** simp only [mul_one, Finset.sum_mul, Algebra.smul_mul_assoc, average, MonoidAlgebra.of_apply,\n Finset.sum_congr, MonoidAlgebra.single_mul_single] ** k : Type u_1 G : Type u_2 inst\u271d\u00b3 : CommSemiring k inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G inst\u271d : Invertible \u2191(Fintype.card G) g : G \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 ** set f : G \u2192 MonoidAlgebra k G := fun x => Finsupp.single x 1 ** k : Type u_1 G : Type u_2 inst\u271d\u00b3 : CommSemiring k inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G inst\u271d : Invertible \u2191(Fintype.card G) g : G f : G \u2192 MonoidAlgebra k G := fun x => fun\u2080 | x => 1 \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 ** show \u215f (Fintype.card G : k) \u2022 \u2211 x : G, f (x * g) = \u215f (Fintype.card G : k) \u2022 \u2211 x : G, f x ** k : Type u_1 G : Type u_2 inst\u271d\u00b3 : CommSemiring k inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G inst\u271d : Invertible \u2191(Fintype.card G) g : G f : G \u2192 MonoidAlgebra k G := fun x => fun\u2080 | x => 1 \u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, f (x * g) = \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, f x ** rw [Function.Bijective.sum_comp (Group.mulRight_bijective g) _] ** Qed", + "informal": "" + }, + { + "formal": "Int.mul_emod_right ** a b : Int \u22a2 a * b % a = 0 ** rw [Int.mul_comm, mul_emod_left] ** Qed", + "informal": "" + }, + { + "formal": "List.count_singleton' ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a b : \u03b1 \u22a2 count a [b] = if a = b then 1 else 0 ** simp [count_cons] ** Qed", + "informal": "" + }, + { + "formal": "birkhoffAverage_one ** R : Type u_1 \u03b1 : Type u_2 M : Type u_3 inst\u271d\u00b2 : DivisionSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M f : \u03b1 \u2192 \u03b1 g : \u03b1 \u2192 M x : \u03b1 \u22a2 birkhoffAverage R f g 1 x = g x ** simp [birkhoffAverage] ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.kernel.set_lintegral_compProd_univ_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e hf : Measurable f t : Set \u03b3 ht : MeasurableSet t \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 ** simp_rw [set_lintegral_compProd \u03ba \u03b7 a hf MeasurableSet.univ ht, Measure.restrict_univ] ** Qed", + "informal": "" + }, + { + "formal": "Fin.cycleRange_of_eq ** n : \u2115 i j : Fin (Nat.succ n) h : j = i \u22a2 \u2191(cycleRange i) j = 0 ** rw [cycleRange_of_le h.le, if_pos h] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.NatTrans.mono_of_mono_app ** C : Type u\u2081 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b2 : Category.{v\u2082, u\u2082} D E : Type u\u2083 inst\u271d\u00b9 : Category.{v\u2083, u\u2083} E F G H I : C \u2964 D \u03b1 : F \u27f6 G inst\u271d : \u2200 (X : C), Mono (\u03b1.app X) Z\u271d : C \u2964 D g h : Z\u271d \u27f6 F eq : g \u226b \u03b1 = h \u226b \u03b1 \u22a2 g = h ** ext X ** case w.h C : Type u\u2081 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b2 : Category.{v\u2082, u\u2082} D E : Type u\u2083 inst\u271d\u00b9 : Category.{v\u2083, u\u2083} E F G H I : C \u2964 D \u03b1 : F \u27f6 G inst\u271d : \u2200 (X : C), Mono (\u03b1.app X) Z\u271d : C \u2964 D g h : Z\u271d \u27f6 F eq : g \u226b \u03b1 = h \u226b \u03b1 X : C \u22a2 g.app X = h.app X ** rw [\u2190 cancel_mono (\u03b1.app X), \u2190 comp_app, eq, comp_app] ** Qed", + "informal": "" + }, + { + "formal": "Stream'.WSeq.tail_think ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : WSeq \u03b1 \u22a2 tail (think s) = think (tail s) ** simp [tail] ** Qed", + "informal": "" + }, + { + "formal": "ClassGroup.mk_canonicalEquiv ** R : Type u_1 K : Type u_2 L : Type u_3 inst\u271d\u00b9\u00b3 : CommRing R inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : DecidableEq L inst\u271d\u2079 : Algebra R K inst\u271d\u2078 : IsFractionRing R K inst\u271d\u2077 : Algebra K L inst\u271d\u2076 : FiniteDimensional K L inst\u271d\u2075 : Algebra R L inst\u271d\u2074 : IsScalarTower R K L inst\u271d\u00b3 : IsDomain R K' : Type u_4 inst\u271d\u00b2 : Field K' inst\u271d\u00b9 : Algebra R K' inst\u271d : IsFractionRing R K' I : (FractionalIdeal R\u2070 K)\u02e3 \u22a2 \u2191mk (\u2191(Units.map \u2191(canonicalEquiv R\u2070 K K')) I) = \u2191mk I ** erw [ClassGroup.mk, MonoidHom.comp_apply, \u2190 MonoidHom.comp_apply (Units.map _),\n \u2190 Units.map_comp, \u2190 RingEquiv.coe_monoidHom_trans,\n FractionalIdeal.canonicalEquiv_trans_canonicalEquiv] ** R : Type u_1 K : Type u_2 L : Type u_3 inst\u271d\u00b9\u00b3 : CommRing R inst\u271d\u00b9\u00b2 : Field K inst\u271d\u00b9\u00b9 : Field L inst\u271d\u00b9\u2070 : DecidableEq L inst\u271d\u2079 : Algebra R K inst\u271d\u2078 : IsFractionRing R K inst\u271d\u2077 : Algebra K L inst\u271d\u2076 : FiniteDimensional K L inst\u271d\u2075 : Algebra R L inst\u271d\u2074 : IsScalarTower R K L inst\u271d\u00b3 : IsDomain R K' : Type u_4 inst\u271d\u00b2 : Field K' inst\u271d\u00b9 : Algebra R K' inst\u271d : IsFractionRing R K' I : (FractionalIdeal R\u2070 K)\u02e3 \u22a2 \u2191(QuotientGroup.mk' (MonoidHom.range (toPrincipalIdeal R (FractionRing R)))) (\u2191(Units.map \u2191(canonicalEquiv R\u2070 K (FractionRing R))) I) = \u2191mk I ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Multiset.mem_of_mem_nsmul ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 a : \u03b1 s : Multiset \u03b1 n : \u2115 h : a \u2208 n \u2022 s \u22a2 a \u2208 s ** induction' n with n ih ** case zero \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 a : \u03b1 s : Multiset \u03b1 n : \u2115 h\u271d : a \u2208 n \u2022 s h : a \u2208 zero \u2022 s \u22a2 a \u2208 s ** rw [zero_nsmul] at h ** case zero \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 a : \u03b1 s : Multiset \u03b1 n : \u2115 h\u271d : a \u2208 n \u2022 s h : a \u2208 0 \u22a2 a \u2208 s ** exact absurd h (not_mem_zero _) ** case succ \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 a : \u03b1 s : Multiset \u03b1 n\u271d : \u2115 h\u271d : a \u2208 n\u271d \u2022 s n : \u2115 ih : a \u2208 n \u2022 s \u2192 a \u2208 s h : a \u2208 succ n \u2022 s \u22a2 a \u2208 s ** rw [succ_nsmul, mem_add] at h ** case succ \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 a : \u03b1 s : Multiset \u03b1 n\u271d : \u2115 h\u271d : a \u2208 n\u271d \u2022 s n : \u2115 ih : a \u2208 n \u2022 s \u2192 a \u2208 s h : a \u2208 s \u2228 a \u2208 n \u2022 s \u22a2 a \u2208 s ** exact h.elim id ih ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.splits_of_map_degree_eq_one ** F : Type u K : Type v L : Type w inst\u271d\u00b2 : CommRing K inst\u271d\u00b9 : Field L inst\u271d : Field F i : K \u2192+* L f : K[X] hf : degree (map i f) = 1 g\u271d : L[X] hg : Irreducible g\u271d x\u271d : g\u271d \u2223 map i f p : L[X] hp : map i f = g\u271d * p \u22a2 degree g\u271d = 1 ** have := congr_arg degree hp ** F : Type u K : Type v L : Type w inst\u271d\u00b2 : CommRing K inst\u271d\u00b9 : Field L inst\u271d : Field F i : K \u2192+* L f : K[X] hf : degree (map i f) = 1 g\u271d : L[X] hg : Irreducible g\u271d x\u271d : g\u271d \u2223 map i f p : L[X] hp : map i f = g\u271d * p this : degree (map i f) = degree (g\u271d * p) \u22a2 degree g\u271d = 1 ** simp [Nat.WithBot.add_eq_one_iff, hf, @eq_comm (WithBot \u2115) 1,\n mt isUnit_iff_degree_eq_zero.2 hg.1] at this ** F : Type u K : Type v L : Type w inst\u271d\u00b2 : CommRing K inst\u271d\u00b9 : Field L inst\u271d : Field F i : K \u2192+* L f : K[X] hf : degree (map i f) = 1 g\u271d : L[X] hg : Irreducible g\u271d x\u271d : g\u271d \u2223 map i f p : L[X] hp : map i f = g\u271d * p this : degree g\u271d = 1 \u2227 degree p = 0 \u22a2 degree g\u271d = 1 ** tauto ** Qed", + "informal": "" + }, + { + "formal": "EuclideanSpace.orthonormal_single ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 \u22a2 Orthonormal \ud835\udd5c fun i => single i 1 ** simp_rw [orthonormal_iff_ite, EuclideanSpace.inner_single_left, map_one, one_mul,\n EuclideanSpace.single_apply] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 \u22a2 \u03b9 \u2192 \u03b9 \u2192 True ** intros ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 i\u271d j\u271d : \u03b9 \u22a2 True ** trivial ** Qed", + "informal": "" + }, + { + "formal": "Nat.Partrec'.rfindOpt ** n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b : \u2115 \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 b \u2208 Nat.rfindOpt fun a => ofNat (Option \u2115) (f (a ::\u1d65 v)) ** 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] ** n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b : \u2115 \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 \u2203 a, (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) \u2227 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b ** refine'\n exists_congr fun a => (and_congr (iff_of_eq _) Iff.rfl).trans (and_congr_right fun h => _) ** case refine'_1 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 \u22a2 (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) = (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) ** congr ** case refine'_1.e_a.e_p n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 \u22a2 (fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) = fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) ** funext n ** case refine'_1.e_a.e_p.h n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n\u271d b a n : \u2115 \u22a2 Part.some (decide (1 \u2264 f (n ::\u1d65 v))) = \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n ::\u1d65 v))))) ** cases f (n ::\u1d65 v) <;> simp [Nat.succ_le_succ] ** case refine'_1.e_a.e_p.h.succ n\u271d\u00b9 : \u2115 f : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n\u271d\u00b9 b a n n\u271d : \u2115 \u22a2 true = Option.isSome (ofNat (Option \u2115) (succ n\u271d)) ** rfl ** case refine'_2 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) \u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b ** have := Nat.rfind_spec h ** case refine'_2 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) this : true \u2208 \u2191(some (Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))))) \u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b ** simp only [Part.coe_some, Part.mem_some_iff] at this ** case refine'_2 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) this : true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b ** revert this ** case refine'_2 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) \u22a2 true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u2192 (b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b) ** cases' f (a ::\u1d65 v) with c <;> intro this ** case refine'_2.succ n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) c : \u2115 this : true = Option.isSome (ofNat (Option \u2115) (succ c)) \u22a2 b = pred (succ c) \u2194 ofNat (Option \u2115) (succ c) = some b ** rw [\u2190 Option.some_inj, eq_comm] ** case refine'_2.succ n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) c : \u2115 this : true = Option.isSome (ofNat (Option \u2115) (succ c)) \u22a2 some (pred (succ c)) = some b \u2194 ofNat (Option \u2115) (succ c) = some b ** rfl ** case refine'_2.zero n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) this : true = Option.isSome (ofNat (Option \u2115) zero) \u22a2 b = pred zero \u2194 ofNat (Option \u2115) zero = some b ** cases this ** Qed", + "informal": "" + }, + { + "formal": "Rat.add_def' ** a b : Rat \u22a2 a + b = mkRat (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) ** rw [add_def, normalize_eq_mkRat] ** Qed", + "informal": "" + }, + { + "formal": "Function.Involutive.iterate_bit0 ** \u03b1 : Type u_1 f : \u03b1 \u2192 \u03b1 n\u271d : \u2115 hf : Involutive f n : \u2115 \u22a2 f^[bit0 n] = id ** rw [bit0, \u2190 two_mul, iterate_mul, involutive_iff_iter_2_eq_id.1 hf, iterate_id] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearMap.strongTopology.t2Space ** \ud835\udd5c\u2081 : Type u_1 \ud835\udd5c\u2082 : Type u_2 inst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081 inst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082 \u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082 E : Type u_3 E' : Type u_4 F : Type u_5 F' : Type u_6 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E inst\u271d\u00b9\u2070 : AddCommGroup E' inst\u271d\u2079 : Module \u211d E' inst\u271d\u2078 : AddCommGroup F inst\u271d\u2077 : Module \ud835\udd5c\u2082 F inst\u271d\u2076 : AddCommGroup F' inst\u271d\u2075 : Module \u211d F' inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : TopologicalSpace E' inst\u271d\u00b2 : TopologicalSpace F inst\u271d\u00b9 : TopologicalAddGroup F inst\u271d : T2Space F \ud835\udd16 : Set (Set E) h\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ \u22a2 T2Space (E \u2192SL[\u03c3] F) ** letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F ** \ud835\udd5c\u2081 : Type u_1 \ud835\udd5c\u2082 : Type u_2 inst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081 inst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082 \u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082 E : Type u_3 E' : Type u_4 F : Type u_5 F' : Type u_6 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E inst\u271d\u00b9\u2070 : AddCommGroup E' inst\u271d\u2079 : Module \u211d E' inst\u271d\u2078 : AddCommGroup F inst\u271d\u2077 : Module \ud835\udd5c\u2082 F inst\u271d\u2076 : AddCommGroup F' inst\u271d\u2075 : Module \u211d F' inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : TopologicalSpace E' inst\u271d\u00b2 : TopologicalSpace F inst\u271d\u00b9 : TopologicalAddGroup F inst\u271d : T2Space F \ud835\udd16 : Set (Set E) h\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ this : UniformSpace F := TopologicalAddGroup.toUniformSpace F \u22a2 T2Space (E \u2192SL[\u03c3] F) ** haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform ** \ud835\udd5c\u2081 : Type u_1 \ud835\udd5c\u2082 : Type u_2 inst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081 inst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082 \u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082 E : Type u_3 E' : Type u_4 F : Type u_5 F' : Type u_6 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E inst\u271d\u00b9\u2070 : AddCommGroup E' inst\u271d\u2079 : Module \u211d E' inst\u271d\u2078 : AddCommGroup F inst\u271d\u2077 : Module \ud835\udd5c\u2082 F inst\u271d\u2076 : AddCommGroup F' inst\u271d\u2075 : Module \u211d F' inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : TopologicalSpace E' inst\u271d\u00b2 : TopologicalSpace F inst\u271d\u00b9 : TopologicalAddGroup F inst\u271d : T2Space F \ud835\udd16 : Set (Set E) h\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ this\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F this : UniformAddGroup F \u22a2 T2Space (E \u2192SL[\u03c3] F) ** letI : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16 ** \ud835\udd5c\u2081 : Type u_1 \ud835\udd5c\u2082 : Type u_2 inst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081 inst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082 \u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082 E : Type u_3 E' : Type u_4 F : Type u_5 F' : Type u_6 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E inst\u271d\u00b9\u2070 : AddCommGroup E' inst\u271d\u2079 : Module \u211d E' inst\u271d\u2078 : AddCommGroup F inst\u271d\u2077 : Module \ud835\udd5c\u2082 F inst\u271d\u2076 : AddCommGroup F' inst\u271d\u2075 : Module \u211d F' inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : TopologicalSpace E' inst\u271d\u00b2 : TopologicalSpace F inst\u271d\u00b9 : TopologicalAddGroup F inst\u271d : T2Space F \ud835\udd16 : Set (Set E) h\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ this\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F this\u271d : UniformAddGroup F this : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16 \u22a2 T2Space (E \u2192SL[\u03c3] F) ** haveI : T2Space (E \u2192\u1d64[\ud835\udd16] F) := UniformOnFun.t2Space_of_covering h\ud835\udd16 ** \ud835\udd5c\u2081 : Type u_1 \ud835\udd5c\u2082 : Type u_2 inst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081 inst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082 \u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082 E : Type u_3 E' : Type u_4 F : Type u_5 F' : Type u_6 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E inst\u271d\u00b9\u2070 : AddCommGroup E' inst\u271d\u2079 : Module \u211d E' inst\u271d\u2078 : AddCommGroup F inst\u271d\u2077 : Module \ud835\udd5c\u2082 F inst\u271d\u2076 : AddCommGroup F' inst\u271d\u2075 : Module \u211d F' inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : TopologicalSpace E' inst\u271d\u00b2 : TopologicalSpace F inst\u271d\u00b9 : TopologicalAddGroup F inst\u271d : T2Space F \ud835\udd16 : Set (Set E) h\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ this\u271d\u00b2 : UniformSpace F := TopologicalAddGroup.toUniformSpace F this\u271d\u00b9 : UniformAddGroup F this\u271d : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16 this : T2Space (E \u2192\u1d64[\ud835\udd16] F) \u22a2 T2Space (E \u2192SL[\u03c3] F) ** exact (strongTopology.embedding_coeFn \u03c3 F \ud835\udd16).t2Space ** Qed", + "informal": "" + }, + { + "formal": "PowerSeries.order_eq_multiplicity_X ** R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 \u22a2 order \u03c6 = multiplicity X \u03c6 ** rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6) ** case inr R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 \u22a2 order \u03c6 = multiplicity X \u03c6 ** induction' ho : order \u03c6 using PartENat.casesOn with n ** case inr.a R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n \u22a2 \u2191n = multiplicity X \u03c6 ** have hn : \u03c6.order.get (order_finite_iff_ne_zero.mpr h\u03c6) = n := by simp [ho] ** case inr.a R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u22a2 \u2191n = multiplicity X \u03c6 ** rw [\u2190 hn] ** case inr.a R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u22a2 \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom)) = multiplicity X \u03c6 ** 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 _ _ _) ** case inr.a R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u22a2 \u00acX ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) \u2223 \u03c6 ** rintro \u27e8\u03c8, H\u27e9 ** case inr.a.intro R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u03c8 : R\u27e6X\u27e7 H : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8 \u22a2 False ** have := congr_arg (coeff R n) H ** case inr.a.intro R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u03c8 : R\u27e6X\u27e7 H : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8 this : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8) \u22a2 False ** rw [\u2190 (\u03c8.commute_X.pow_right _).eq, coeff_mul_of_lt_order, \u2190 hn] at this ** case inl R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6 : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u22a2 order 0 = multiplicity X 0 ** simp ** case inr.a R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d ho : order \u03c6 = \u22a4 \u22a2 \u22a4 = multiplicity X \u03c6 ** simp [h\u03c6] at ho ** R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n \u22a2 Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n ** simp [ho] ** case inr.a.intro R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u03c8 : R\u27e6X\u27e7 H : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8 this : \u2191(coeff R (Part.get (order \u03c6) (_ : (order \u03c6).Dom))) \u03c6 = 0 \u22a2 False ** exact coeff_order _ this ** case inr.a.intro R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u03c8 : R\u27e6X\u27e7 H : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8 this : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)) \u22a2 \u2191n < order (X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)) ** rw [X_pow_eq, order_monomial] ** case inr.a.intro R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u03c8 : R\u27e6X\u27e7 H : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8 this : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)) \u22a2 \u2191n < if 1 = 0 then \u22a4 else \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) ** split_ifs ** case pos R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u03c8 : R\u27e6X\u27e7 H : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8 this : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)) h\u271d : 1 = 0 \u22a2 \u2191n < \u22a4 ** exact PartENat.natCast_lt_top _ ** case neg R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u03c8 : R\u27e6X\u27e7 H : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8 this : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)) h\u271d : \u00ac1 = 0 \u22a2 \u2191n < \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) ** rw [\u2190 hn, PartENat.coe_lt_coe] ** case neg R\u271d : Type u_1 inst\u271d\u00b2 : Semiring R\u271d \u03c6\u271d : R\u271d\u27e6X\u27e7 R : Type u_2 inst\u271d\u00b9 : Semiring R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03c6 : R\u27e6X\u27e7 h\u03c6 : \u03c6 \u2260 0 x\u271d : PartENat ho\u271d : order \u03c6 = x\u271d n : \u2115 ho : order \u03c6 = \u2191n hn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n \u03c8 : R\u27e6X\u27e7 H : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8 this : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)) h\u271d : \u00ac1 = 0 \u22a2 Part.get (order \u03c6) (_ : (order \u03c6).Dom) < Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1 ** exact Nat.lt_succ_self _ ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.limit_map_limitObjIsoLimitCompEvaluation_hom ** C : Type u inst\u271d\u2074 : Category.{v, u} C D : Type u' inst\u271d\u00b3 : Category.{v', u'} D J : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} J K : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} K inst\u271d : HasLimitsOfShape J C i j : K F : J \u2964 K \u2964 C f : i \u27f6 j \u22a2 (limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom = (limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f)) ** ext ** case w C : Type u inst\u271d\u2074 : Category.{v, u} C D : Type u' inst\u271d\u00b3 : Category.{v', u'} D J : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} J K : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} K inst\u271d : HasLimitsOfShape J C i j : K F : J \u2964 K \u2964 C f : i \u27f6 j j\u271d : J \u22a2 ((limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom) \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d = ((limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f))) \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d ** simp ** Qed", + "informal": "" + }, + { + "formal": "Nat.mod_add_mod ** m n k : Nat \u22a2 (m % n + k) % n = (m + k) % n ** have := (add_mul_mod_self_left (m % n + k) n (m / n)).symm ** m n k : Nat this : (m % n + k) % n = (m % n + k + n * (m / n)) % n \u22a2 (m % n + k) % n = (m + k) % n ** rwa [Nat.add_right_comm, mod_add_div] at this ** Qed", + "informal": "" + }, + { + "formal": "InnerProductGeometry.norm_sub_eq_abs_sub_norm_iff_angle_eq_zero ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 \u22a2 \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| \u2194 angle x y = 0 ** refine' \u27e8fun h => _, norm_sub_eq_abs_sub_norm_of_angle_eq_zero\u27e9 ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 h : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| \u22a2 angle x y = 0 ** rw [\u2190 inner_eq_mul_norm_iff_angle_eq_zero hx hy] ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 h : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| \u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016 ** have h1 : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 := by\n rw [h]\n exact sq_abs (\u2016x\u2016 - \u2016y\u2016) ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 h : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| h1 : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 \u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016 ** rw [norm_sub_pow_two_real] at h1 ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 h : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| h1\u271d : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 h1 : \u2016x\u2016 ^ 2 - 2 * inner x y + \u2016y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 \u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016 ** 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 ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 h : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| \u22a2 \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 ** rw [h] ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 h : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| \u22a2 |\u2016x\u2016 - \u2016y\u2016| ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 ** exact sq_abs (\u2016x\u2016 - \u2016y\u2016) ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 h : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| h1\u271d : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 h1 : \u2016x\u2016 ^ 2 - 2 * inner x y + \u2016y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 \u22a2 inner x y = ((\u2016x\u2016 + \u2016y\u2016) ^ 2 - \u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2) / 2 ** linarith ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V hx : x \u2260 0 hy : y \u2260 0 h : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| h1\u271d : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 h1 : \u2016x\u2016 ^ 2 - 2 * inner x y + \u2016y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 \u22a2 ((\u2016x\u2016 + \u2016y\u2016) ^ 2 - \u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2) / 2 = \u2016x\u2016 * \u2016y\u2016 ** ring ** Qed", + "informal": "" + } +] \ No newline at end of file