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Mathlib/MeasureTheory/Integral/CircleIntegral.lean | continuous_circleMap_inv | [
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222,
45
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Mathlib/Data/Sym/Basic.lean | Sym.cast_cast | [] | [
480,
6
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478,
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Mathlib/Data/Finset/Basic.lean | Finset.nonempty_iff_ne_empty | [] | [
608,
44
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607,
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Mathlib/LinearAlgebra/ProjectiveSpace/Subspace.lean | Projectivization.Subspace.span_union | [] | [
183,
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Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | Complex.sin_eq_tsum | [] | [
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Mathlib/Combinatorics/Quiver/Path.lean | Quiver.Path.length_comp | [] | [
124,
56
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Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | Complex.differentiableAt_cosh | [] | [
158,
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157,
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Mathlib/CategoryTheory/Types.lean | CategoryTheory.types_id | [] | [
66,
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Mathlib/Topology/Instances/NNReal.lean | NNReal.summable_nat_add | [] | [
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Mathlib/Data/UnionFind.lean | UnionFind.lt_rankMax | [
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195,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/LinearAlgebra/LinearIndependent.lean | linearIndependent_iff_injective_total | [] | [
479,
76
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Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.setToFun_finset_sum' | [
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},
{
"state_after": "case refine'_2\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1412380\nG : Type ?u.1412383\n𝕜 : Type ?u.1412386\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns✝ : Finset ι\nf : ι → α → E\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Integrable (f i_1)\n⊢ setToFun μ T hT (f i) + setToFun μ T hT (∑ i in s, f i) = setToFun μ T hT (f i) + ∑ i in s, setToFun μ T hT (f i)\n\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1412380\nG : Type ?u.1412383\n𝕜 : Type ?u.1412386\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns✝ : Finset ι\nf : ι → α → E\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Integrable (f i_1)\n⊢ Integrable (∑ i in s, f i)",
"state_before": "case refine'_2\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1412380\nG : Type ?u.1412383\n𝕜 : Type ?u.1412386\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns✝ : Finset ι\nf : ι → α → E\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Integrable (f i_1)\n⊢ setToFun μ T hT (f i + ∑ i in s, f i) = setToFun μ T hT (f i) + ∑ i in s, setToFun μ T hT (f i)",
"tactic": "rw [setToFun_add hT (hf i (Finset.mem_insert_self i s)) _]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1412380\nG : Type ?u.1412383\n𝕜 : Type ?u.1412386\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns✝ : Finset ι\nf : ι → α → E\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Integrable (f i_1)\n⊢ setToFun μ T hT (f i) + setToFun μ T hT (∑ i in s, f i) = setToFun μ T hT (f i) + ∑ i in s, setToFun μ T hT (f i)",
"tactic": "rw [ih fun i hi => hf i (Finset.mem_insert_of_mem hi)]"
},
{
"state_after": "case h.e'_5.h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1412380\nG : Type ?u.1412383\n𝕜 : Type ?u.1412386\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns✝ : Finset ι\nf : ι → α → E\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Integrable (f i_1)\nx : α\n⊢ Finset.sum s (fun i => f i) x = ∑ i in s, f i x",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1412380\nG : Type ?u.1412383\n𝕜 : Type ?u.1412386\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns✝ : Finset ι\nf : ι → α → E\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Integrable (f i_1)\n⊢ Integrable (∑ i in s, f i)",
"tactic": "convert integrable_finset_sum s fun i hi => hf i (Finset.mem_insert_of_mem hi) with x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1412380\nG : Type ?u.1412383\n𝕜 : Type ?u.1412386\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns✝ : Finset ι\nf : ι → α → E\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Integrable (f i_1)\nx : α\n⊢ Finset.sum s (fun i => f i) x = ∑ i in s, f i x",
"tactic": "simp"
}
] | [
1389,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1377,
1
] |
Mathlib/Order/WithBot.lean | WithBot.exists | [] | [
75,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
11
] |
Mathlib/Data/Set/Basic.lean | Set.insert_comm | [] | [
1182,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1181,
1
] |
Mathlib/Topology/UniformSpace/CompactConvergence.lean | ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd | [
{
"state_after": "case intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "case intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "let U : α → Set α := fun x => f ⁻¹' ball (f x) Z"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "have hUK : K ⊆ ⋃ x : K, U (x : K) := by\n intro x hx\n simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]\n exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "have hC : K ⊆ ⋃ i, C i := by\n rw [← K.inter_iUnion, subset_inter_iff]\n refine' ⟨Subset.rfl, ht.trans _⟩\n simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]\n exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W := by\n simp only [← image_subset_iff, ← mem_preimage]\n rintro ⟨⟨x, hx₁⟩, hx₂⟩\n have hZW : closure (ball (f x) Z) ⊆ ball (f x) W := by\n intro y hy\n obtain ⟨z, hz₁, hz₂⟩ := UniformSpace.mem_closure_iff_ball.mp hy hZ₁\n exact ball_mono hZ₃ _ (mem_ball_comp hz₂ ((mem_ball_symmetry hZ₂).mp hz₁))\n calc\n f '' (K ∩ closure (U x)) ⊆ f '' closure (U x) := image_subset _ (inter_subset_right _ _)\n _ ⊆ closure (f '' U x) := f.continuous.continuousOn.image_closure\n _ ⊆ closure (ball (f x) Z) := by\n apply closure_mono\n simp only [image_subset_iff]\n rfl\n _ ⊆ ball (f x) W := hZW"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\ng : C(α, β)\nhg : g ∈ ⋂ (i : { x // x ∈ t }), CompactOpen.gen (C i) ((fun i => ball (↑f ↑↑i) W) i)\nx : α\nhx : x ∈ K\n⊢ ∃ z, (↑f x, z) ∈ W ∧ (z, ↑g x) ∈ W",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\n⊢ ∃ ι x C _hC U _hU,\n (f ∈ ⋂ (i : ι), CompactOpen.gen (C i) (U i)) ∧ (⋂ (i : ι), CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f",
"tactic": "refine'\n ⟨t, t.fintypeCoeSort, C, fun i => hK.inter_right isClosed_closure, fun i =>\n ball (f ((i : K) : α)) W, fun i => isOpen_ball _ hW₄, by simp [CompactOpen.gen, hfC],\n fun g hg x hx => hW₃ (mem_compRel.mpr _)⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\ng : C(α, β)\nx : α\nhx : x ∈ K\nhg : ∀ (i : { x // x ∈ t }), K ∩ closure (↑f ⁻¹' ball (↑f ↑↑i) Z) ⊆ (fun a => ↑g a) ⁻¹' ball (↑f ↑↑i) W\n⊢ ∃ z, (↑f x, z) ∈ W ∧ (z, ↑g x) ∈ W",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\ng : C(α, β)\nhg : g ∈ ⋂ (i : { x // x ∈ t }), CompactOpen.gen (C i) ((fun i => ball (↑f ↑↑i) W) i)\nx : α\nhx : x ∈ K\n⊢ ∃ z, (↑f x, z) ∈ W ∧ (z, ↑g x) ∈ W",
"tactic": "simp only [mem_iInter, CompactOpen.gen, mem_setOf_eq, image_subset_iff] at hg"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\ng : C(α, β)\nx : α\nhx : x ∈ K\nhg : ∀ (i : { x // x ∈ t }), K ∩ closure (↑f ⁻¹' ball (↑f ↑↑i) Z) ⊆ (fun a => ↑g a) ⁻¹' ball (↑f ↑↑i) W\ny : { x // x ∈ t }\nhy : x ∈ C y\n⊢ ∃ z, (↑f x, z) ∈ W ∧ (z, ↑g x) ∈ W",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\ng : C(α, β)\nx : α\nhx : x ∈ K\nhg : ∀ (i : { x // x ∈ t }), K ∩ closure (↑f ⁻¹' ball (↑f ↑↑i) Z) ⊆ (fun a => ↑g a) ⁻¹' ball (↑f ↑↑i) W\n⊢ ∃ z, (↑f x, z) ∈ W ∧ (z, ↑g x) ∈ W",
"tactic": "obtain ⟨y, hy⟩ := mem_iUnion.mp (hC hx)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\ng : C(α, β)\nx : α\nhx : x ∈ K\nhg : ∀ (i : { x // x ∈ t }), K ∩ closure (↑f ⁻¹' ball (↑f ↑↑i) Z) ⊆ (fun a => ↑g a) ⁻¹' ball (↑f ↑↑i) W\ny : { x // x ∈ t }\nhy : x ∈ C y\n⊢ ∃ z, (↑f x, z) ∈ W ∧ (z, ↑g x) ∈ W",
"tactic": "exact ⟨f y, (mem_ball_symmetry hW₂).mp (hfC y hy), mem_preimage.mp (hg y hy)⟩"
},
{
"state_after": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nx : α\nhx : x ∈ K\n⊢ x ∈ ⋃ (x : ↑K), U ↑x",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\n⊢ K ⊆ ⋃ (x : ↑K), U ↑x",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nx : α\nhx : x ∈ K\n⊢ ∃ i, i ∈ K ∧ ↑f x ∈ ball (↑f i) Z",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nx : α\nhx : x ∈ K\n⊢ x ∈ ⋃ (x : ↑K), U ↑x",
"tactic": "simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]"
},
{
"state_after": "no goals",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nx : α\nhx : x ∈ K\n⊢ ∃ i, i ∈ K ∧ ↑f x ∈ ball (↑f i) Z",
"tactic": "exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nx : α\nhx : x ∈ K\n⊢ ↑{ val := x, property := hx } ∈ K ∧ ↑f x ∈ ball (↑f ↑{ val := x, property := hx }) Z",
"tactic": "simp [hx, mem_ball_self (f x) hZ₁]"
},
{
"state_after": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ K ⊆ K ∧ K ⊆ ⋃ (i : { x // x ∈ t }), closure (U ↑↑i)",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ K ⊆ ⋃ (i : { x // x ∈ t }), C i",
"tactic": "rw [← K.inter_iUnion, subset_inter_iff]"
},
{
"state_after": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ (⋃ (i : ↑K) (_ : i ∈ t), U ↑i) ⊆ ⋃ (i : { x // x ∈ t }), closure (U ↑↑i)",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ K ⊆ K ∧ K ⊆ ⋃ (i : { x // x ∈ t }), closure (U ↑↑i)",
"tactic": "refine' ⟨Subset.rfl, ht.trans _⟩"
},
{
"state_after": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ ∀ (x : α) (h : x ∈ K),\n { val := x, property := h } ∈ t → ↑f ⁻¹' ball (↑f x) Z ⊆ ⋃ (i : { x // x ∈ t }), closure (↑f ⁻¹' ball (↑f ↑↑i) Z)",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ (⋃ (i : ↑K) (_ : i ∈ t), U ↑i) ⊆ ⋃ (i : { x // x ∈ t }), closure (U ↑↑i)",
"tactic": "simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\n⊢ ∀ (x : α) (h : x ∈ K),\n { val := x, property := h } ∈ t → ↑f ⁻¹' ball (↑f x) Z ⊆ ⋃ (i : { x // x ∈ t }), closure (↑f ⁻¹' ball (↑f ↑↑i) Z)",
"tactic": "exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])"
},
{
"state_after": "no goals",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\n⊢ ↑f ⁻¹' ball (↑f x) Z ⊆ closure (↑f ⁻¹' ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) Z)",
"tactic": "simp [subset_closure]"
},
{
"state_after": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\n⊢ ∀ (i : { x // x ∈ t }), ↑f '' (K ∩ closure (↑f ⁻¹' ball (↑f ↑↑i) Z)) ⊆ ball (↑f ↑↑i) W",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\n⊢ ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W",
"tactic": "simp only [← image_subset_iff, ← mem_preimage]"
},
{
"state_after": "case mk.mk\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\n⊢ ↑f '' (K ∩ closure (↑f ⁻¹' ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) Z)) ⊆\n ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) W",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\n⊢ ∀ (i : { x // x ∈ t }), ↑f '' (K ∩ closure (↑f ⁻¹' ball (↑f ↑↑i) Z)) ⊆ ball (↑f ↑↑i) W",
"tactic": "rintro ⟨⟨x, hx₁⟩, hx₂⟩"
},
{
"state_after": "case mk.mk\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\nhZW : closure (ball (↑f x) Z) ⊆ ball (↑f x) W\n⊢ ↑f '' (K ∩ closure (↑f ⁻¹' ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) Z)) ⊆\n ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) W",
"state_before": "case mk.mk\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\n⊢ ↑f '' (K ∩ closure (↑f ⁻¹' ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) Z)) ⊆\n ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) W",
"tactic": "have hZW : closure (ball (f x) Z) ⊆ ball (f x) W := by\n intro y hy\n obtain ⟨z, hz₁, hz₂⟩ := UniformSpace.mem_closure_iff_ball.mp hy hZ₁\n exact ball_mono hZ₃ _ (mem_ball_comp hz₂ ((mem_ball_symmetry hZ₂).mp hz₁))"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\nhZW : closure (ball (↑f x) Z) ⊆ ball (↑f x) W\n⊢ ↑f '' (K ∩ closure (↑f ⁻¹' ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) Z)) ⊆\n ball (↑f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) W",
"tactic": "calc\n f '' (K ∩ closure (U x)) ⊆ f '' closure (U x) := image_subset _ (inter_subset_right _ _)\n _ ⊆ closure (f '' U x) := f.continuous.continuousOn.image_closure\n _ ⊆ closure (ball (f x) Z) := by\n apply closure_mono\n simp only [image_subset_iff]\n rfl\n _ ⊆ ball (f x) W := hZW"
},
{
"state_after": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\ny : β\nhy : y ∈ closure (ball (↑f x) Z)\n⊢ y ∈ ball (↑f x) W",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\n⊢ closure (ball (↑f x) Z) ⊆ ball (↑f x) W",
"tactic": "intro y hy"
},
{
"state_after": "case intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\ny : β\nhy : y ∈ closure (ball (↑f x) Z)\nz : β\nhz₁ : z ∈ ball y Z\nhz₂ : z ∈ ball (↑f x) Z\n⊢ y ∈ ball (↑f x) W",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\ny : β\nhy : y ∈ closure (ball (↑f x) Z)\n⊢ y ∈ ball (↑f x) W",
"tactic": "obtain ⟨z, hz₁, hz₂⟩ := UniformSpace.mem_closure_iff_ball.mp hy hZ₁"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\ny : β\nhy : y ∈ closure (ball (↑f x) Z)\nz : β\nhz₁ : z ∈ ball y Z\nhz₂ : z ∈ ball (↑f x) Z\n⊢ y ∈ ball (↑f x) W",
"tactic": "exact ball_mono hZ₃ _ (mem_ball_comp hz₂ ((mem_ball_symmetry hZ₂).mp hz₁))"
},
{
"state_after": "case h\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\nhZW : closure (ball (↑f x) Z) ⊆ ball (↑f x) W\n⊢ ↑f '' U x ⊆ ball (↑f x) Z",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\nhZW : closure (ball (↑f x) Z) ⊆ ball (↑f x) W\n⊢ closure (↑f '' U x) ⊆ closure (ball (↑f x) Z)",
"tactic": "apply closure_mono"
},
{
"state_after": "case h\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\nhZW : closure (ball (↑f x) Z) ⊆ ball (↑f x) W\n⊢ ↑f ⁻¹' ball (↑f x) Z ⊆ (fun a => ↑f a) ⁻¹' ball (↑f x) Z",
"state_before": "case h\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\nhZW : closure (ball (↑f x) Z) ⊆ ball (↑f x) W\n⊢ ↑f '' U x ⊆ ball (↑f x) Z",
"tactic": "simp only [image_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nx : α\nhx₁ : x ∈ K\nhx₂ : { val := x, property := hx₁ } ∈ t\nhZW : closure (ball (↑f x) Z) ⊆ ball (↑f x) W\n⊢ ↑f ⁻¹' ball (↑f x) Z ⊆ (fun a => ↑f a) ⁻¹' ball (↑f x) Z",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nhK : IsCompact K\nhV : V ∈ 𝓤 β\nW : Set (β × β)\nhW₁ : W ∈ 𝓤 β\nhW₄ : IsOpen W\nhW₂ : SymmetricRel W\nhW₃ : W ○ W ⊆ V\nZ : Set (β × β)\nhZ₁ : Z ∈ 𝓤 β\nhZ₄ : IsOpen Z\nhZ₂ : SymmetricRel Z\nhZ₃ : Z ○ Z ⊆ W\nU : α → Set α := fun x => ↑f ⁻¹' ball (↑f x) Z\nhU : ∀ (x : α), IsOpen (U x)\nhUK : K ⊆ ⋃ (x : ↑K), U ↑x\nt : Finset ↑K\nht : K ⊆ ⋃ (i : ↑K) (_ : i ∈ t), U ↑i\nC : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)\nhC : K ⊆ ⋃ (i : { x // x ∈ t }), C i\nhfC : ∀ (i : { x // x ∈ t }), C i ⊆ ↑f ⁻¹' ball (↑f ↑↑i) W\n⊢ f ∈ ⋂ (i : { x // x ∈ t }), CompactOpen.gen (C i) ((fun i => ball (↑f ↑↑i) W) i)",
"tactic": "simp [CompactOpen.gen, hfC]"
}
] | [
267,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
225,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean | coe_frobeniusEquiv | [] | [
57,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
56,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean | integral_cos_mul_complex | [
{
"state_after": "case hderiv\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x\n\ncase hint\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ IntervalIntegrable (fun y => Complex.cos (z * ↑y)) MeasureTheory.volume a b",
"state_before": "a✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ (∫ (x : ℝ) in a..b, Complex.cos (z * ↑x)) = Complex.sin (z * ↑b) / z - Complex.sin (z * ↑a) / z",
"tactic": "apply integral_eq_sub_of_hasDerivAt"
},
{
"state_after": "case hint\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ IntervalIntegrable (fun y => Complex.cos (z * ↑y)) MeasureTheory.volume a b\n\ncase hderiv\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"state_before": "case hderiv\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x\n\ncase hint\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ IntervalIntegrable (fun y => Complex.cos (z * ↑y)) MeasureTheory.volume a b",
"tactic": "swap"
},
{
"state_after": "case hderiv\na✝¹ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b x : ℝ\na✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"state_before": "case hderiv\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"tactic": "intro x _"
},
{
"state_after": "case hderiv\na✝² b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b x : ℝ\na✝ : x ∈ [[a✝¹, b]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"state_before": "case hderiv\na✝¹ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b x : ℝ\na✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"tactic": "have a := Complex.hasDerivAt_sin (↑x * z)"
},
{
"state_after": "case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"state_before": "case hderiv\na✝² b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b x : ℝ\na✝ : x ∈ [[a✝¹, b]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"tactic": "have b : HasDerivAt (fun y => y * z : ℂ → ℂ) z ↑x := hasDerivAt_mul_const _"
},
{
"state_after": "case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"state_before": "case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"tactic": "have c : HasDerivAt (fun y : ℂ => Complex.sin (y * z)) _ ↑x := HasDerivAt.comp (𝕜 := ℂ) x a b"
},
{
"state_after": "case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (↑y * z) / z) (Complex.cos (↑x * z) * z / z) x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"state_before": "case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"tactic": "have d := HasDerivAt.comp_ofReal (c.div_const z)"
},
{
"state_after": "case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"state_before": "case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (↑y * z) / z) (Complex.cos (↑x * z) * z / z) x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"tactic": "simp only [mul_comm] at d"
},
{
"state_after": "case h.e'_7\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ Complex.cos (z * ↑x) = z * Complex.cos (z * ↑x) / z",
"state_before": "case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x",
"tactic": "convert d using 1"
},
{
"state_after": "case h.e'_7\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ Complex.cos (z * ↑x) = Complex.cos (z * ↑x) * z / z",
"state_before": "case h.e'_7\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ Complex.cos (z * ↑x) = z * Complex.cos (z * ↑x) / z",
"tactic": "conv_rhs => arg 1; rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ Complex.cos (z * ↑x) = Complex.cos (z * ↑x) * z / z",
"tactic": "rw [mul_div_cancel _ hz]"
},
{
"state_after": "case hint.hu\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ Continuous fun y => Complex.cos (z * ↑y)",
"state_before": "case hint\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ IntervalIntegrable (fun y => Complex.cos (z * ↑y)) MeasureTheory.volume a b",
"tactic": "apply Continuous.intervalIntegrable"
},
{
"state_after": "no goals",
"state_before": "case hint.hu\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ Continuous fun y => Complex.cos (z * ↑y)",
"tactic": "exact Complex.continuous_cos.comp (continuous_const.mul Complex.continuous_ofReal)"
}
] | [
531,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
517,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean | Real.le_logb_iff_rpow_le | [
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nhb : 1 < b\nhy : 0 < y\n⊢ x ≤ logb b y ↔ b ^ x ≤ y",
"tactic": "rw [← rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hy]"
}
] | [
172,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean | PiLp.equiv_symm_apply | [] | [
112,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | MeasureTheory.MeasurePreserving.iterate | [] | [
136,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
11
] |
Mathlib/Algebra/Support.lean | Function.support_inv | [] | [
385,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
384,
1
] |
Mathlib/Analysis/Convex/Between.lean | Wbtw.rotate_iff | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.222353\nP : Type u_3\nP' : Type ?u.222359\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y z : P\nh : Wbtw R x y z\n⊢ Wbtw R z x y ↔ x = y",
"tactic": "rw [← wbtw_rotate_iff R x, and_iff_right h]"
}
] | [
400,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
399,
1
] |
Mathlib/Order/Chain.lean | chainClosure_succ_total | [
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s t : Set α\na b x y : α\ns✝ : Set α\na✝ : ChainClosure r s✝\na_ih✝ : ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ s✝ → s✝ = c₁ ∨ SuccChain r c₁ ⊆ s✝\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ SuccChain r s✝\n⊢ SuccChain r s✝ = c₁ ∨ SuccChain r c₁ ⊆ SuccChain r s✝\n\ncase union\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s t : Set α\na b x y : α\ns✝ : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s✝ → ChainClosure r a\na_ih✝ : ∀ (a : Set α), a ∈ s✝ → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s✝\n⊢ ⋃₀ s✝ = c₁ ∨ SuccChain r c₁ ⊆ ⋃₀ s✝",
"state_before": "α : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₁ c₂ c₃ s t : Set α\na b x y : α\nhc₁ : ChainClosure r c₁\nhc₂ : ChainClosure r c₂\nh : c₁ ⊆ c₂\n⊢ c₂ = c₁ ∨ SuccChain r c₁ ⊆ c₂",
"tactic": "induction hc₂ generalizing c₁ hc₁"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂✝ c₃ s t : Set α\na b x y : α\nc₂ : Set α\na✝ : ChainClosure r c₂\nih : ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ c₂ → c₂ = c₁ ∨ SuccChain r c₁ ⊆ c₂\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ SuccChain r c₂\nh₁ : c₁ ⊆ c₂\n⊢ SuccChain r c₁ ⊆ SuccChain r c₂",
"state_before": "α : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂✝ c₃ s t : Set α\na b x y : α\nc₂ : Set α\na✝ : ChainClosure r c₂\nih : ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ c₂ → c₂ = c₁ ∨ SuccChain r c₁ ⊆ c₂\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ SuccChain r c₂\n⊢ SuccChain r c₂ = c₁ ∨ SuccChain r c₁ ⊆ SuccChain r c₂",
"tactic": "refine' ((chainClosure_succ_total_aux hc₁) fun c₁ => ih).imp h.antisymm' fun h₁ => _"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂✝ c₃ s t : Set α\na b x y : α\nc₂ : Set α\na✝ : ChainClosure r c₂\nih : ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ c₂ → c₂ = c₁ ∨ SuccChain r c₁ ⊆ c₂\nhc₁ : ChainClosure r c₂\nh : c₂ ⊆ SuccChain r c₂\nh₁ : c₂ ⊆ c₂\n⊢ SuccChain r c₂ ⊆ SuccChain r c₂\n\ncase inr\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂✝ c₃ s t : Set α\na b x y : α\nc₂ : Set α\na✝ : ChainClosure r c₂\nih : ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ c₂ → c₂ = c₁ ∨ SuccChain r c₁ ⊆ c₂\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ SuccChain r c₂\nh₁ : c₁ ⊆ c₂\nh₂ : SuccChain r c₁ ⊆ c₂\n⊢ SuccChain r c₁ ⊆ SuccChain r c₂",
"state_before": "α : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂✝ c₃ s t : Set α\na b x y : α\nc₂ : Set α\na✝ : ChainClosure r c₂\nih : ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ c₂ → c₂ = c₁ ∨ SuccChain r c₁ ⊆ c₂\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ SuccChain r c₂\nh₁ : c₁ ⊆ c₂\n⊢ SuccChain r c₁ ⊆ SuccChain r c₂",
"tactic": "obtain rfl | h₂ := ih hc₁ h₁"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂✝ c₃ s t : Set α\na b x y : α\nc₂ : Set α\na✝ : ChainClosure r c₂\nih : ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ c₂ → c₂ = c₁ ∨ SuccChain r c₁ ⊆ c₂\nhc₁ : ChainClosure r c₂\nh : c₂ ⊆ SuccChain r c₂\nh₁ : c₂ ⊆ c₂\n⊢ SuccChain r c₂ ⊆ SuccChain r c₂",
"tactic": "exact Subset.rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂✝ c₃ s t : Set α\na b x y : α\nc₂ : Set α\na✝ : ChainClosure r c₂\nih : ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ c₂ → c₂ = c₁ ∨ SuccChain r c₁ ⊆ c₂\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ SuccChain r c₂\nh₁ : c₁ ⊆ c₂\nh₂ : SuccChain r c₁ ⊆ c₂\n⊢ SuccChain r c₁ ⊆ SuccChain r c₂",
"tactic": "exact h₂.trans subset_succChain"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\n⊢ ⋃₀ s ⊆ c₁ ∨ SuccChain r c₁ ⊆ ⋃₀ s",
"state_before": "α : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\n⊢ ⋃₀ s = c₁ ∨ SuccChain r c₁ ⊆ ⋃₀ s",
"tactic": "apply Or.imp_left h.antisymm'"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\n⊢ ¬(⋃₀ s ⊆ c₁ ∨ SuccChain r c₁ ⊆ ⋃₀ s) → False",
"state_before": "α : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\n⊢ ⋃₀ s ⊆ c₁ ∨ SuccChain r c₁ ⊆ ⋃₀ s",
"tactic": "apply by_contradiction"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\n⊢ ∀ (x : Set α), x ∈ s → ¬x ⊆ c₁ → ¬SuccChain r c₁ ⊆ ⋃₀ s → False",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\n⊢ ¬(⋃₀ s ⊆ c₁ ∨ SuccChain r c₁ ⊆ ⋃₀ s) → False",
"tactic": "simp only [sUnion_subset_iff, not_or, not_forall, exists_prop, and_imp, forall_exists_index]"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\n⊢ False",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\n⊢ ∀ (x : Set α), x ∈ s → ¬x ⊆ c₁ → ¬SuccChain r c₁ ⊆ ⋃₀ s → False",
"tactic": "intro c₃ hc₃ h₁ h₂"
},
{
"state_after": "case a.inl\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh✝ : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\nh : SuccChain r c₃ ⊆ c₁\n⊢ False\n\ncase a.inr\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh✝ : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\nh : c₁ ⊆ c₃\n⊢ False",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\n⊢ False",
"tactic": "obtain h | h := chainClosure_succ_total_aux hc₁ fun c₄ => ih _ hc₃"
},
{
"state_after": "case a.inr.inl\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh✝ : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\nh : c₁ ⊆ c₃\nh' : c₃ = c₁\n⊢ False\n\ncase a.inr.inr\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh✝ : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\nh : c₁ ⊆ c₃\nh' : SuccChain r c₁ ⊆ c₃\n⊢ False",
"state_before": "case a.inr\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh✝ : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\nh : c₁ ⊆ c₃\n⊢ False",
"tactic": "obtain h' | h' := ih c₃ hc₃ hc₁ h"
},
{
"state_after": "no goals",
"state_before": "case a.inl\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh✝ : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\nh : SuccChain r c₃ ⊆ c₁\n⊢ False",
"tactic": "exact h₁ (subset_succChain.trans h)"
},
{
"state_after": "no goals",
"state_before": "case a.inr.inl\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh✝ : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\nh : c₁ ⊆ c₃\nh' : c₃ = c₁\n⊢ False",
"tactic": "exact h₁ h'.subset"
},
{
"state_after": "no goals",
"state_before": "case a.inr.inr\nα : Type u_1\nβ : Type ?u.11677\nr : α → α → Prop\nc c₂ c₃✝ s✝ t : Set α\na b x y : α\ns : Set (Set α)\na✝ : ∀ (a : Set α), a ∈ s → ChainClosure r a\nih : ∀ (a : Set α), a ∈ s → ∀ {c₁ : Set α}, ChainClosure r c₁ → c₁ ⊆ a → a = c₁ ∨ SuccChain r c₁ ⊆ a\nc₁ : Set α\nhc₁ : ChainClosure r c₁\nh✝ : c₁ ⊆ ⋃₀ s\nc₃ : Set α\nhc₃ : c₃ ∈ s\nh₁ : ¬c₃ ⊆ c₁\nh₂ : ¬SuccChain r c₁ ⊆ ⋃₀ s\nh : c₁ ⊆ c₃\nh' : SuccChain r c₁ ⊆ c₃\n⊢ False",
"tactic": "exact h₂ (h'.trans <| subset_sUnion_of_mem hc₃)"
}
] | [
243,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
226,
9
] |
Mathlib/Data/Set/Sups.lean | Set.sups_subset_right | [] | [
94,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean | AddSubmonoid.pointwise_smul_le_pointwise_smul_iff₀ | [] | [
466,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
464,
1
] |
Mathlib/Data/Set/Function.lean | Set.BijOn.image_eq | [] | [
970,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
969,
1
] |
Std/Data/Int/DivMod.lean | Int.eq_div_of_mul_eq_right | [] | [
749,
47
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
747,
11
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean | Matrix.isSymm_zero | [] | [
78,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Data/Nat/Totient.lean | Nat.totient_pos | [
{
"state_after": "no goals",
"state_before": "⊢ 0 < 0 → 0 < φ 0",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "⊢ 0 < 1 → 0 < φ 1",
"tactic": "simp [totient]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nx✝ : 0 < n + 2\n⊢ 1 < n + 2",
"tactic": "simp"
}
] | [
76,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.le_sub_of_add_le_right | [] | [
1165,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1164,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | collinear_insert_insert_of_mem_affineSpan_pair | [
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.323622\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p₄ : P\nh₁ : p₁ ∈ affineSpan k {p₃, p₄}\nh₂ : p₂ ∈ affineSpan k {p₃, p₄}\n⊢ Collinear k {p₃, p₄}",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.323622\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p₄ : P\nh₁ : p₁ ∈ affineSpan k {p₃, p₄}\nh₂ : p₂ ∈ affineSpan k {p₃, p₄}\n⊢ Collinear k {p₁, p₂, p₃, p₄}",
"tactic": "rw [collinear_insert_iff_of_mem_affineSpan\n ((AffineSubspace.le_def' _ _).1 (affineSpan_mono k (Set.subset_insert _ _)) _ h₁),\n collinear_insert_iff_of_mem_affineSpan h₂]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.323622\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p₄ : P\nh₁ : p₁ ∈ affineSpan k {p₃, p₄}\nh₂ : p₂ ∈ affineSpan k {p₃, p₄}\n⊢ Collinear k {p₃, p₄}",
"tactic": "exact collinear_pair _ _ _"
}
] | [
568,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
563,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.sign_coe_neg_pi_div_two | [
{
"state_after": "no goals",
"state_before": "⊢ sign ↑(-π / 2) = -1",
"tactic": "rw [sign, sin_coe, neg_div, Real.sin_neg, sin_pi_div_two, Left.sign_neg, sign_one]"
}
] | [
976,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
975,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.add_lt_add_right_coe | [] | [
689,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
688,
1
] |
Mathlib/Data/Finset/Image.lean | Finset.map_disjiUnion | [] | [
281,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | QuadraticForm.coe_copy | [] | [
194,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_left_pure_ceil | [] | [
83,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Mathlib/LinearAlgebra/TensorPower.lean | TensorPower.algebraMap₀_eq_smul_one | [
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\n⊢ r • ↑(tprod R) isEmptyElim = r • GradedMonoid.GOne.one",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\n⊢ ↑algebraMap₀ r = r • GradedMonoid.GOne.one",
"tactic": "simp [algebraMap₀]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\n⊢ r • ↑(tprod R) isEmptyElim = r • GradedMonoid.GOne.one",
"tactic": "congr"
}
] | [
237,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
236,
1
] |
Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | le_mul_right | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : CanonicallyOrderedMonoid α\na b c d : α\nh : a ≤ b\n⊢ a = a * 1",
"tactic": "simp"
}
] | [
283,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
280,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean | circleIntegral.integral_eq_zero_of_hasDerivWithinAt | [] | [
487,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
485,
1
] |
Mathlib/CategoryTheory/Functor/Category.lean | CategoryTheory.NatTrans.id_hcomp_app | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G H✝ I : C ⥤ D\nH : E ⥤ C\nα : F ⟶ G\nX : E\n⊢ (𝟙 H ◫ α).app X = α.app (H.obj X)",
"tactic": "simp"
}
] | [
124,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.IsImage.iff_symm_preimage_eq | [] | [
519,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
518,
1
] |
Mathlib/Topology/Category/Profinite/Basic.lean | FintypeCat.discreteTopology | [] | [
235,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
234,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.pushout.inr_desc | [] | [
1194,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1192,
1
] |
Mathlib/Order/RelClasses.lean | WellFoundedGT.induction | [] | [
440,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
439,
1
] |
Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | ModuleCat.Free.ε_apply | [] | [
83,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.comap' | [] | [
162,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean | MvPolynomial.vars_prod | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.166872\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ vars (∏ i in s, f i) ⊆ Finset.biUnion s fun i => vars (f i)",
"tactic": "classical\ninduction s using Finset.induction_on with\n| empty => simp\n| insert hs hsub =>\n simp only [hs, Finset.biUnion_insert, Finset.prod_insert, not_false_iff]\n apply Finset.Subset.trans (vars_mul _ _)\n exact Finset.union_subset_union (Finset.Subset.refl _) hsub"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.166872\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ vars (∏ i in s, f i) ⊆ Finset.biUnion s fun i => vars (f i)",
"tactic": "induction s using Finset.induction_on with\n| empty => simp\n| insert hs hsub =>\n simp only [hs, Finset.biUnion_insert, Finset.prod_insert, not_false_iff]\n apply Finset.Subset.trans (vars_mul _ _)\n exact Finset.union_subset_union (Finset.Subset.refl _) hsub"
},
{
"state_after": "no goals",
"state_before": "case empty\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.166872\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\nf : ι → MvPolynomial σ R\n⊢ vars (∏ i in ∅, f i) ⊆ Finset.biUnion ∅ fun i => vars (f i)",
"tactic": "simp"
},
{
"state_after": "case insert\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.166872\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\nf : ι → MvPolynomial σ R\na✝ : ι\ns✝ : Finset ι\nhs : ¬a✝ ∈ s✝\nhsub : vars (∏ i in s✝, f i) ⊆ Finset.biUnion s✝ fun i => vars (f i)\n⊢ vars (f a✝ * ∏ i in s✝, f i) ⊆ vars (f a✝) ∪ Finset.biUnion s✝ fun i => vars (f i)",
"state_before": "case insert\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.166872\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\nf : ι → MvPolynomial σ R\na✝ : ι\ns✝ : Finset ι\nhs : ¬a✝ ∈ s✝\nhsub : vars (∏ i in s✝, f i) ⊆ Finset.biUnion s✝ fun i => vars (f i)\n⊢ vars (∏ i in insert a✝ s✝, f i) ⊆ Finset.biUnion (insert a✝ s✝) fun i => vars (f i)",
"tactic": "simp only [hs, Finset.biUnion_insert, Finset.prod_insert, not_false_iff]"
},
{
"state_after": "case insert\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.166872\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\nf : ι → MvPolynomial σ R\na✝ : ι\ns✝ : Finset ι\nhs : ¬a✝ ∈ s✝\nhsub : vars (∏ i in s✝, f i) ⊆ Finset.biUnion s✝ fun i => vars (f i)\n⊢ vars (f a✝) ∪ vars (∏ i in s✝, f i) ⊆ vars (f a✝) ∪ Finset.biUnion s✝ fun i => vars (f i)",
"state_before": "case insert\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.166872\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\nf : ι → MvPolynomial σ R\na✝ : ι\ns✝ : Finset ι\nhs : ¬a✝ ∈ s✝\nhsub : vars (∏ i in s✝, f i) ⊆ Finset.biUnion s✝ fun i => vars (f i)\n⊢ vars (f a✝ * ∏ i in s✝, f i) ⊆ vars (f a✝) ∪ Finset.biUnion s✝ fun i => vars (f i)",
"tactic": "apply Finset.Subset.trans (vars_mul _ _)"
},
{
"state_after": "no goals",
"state_before": "case insert\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.166872\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\nf : ι → MvPolynomial σ R\na✝ : ι\ns✝ : Finset ι\nhs : ¬a✝ ∈ s✝\nhsub : vars (∏ i in s✝, f i) ⊆ Finset.biUnion s✝ fun i => vars (f i)\n⊢ vars (f a✝) ∪ vars (∏ i in s✝, f i) ⊆ vars (f a✝) ∪ Finset.biUnion s✝ fun i => vars (f i)",
"tactic": "exact Finset.union_subset_union (Finset.Subset.refl _) hsub"
}
] | [
392,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
384,
1
] |
Mathlib/Order/Concept.lean | intentClosure_empty | [] | [
87,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/Analysis/Complex/Isometry.lean | rotationOf_rotation | [
{
"state_after": "no goals",
"state_before": "a : { x // x ∈ circle }\n⊢ ↑(rotationOf (↑rotation a)) = ↑a",
"tactic": "simp"
}
] | [
86,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
] |
Mathlib/RingTheory/Algebraic.lean | Algebra.isAlgebraic_iff_isIntegral | [] | [
200,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
11
] |
Mathlib/Topology/LocalAtTarget.lean | Set.restrictPreimage_openEmbedding | [] | [
55,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | BoxIntegral.TaggedPrepartition.IsSubordinate.infPrepartition | [] | [
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271,
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Mathlib/Order/RelIso/Basic.lean | RelIso.cast_refl | [] | [
764,
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762,
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Mathlib/Algebra/Group/Units.lean | IsUnit.mul_val_inv | [
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Mathlib/CategoryTheory/Products/Bifunctor.lean | CategoryTheory.Bifunctor.map_id_comp | [
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Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineMap.span_eq_top_of_surjective | [
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1586,
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Mathlib/SetTheory/Game/PGame.lean | PGame.neg_lf_zero_iff | [
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Mathlib/Data/List/Duplicate.lean | List.duplicate_cons_iff | [
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Mathlib/Topology/Bornology/Basic.lean | Bornology.isCobounded_biInter_finset | [] | [
270,
23
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268,
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Std/Data/List/Lemmas.lean | List.next?_cons | [] | [
457,
74
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457,
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Mathlib/Data/Real/Hyperreal.lean | Hyperreal.infinite_iff_not_exists_st | [] | [
342,
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341,
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Mathlib/SetTheory/Cardinal/Cofinality.lean | Cardinal.derivBFamily_lt_ord | [
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Mathlib/Algebra/Lie/Submodule.lean | LieSubmodule.submodule_span_le_lieSpan | [
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624,
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Mathlib/Algebra/Associated.lean | Associates.quot_mk_eq_mk | [] | [
758,
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757,
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Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | SimpleGraph.sum_degrees_eq_twice_card_edges | [] | [
116,
72
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115,
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Mathlib/FieldTheory/PerfectClosure.lean | PerfectClosure.quot_mk_eq_mk | [] | [
163,
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162,
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Mathlib/Computability/Partrec.lean | Computable.fin_app | [] | [
399,
26
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398,
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Mathlib/Analysis/InnerProductSpace/Projection.lean | maximal_orthonormal_iff_orthogonalComplement_eq_bot | [
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{
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"tactic": "constructor"
},
{
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"tactic": "contrapose!"
},
{
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"tactic": "rintro ⟨x, hx', hx⟩"
},
{
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"tactic": "let e := (‖x‖⁻¹ : 𝕜) • x"
},
{
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"tactic": "have he : ‖e‖ = 1 := by simp [norm_smul_inv_norm hx]"
},
{
"state_after": "case mp.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\n⊢ ∃ u, u ⊇ v ∧ Orthonormal 𝕜 Subtype.val ∧ u ≠ v",
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"tactic": "have he' : e ∈ (span 𝕜 v)ᗮ := smul_mem' _ _ hx'"
},
{
"state_after": "case mp.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\n⊢ ∃ u, u ⊇ v ∧ Orthonormal 𝕜 Subtype.val ∧ u ≠ v",
"state_before": "case mp.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\n⊢ ∃ u, u ⊇ v ∧ Orthonormal 𝕜 Subtype.val ∧ u ≠ v",
"tactic": "have he'' : e ∉ v := by\n intro hev\n have : e = 0 := by\n have : e ∈ span 𝕜 v ⊓ (span 𝕜 v)ᗮ := ⟨subset_span hev, he'⟩\n simpa [(span 𝕜 v).inf_orthogonal_eq_bot] using this\n have : e ≠ 0 := hv.ne_zero ⟨e, hev⟩\n contradiction"
},
{
"state_after": "case mp.intro.intro.refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\n⊢ ∀ (i : { x // x ∈ insert e v }), ‖↑i‖ = 1\n\ncase mp.intro.intro.refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\n⊢ ∀ {i j : { x // x ∈ insert e v }}, i ≠ j → inner ↑i ↑j = 0",
"state_before": "case mp.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\n⊢ ∃ u, u ⊇ v ∧ Orthonormal 𝕜 Subtype.val ∧ u ≠ v",
"tactic": "refine' ⟨insert e v, v.subset_insert e, ⟨_, _⟩, (ne_insert_of_not_mem v he'').symm⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\n⊢ ‖e‖ = 1",
"tactic": "simp [norm_smul_inv_norm hx]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\n⊢ False",
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"tactic": "intro hev"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis : e = 0\n⊢ False",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\n⊢ False",
"tactic": "have : e = 0 := by\n have : e ∈ span 𝕜 v ⊓ (span 𝕜 v)ᗮ := ⟨subset_span hev, he'⟩\n simpa [(span 𝕜 v).inf_orthogonal_eq_bot] using this"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis✝ : e = 0\nthis : e ≠ 0\n⊢ False",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis : e = 0\n⊢ False",
"tactic": "have : e ≠ 0 := hv.ne_zero ⟨e, hev⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis✝ : e = 0\nthis : e ≠ 0\n⊢ False",
"tactic": "contradiction"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis : e ∈ span 𝕜 v ⊓ (span 𝕜 v)ᗮ\n⊢ e = 0",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\n⊢ e = 0",
"tactic": "have : e ∈ span 𝕜 v ⊓ (span 𝕜 v)ᗮ := ⟨subset_span hev, he'⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis : e ∈ span 𝕜 v ⊓ (span 𝕜 v)ᗮ\n⊢ e = 0",
"tactic": "simpa [(span 𝕜 v).inf_orthogonal_eq_bot] using this"
},
{
"state_after": "case mp.intro.intro.refine'_1.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\na : E\nha' : a ∈ insert e v\n⊢ ‖↑{ val := a, property := ha' }‖ = 1",
"state_before": "case mp.intro.intro.refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\n⊢ ∀ (i : { x // x ∈ insert e v }), ‖↑i‖ = 1",
"tactic": "rintro ⟨a, ha'⟩"
},
{
"state_after": "case mp.intro.intro.refine'_1.mk.inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\na : E\nha' : a ∈ insert e v\nha : a = e\n⊢ ‖↑{ val := a, property := ha' }‖ = 1\n\ncase mp.intro.intro.refine'_1.mk.inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\na : E\nha' : a ∈ insert e v\nha : a ∈ v\n⊢ ‖↑{ val := a, property := ha' }‖ = 1",
"state_before": "case mp.intro.intro.refine'_1.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\na : E\nha' : a ∈ insert e v\n⊢ ‖↑{ val := a, property := ha' }‖ = 1",
"tactic": "cases' eq_or_mem_of_mem_insert ha' with ha ha"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.refine'_1.mk.inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\na : E\nha' : a ∈ insert e v\nha : a = e\n⊢ ‖↑{ val := a, property := ha' }‖ = 1",
"tactic": "simp [ha, he]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.refine'_1.mk.inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\na : E\nha' : a ∈ insert e v\nha : a ∈ v\n⊢ ‖↑{ val := a, property := ha' }‖ = 1",
"tactic": "exact hv.1 ⟨a, ha⟩"
},
{
"state_after": "case mp.intro.intro.refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\nh_end : ∀ (a : E), a ∈ v → inner a e = 0\n⊢ ∀ {i j : { x // x ∈ insert e v }}, i ≠ j → inner ↑i ↑j = 0",
"state_before": "case mp.intro.intro.refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\n⊢ ∀ {i j : { x // x ∈ insert e v }}, i ≠ j → inner ↑i ↑j = 0",
"tactic": "have h_end : ∀ a ∈ v, ⟪a, e⟫ = 0 := by\n intro a ha\n exact he' a (Submodule.subset_span ha)"
},
{
"state_after": "case mp.intro.intro.refine'_2.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : ¬e ∈ v\nh_end : ∀ (a : E), a ∈ v → inner a e = 0\na : E\nha' : a ∈ insert e v\n⊢ ∀ {j : { x // x ∈ insert e v }}, { val := a, property := ha' } ≠ j → inner ↑{ val := a, property := ha' } ↑j = 0",
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"tactic": "rintro ⟨a, ha'⟩"
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"tactic": "have : (⟨a, ha⟩ : v) ≠ ⟨b, hb⟩ := by\n intro hab''\n apply hab'\n simpa using hab''"
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"tactic": "exact he' a (Submodule.subset_span ha)"
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"tactic": "have hb : b ∈ v := by\n refine' mem_of_mem_insert_of_ne hb' _\n intro hbe'\n apply hab'\n simp [ha, hbe']"
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"tactic": "rw [inner_eq_zero_symm]"
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"state_before": "case mpr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\n⊢ ¬x ∈ v → x ∈ (span 𝕜 v)ᗮ",
"tactic": "intro hxv y hy"
},
{
"state_after": "case mpr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\ny : E\nhy : y ∈ span 𝕜 v\nhxv' : ¬{ val := x, property := hxu } ∈ Subtype.val ⁻¹' v\n⊢ inner y x = 0",
"state_before": "case mpr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\ny : E\nhy : y ∈ span 𝕜 v\n⊢ inner y x = 0",
"tactic": "have hxv' : (⟨x, hxu⟩ : u) ∉ ((↑) ⁻¹' v : Set u) := by simp [huv, hxv]"
},
{
"state_after": "case mpr.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\nhxv' : ¬{ val := x, property := hxu } ∈ Subtype.val ⁻¹' v\nl : ↑u →₀ 𝕜\nhy : ↑(Finsupp.total (↑u) E 𝕜 Subtype.val) l ∈ span 𝕜 v\nhl : l ∈ Finsupp.supported 𝕜 𝕜 (?m.1124382 (↑(Finsupp.total (↑u) E 𝕜 Subtype.val) l) hy l ⁻¹' v)\n⊢ inner (↑(Finsupp.total (↑u) E 𝕜 Subtype.val) l) x = 0",
"state_before": "case mpr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\ny : E\nhy : y ∈ span 𝕜 v\nhxv' : ¬{ val := x, property := hxu } ∈ Subtype.val ⁻¹' v\n⊢ inner y x = 0",
"tactic": "obtain ⟨l, hl, rfl⟩ :\n ∃ l ∈ Finsupp.supported 𝕜 𝕜 ((↑) ⁻¹' v : Set u), (Finsupp.total (↥u) E 𝕜 (↑)) l = y := by\n rw [← Finsupp.mem_span_image_iff_total]\n simp [huv, inter_eq_self_of_subset_left, hy]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\nhxv' : ¬{ val := x, property := hxu } ∈ Subtype.val ⁻¹' v\nl : ↑u →₀ 𝕜\nhy : ↑(Finsupp.total (↑u) E 𝕜 Subtype.val) l ∈ span 𝕜 v\nhl : l ∈ Finsupp.supported 𝕜 𝕜 (?m.1124382 (↑(Finsupp.total (↑u) E 𝕜 Subtype.val) l) hy l ⁻¹' v)\n⊢ inner (↑(Finsupp.total (↑u) E 𝕜 Subtype.val) l) x = 0",
"tactic": "exact hu.inner_finsupp_eq_zero hxv' hl"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\ny : E\nhy : y ∈ span 𝕜 v\n⊢ ¬{ val := x, property := hxu } ∈ Subtype.val ⁻¹' v",
"tactic": "simp [huv, hxv]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\ny : E\nhy : y ∈ span 𝕜 v\nhxv' : ¬{ val := x, property := hxu } ∈ Subtype.val ⁻¹' v\n⊢ y ∈ span 𝕜 (Subtype.val '' (Subtype.val ⁻¹' v))",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\ny : E\nhy : y ∈ span 𝕜 v\nhxv' : ¬{ val := x, property := hxu } ∈ Subtype.val ⁻¹' v\n⊢ ∃ l, l ∈ Finsupp.supported 𝕜 𝕜 (Subtype.val ⁻¹' v) ∧ ↑(Finsupp.total (↑u) E 𝕜 Subtype.val) l = y",
"tactic": "rw [← Finsupp.mem_span_image_iff_total]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1095350\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nh : ∀ (x : E), x ∈ (span 𝕜 v)ᗮ → x = 0\nu : Set E\nhuv : v ⊆ u\nhu : Orthonormal 𝕜 Subtype.val\nx : E\nhxu : x ∈ u\nhxv : ¬x ∈ v\ny : E\nhy : y ∈ span 𝕜 v\nhxv' : ¬{ val := x, property := hxu } ∈ Subtype.val ⁻¹' v\n⊢ y ∈ span 𝕜 (Subtype.val '' (Subtype.val ⁻¹' v))",
"tactic": "simp [huv, inter_eq_self_of_subset_left, hy]"
}
] | [
1328,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1269,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean | Polynomial.dvd_term_of_isRoot_of_dvd_terms | [] | [
469,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
467,
1
] |
Std/Data/Int/Lemmas.lean | Int.add_le_of_le_sub_right | [
{
"state_after": "a b c : Int\nh✝ : a ≤ c - b\nh : a + b ≤ c - b + b\n⊢ a + b ≤ c",
"state_before": "a b c : Int\nh : a ≤ c - b\n⊢ a + b ≤ c",
"tactic": "have h := Int.add_le_add_right h b"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh✝ : a ≤ c - b\nh : a + b ≤ c - b + b\n⊢ a + b ≤ c",
"tactic": "rwa [Int.sub_add_cancel] at h"
}
] | [
968,
32
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
966,
11
] |
Mathlib/LinearAlgebra/Dimension.lean | rank_pos_iff_exists_ne_zero | [
{
"state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.253674\nR : Type u\nM : Type v\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : NoZeroSMulDivisors R M\ninst✝ : Nontrivial R\n⊢ ¬0 < Module.rank R M ↔ ¬∃ x, x ≠ 0",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.253674\nR : Type u\nM : Type v\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : NoZeroSMulDivisors R M\ninst✝ : Nontrivial R\n⊢ 0 < Module.rank R M ↔ ∃ x, x ≠ 0",
"tactic": "rw [← not_iff_not]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.253674\nR : Type u\nM : Type v\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : NoZeroSMulDivisors R M\ninst✝ : Nontrivial R\n⊢ ¬0 < Module.rank R M ↔ ¬∃ x, x ≠ 0",
"tactic": "simpa using rank_zero_iff_forall_zero"
}
] | [
522,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
520,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean | Polynomial.coeff_restriction | [
{
"state_after": "R : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\n⊢ ↑(if coeff p n = 0 then 0 else { val := coeff p n, property := (_ : coeff p n ∈ Subring.closure ↑(frange p)) }) =\n coeff p n",
"state_before": "R : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\n⊢ ↑(coeff (restriction p) n) = coeff p n",
"tactic": "simp only [restriction, coeff_monomial, finset_sum_coeff, mem_support_iff, Finset.sum_ite_eq',\n Ne.def, ite_not]"
},
{
"state_after": "case inl\nR : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\nh : coeff p n = 0\n⊢ ↑0 = coeff p n\n\ncase inr\nR : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\nh : ¬coeff p n = 0\n⊢ ↑{ val := coeff p n, property := (_ : coeff p n ∈ Subring.closure ↑(frange p)) } = coeff p n",
"state_before": "R : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\n⊢ ↑(if coeff p n = 0 then 0 else { val := coeff p n, property := (_ : coeff p n ∈ Subring.closure ↑(frange p)) }) =\n coeff p n",
"tactic": "split_ifs with h"
},
{
"state_after": "case inl\nR : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\nh : coeff p n = 0\n⊢ ↑0 = 0",
"state_before": "case inl\nR : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\nh : coeff p n = 0\n⊢ ↑0 = coeff p n",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\nh : coeff p n = 0\n⊢ ↑0 = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u\nS : Type ?u.66016\ninst✝ : Ring R\np : R[X]\nn : ℕ\nh : ¬coeff p n = 0\n⊢ ↑{ val := coeff p n, property := (_ : coeff p n ∈ Subring.closure ↑(frange p)) } = coeff p n",
"tactic": "rfl"
}
] | [
283,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | SimpleGraph.sum_incMatrix_apply_of_mem_edgeSet | [
{
"state_after": "no goals",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na b : α\ne : Sym2 α\n⊢ e ∈ edgeSet G → ∑ a : α, incMatrix R G a e = 2",
"tactic": "classical\n refine' e.ind _\n intro a b h\n rw [mem_edgeSet] at h\n rw [← Nat.cast_two, ← card_doubleton h.ne]\n simp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h, true_and_iff]\n congr 2\n ext e\n simp only [mem_filter, mem_univ, true_and_iff, mem_insert, mem_singleton]"
},
{
"state_after": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na b : α\ne : Sym2 α\n⊢ ∀ (x y : α),\n Quotient.mk (Rel.setoid α) (x, y) ∈ edgeSet G → ∑ a : α, incMatrix R G a (Quotient.mk (Rel.setoid α) (x, y)) = 2",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na b : α\ne : Sym2 α\n⊢ e ∈ edgeSet G → ∑ a : α, incMatrix R G a e = 2",
"tactic": "refine' e.ind _"
},
{
"state_after": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Quotient.mk (Rel.setoid α) (a, b) ∈ edgeSet G\n⊢ ∑ a_1 : α, incMatrix R G a_1 (Quotient.mk (Rel.setoid α) (a, b)) = 2",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na b : α\ne : Sym2 α\n⊢ ∀ (x y : α),\n Quotient.mk (Rel.setoid α) (x, y) ∈ edgeSet G → ∑ a : α, incMatrix R G a (Quotient.mk (Rel.setoid α) (x, y)) = 2",
"tactic": "intro a b h"
},
{
"state_after": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Adj G a b\n⊢ ∑ a_1 : α, incMatrix R G a_1 (Quotient.mk (Rel.setoid α) (a, b)) = 2",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Quotient.mk (Rel.setoid α) (a, b) ∈ edgeSet G\n⊢ ∑ a_1 : α, incMatrix R G a_1 (Quotient.mk (Rel.setoid α) (a, b)) = 2",
"tactic": "rw [mem_edgeSet] at h"
},
{
"state_after": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Adj G a b\n⊢ ∑ a_1 : α, incMatrix R G a_1 (Quotient.mk (Rel.setoid α) (a, b)) = ↑(card {a, b})",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Adj G a b\n⊢ ∑ a_1 : α, incMatrix R G a_1 (Quotient.mk (Rel.setoid α) (a, b)) = 2",
"tactic": "rw [← Nat.cast_two, ← card_doubleton h.ne]"
},
{
"state_after": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Adj G a b\n⊢ ↑(card (filter (fun x => x = a ∨ x = b) univ)) = ↑(card {a, b})",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Adj G a b\n⊢ ∑ a_1 : α, incMatrix R G a_1 (Quotient.mk (Rel.setoid α) (a, b)) = ↑(card {a, b})",
"tactic": "simp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h, true_and_iff]"
},
{
"state_after": "case e_a.e_s\nR : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Adj G a b\n⊢ filter (fun x => x = a ∨ x = b) univ = {a, b}",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Adj G a b\n⊢ ↑(card (filter (fun x => x = a ∨ x = b) univ)) = ↑(card {a, b})",
"tactic": "congr 2"
},
{
"state_after": "case e_a.e_s.a\nR : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne✝ : Sym2 α\na b : α\nh : Adj G a b\ne : α\n⊢ e ∈ filter (fun x => x = a ∨ x = b) univ ↔ e ∈ {a, b}",
"state_before": "case e_a.e_s\nR : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne : Sym2 α\na b : α\nh : Adj G a b\n⊢ filter (fun x => x = a ∨ x = b) univ = {a, b}",
"tactic": "ext e"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_s.a\nR : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Fintype α\ninst✝ : NonAssocSemiring R\na✝ b✝ : α\ne✝ : Sym2 α\na b : α\nh : Adj G a b\ne : α\n⊢ e ∈ filter (fun x => x = a ∨ x = b) univ ↔ e ∈ {a, b}",
"tactic": "simp only [mem_filter, mem_univ, true_and_iff, mem_insert, mem_singleton]"
}
] | [
145,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
] |
Mathlib/Data/List/Cycle.lean | Cycle.mem_reverse_iff | [] | [
558,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
557,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean | Polynomial.eval₂_gcd_eq_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\ninst✝ : CommSemiring k\nϕ : R →+* k\nf g : R[X]\nα : k\nhf : eval₂ ϕ α f = 0\nhg : eval₂ ϕ α g = 0\n⊢ eval₂ ϕ α (EuclideanDomain.gcd f g) = 0",
"tactic": "rw [EuclideanDomain.gcd_eq_gcd_ab f g, Polynomial.eval₂_add, Polynomial.eval₂_mul,\n Polynomial.eval₂_mul, hf, hg, MulZeroClass.zero_mul, MulZeroClass.zero_mul, zero_add]"
}
] | [
329,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
326,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.mk_ne_zero_of_natDegree_lt | [] | [
224,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Mathlib/FieldTheory/Subfield.lean | Subfield.coe_iSup_of_directed | [
{
"state_after": "no goals",
"state_before": "K : Type u\nL : Type v\nM : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subfield K\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : K\n⊢ x ∈ ↑(⨆ (i : ι), S i) ↔ x ∈ ⋃ (i : ι), ↑(S i)",
"tactic": "simp [mem_iSup_of_directed hS]"
}
] | [
839,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
837,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean | derivWithin_smul | [] | [
82,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
] |
Mathlib/Algebra/ContinuedFractions/Translations.lean | GeneralizedContinuedFraction.convergent_eq_conts_a_div_conts_b | [] | [
110,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean | NormedAddGroupHom.Equalizer.norm_map_le | [] | [
1003,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1001,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean | IsLocalMaxOn.fderivWithin_eq_zero | [
{
"state_after": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nh : IsLocalMaxOn f s a\ny : E\nhy : y ∈ posTangentConeAt s a\nhy' : -y ∈ posTangentConeAt s a\nhf : ¬DifferentiableWithinAt ℝ f s a\n⊢ ↑0 y = 0",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nh : IsLocalMaxOn f s a\ny : E\nhy : y ∈ posTangentConeAt s a\nhy' : -y ∈ posTangentConeAt s a\nhf : ¬DifferentiableWithinAt ℝ f s a\n⊢ ↑(fderivWithin ℝ f s a) y = 0",
"tactic": "rw [fderivWithin_zero_of_not_differentiableWithinAt hf]"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nh : IsLocalMaxOn f s a\ny : E\nhy : y ∈ posTangentConeAt s a\nhy' : -y ∈ posTangentConeAt s a\nhf : ¬DifferentiableWithinAt ℝ f s a\n⊢ ↑0 y = 0",
"tactic": "rfl"
}
] | [
152,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
147,
1
] |
Mathlib/MeasureTheory/Group/Action.lean | MeasureTheory.measure_preimage_smul | [] | [
180,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean | Polynomial.natDegree_C_mul_le | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nι : Type w\na✝ b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\na : R\nf : R[X]\n⊢ natDegree (↑C a) + natDegree f = 0 + natDegree f",
"tactic": "rw [natDegree_C a]"
}
] | [
97,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.sup_direction_le | [
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns1 s2 : AffineSubspace k P\n⊢ Submodule.span k (↑s1 -ᵥ ↑s1) ⊔ Submodule.span k (↑s2 -ᵥ ↑s2) ≤ Submodule.span k (↑(s1 ⊔ s2) -ᵥ ↑(s1 ⊔ s2))",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns1 s2 : AffineSubspace k P\n⊢ direction s1 ⊔ direction s2 ≤ direction (s1 ⊔ s2)",
"tactic": "simp only [direction_eq_vectorSpan, vectorSpan_def]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns1 s2 : AffineSubspace k P\n⊢ Submodule.span k (↑s1 -ᵥ ↑s1) ⊔ Submodule.span k (↑s2 -ᵥ ↑s2) ≤ Submodule.span k (↑(s1 ⊔ s2) -ᵥ ↑(s1 ⊔ s2))",
"tactic": "exact\n sup_le\n (sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_left : s1 ≤ s1 ⊔ s2)) hp)\n (sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_right : s2 ≤ s1 ⊔ s2)) hp)"
}
] | [
951,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
945,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.val_apply | [] | [
446,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
446,
1
] |
Mathlib/LinearAlgebra/Basis.lean | Basis.coord_equivFun_symm | [] | [
1009,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1008,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.mk_multiset_of_isEmpty | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : IsEmpty α\n⊢ (#α →₀ ℕ) = 1",
"tactic": "simp"
}
] | [
1081,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1080,
1
] |
Mathlib/MeasureTheory/Measure/Complex.lean | MeasureTheory.SignedMeasure.toComplexMeasure_apply | [] | [
74,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | InnerProductGeometry.angle_zero_left | [
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y x : V\n⊢ arccos (inner 0 x / (‖0‖ * ‖x‖)) = π / 2",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y x : V\n⊢ angle 0 x = π / 2",
"tactic": "unfold angle"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y x : V\n⊢ arccos (inner 0 x / (‖0‖ * ‖x‖)) = π / 2",
"tactic": "rw [inner_zero_left, zero_div, Real.arccos_zero]"
}
] | [
122,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.filter_subset | [] | [
2653,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2652,
1
] |
Mathlib/GroupTheory/Complement.lean | Subgroup.isComplement'_top_bot | [] | [
176,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
] |
Mathlib/Order/CompleteLattice.lean | inf_eq_iInf | [] | [
1527,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1526,
1
] |
Mathlib/Data/Finset/Card.lean | Finset.card_range | [] | [
169,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean | LatticeOrderedCommGroup.neg_of_le_one | [] | [
507,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
506,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.SameCycle.exists_pow_eq | [
{
"state_after": "case pos\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y\n\ncase neg\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : ¬x ∈ support f\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y",
"state_before": "ι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y",
"tactic": "by_cases hx : x ∈ f.support"
},
{
"state_after": "case pos.intro.intro\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nk : ℕ\nhk : k < card (support (cycleOf f x))\nhk' : ↑(f ^ k) x = y\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y",
"state_before": "case pos\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y",
"tactic": "obtain ⟨k, hk, hk'⟩ := h.exists_pow_eq_of_mem_support hx"
},
{
"state_after": "case pos.intro.intro.zero\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.zero) x = y\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y\n\ncase pos.intro.intro.succ\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nk : ℕ\nhk : Nat.succ k < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.succ k) x = y\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y",
"state_before": "case pos.intro.intro\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nk : ℕ\nhk : k < card (support (cycleOf f x))\nhk' : ↑(f ^ k) x = y\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y",
"tactic": "cases' k with k"
},
{
"state_after": "case pos.intro.intro.zero.refine'_1\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.zero) x = y\n⊢ 0 < card (support (cycleOf f x))\n\ncase pos.intro.intro.zero.refine'_2\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.zero) x = y\n⊢ ↑(f ^ card (support (cycleOf f x))) x = y",
"state_before": "case pos.intro.intro.zero\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.zero) x = y\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y",
"tactic": "refine' ⟨(f.cycleOf x).support.card, _, self_le_add_right _ _, _⟩"
},
{
"state_after": "case pos.intro.intro.zero.refine'_1\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.zero) x = y\n⊢ cycleOf f x ≠ 1",
"state_before": "case pos.intro.intro.zero.refine'_1\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.zero) x = y\n⊢ 0 < card (support (cycleOf f x))",
"tactic": "refine' zero_lt_one.trans (one_lt_card_support_of_ne_one _)"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.zero.refine'_1\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.zero) x = y\n⊢ cycleOf f x ≠ 1",
"tactic": "simpa using hx"
},
{
"state_after": "case pos.intro.intro.zero.refine'_2\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : x = y\n⊢ ↑(f ^ card (support (cycleOf f x))) x = y",
"state_before": "case pos.intro.intro.zero.refine'_2\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.zero) x = y\n⊢ ↑(f ^ card (support (cycleOf f x))) x = y",
"tactic": "simp only [Nat.zero_eq, pow_zero, coe_one, id_eq] at hk'"
},
{
"state_after": "case pos.intro.intro.zero.refine'_2\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx : α\nf : Perm α\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nh : SameCycle f x x\n⊢ ↑(f ^ card (support (cycleOf f x))) x = x",
"state_before": "case pos.intro.intro.zero.refine'_2\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nhk' : x = y\n⊢ ↑(f ^ card (support (cycleOf f x))) x = y",
"tactic": "subst hk'"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.zero.refine'_2\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx : α\nf : Perm α\nhx : x ∈ support f\nhk : Nat.zero < card (support (cycleOf f x))\nh : SameCycle f x x\n⊢ ↑(f ^ card (support (cycleOf f x))) x = x",
"tactic": "rw [← (isCycle_cycleOf _ <| mem_support.1 hx).orderOf, ← cycleOf_pow_apply_self,\n pow_orderOf_eq_one, one_apply]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.succ\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nk : ℕ\nhk : Nat.succ k < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.succ k) x = y\n⊢ ∃ i x_1 x_2, ↑(f ^ i) x = y",
"tactic": "exact ⟨k + 1, by simp, Nat.le_succ_of_le hk.le, hk'⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : x ∈ support f\nk : ℕ\nhk : Nat.succ k < card (support (cycleOf f x))\nhk' : ↑(f ^ Nat.succ k) x = y\n⊢ 0 < k + 1",
"tactic": "simp"
},
{
"state_after": "case neg\nι : Type ?u.2618181\nα : Type u_1\nβ : Type ?u.2618187\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx y : α\nf : Perm α\nh : SameCycle f x y\nhx : ¬x ∈ support f\n⊢ ↑(f ^ 1) x = y",
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Mathlib/Topology/Inseparable.lean | specializes_iff_nhds | [] | [
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Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | Complex.range_sin | [] | [
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Mathlib/CategoryTheory/Limits/Opposites.lean | CategoryTheory.Limits.PullbackCone.op_inr | [
{
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Mathlib/Data/Real/NNReal.lean | NNReal.exists | [] | [
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Mathlib/Logic/Basic.lean | exists_apply_eq_apply' | [] | [
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