file_path stringlengths 11 79 | full_name stringlengths 2 100 | traced_tactics list | end list | commit stringclasses 4
values | url stringclasses 4
values | start list |
|---|---|---|---|---|---|---|
Mathlib/Data/PEquiv.lean | PEquiv.refl_trans | [
{
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"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\n⊢ PEquiv.trans (PEquiv.refl α) f = f",
"tactic": "ext"
},
{
... | [
169,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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1
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Mathlib/SetTheory/Ordinal/Basic.lean | Cardinal.univ_umax | [] | [
1499,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/CategoryTheory/Preadditive/Schur.lean | CategoryTheory.finrank_hom_simple_simple_eq_one_iff | [
{
"state_after": "case mp\nC : Type u_3\ninst✝⁹ : Category C\ninst✝⁸ : Preadditive C\n𝕜 : Type u_1\ninst✝⁷ : Field 𝕜\ninst✝⁶ : IsAlgClosed 𝕜\ninst✝⁵ : Linear 𝕜 C\ninst✝⁴ : HasKernels C\nX Y : C\ninst✝³ : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝² : FiniteDimensional 𝕜 (X ⟶ Y)\ninst✝¹ : Simple X\ninst✝ : Simple Y... | [
199,
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Mathlib/Algebra/Homology/HomologicalComplex.lean | CochainComplex.of_d_ne | [
{
"state_after": "ι : Type ?u.219028\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nα : Type u_1\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX : α → V\nd : (n : α) → X n ⟶ X (n + 1)\nsq : ∀ (n : α), d n ≫ d (n + 1) = 0\ni j : α\nh : i + 1 ≠ j\n⊢ (if h : i + 1 = j ... | [
912,
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910,
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Mathlib/Order/UpperLower/Basic.lean | upperClosure_union | [] | [
1434,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1433,
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Mathlib/GroupTheory/Solvable.lean | Equiv.Perm.fin_5_not_solvable | [
{
"state_after": "G : Type ?u.88563\nG' : Type ?u.88566\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nx : Perm (Fin 5) :=\n { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4],\n left_inv := (_ : Function.LeftInverse ![2, 0, 1, 3, 4] ![1, 2, 0, 3, 4]),\n right_inv := (_ : Function.RightInverse ![... | [
225,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
216,
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src/lean/Init/Data/AC.lean | Lean.Data.AC.Context.unwrap_isNeutral | [
{
"state_after": "α : Sort u_1\nctx : Context α\nx : Nat\n⊢ (match (var ctx x).neutral with\n | some val => true\n | none => false) =\n true →\n IsNeutral ctx.op (var ctx x).value",
"state_before": "α : Sort u_1\nctx : Context α\nx : Nat\n⊢ ContextInformation.isNeutral ctx x = true → IsNeu... | [
276,
33
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
269,
1
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Mathlib/FieldTheory/IsAlgClosed/Basic.lean | IsAlgClosed.exists_aeval_eq_zero_of_injective | [] | [
126,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
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Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean | CategoryTheory.exact_f_d | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\n⊢ f ≫ d f = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : En... | [
246,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
244,
1
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Mathlib/Topology/UnitInterval.lean | unitInterval.add_pos | [] | [
156,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
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Mathlib/Data/Set/Intervals/Basic.lean | Set.Iic_inj | [] | [
973,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
972,
1
] |
Mathlib/Data/Real/CauSeqCompletion.lean | CauSeq.eq_lim_of_const_equiv | [] | [
351,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
350,
1
] |
Mathlib/Data/Finset/Fold.lean | Finset.fold_op_rel_iff_or | [
{
"state_after": "case empty\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.60299\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\nr : β → β → Prop\nhr : ∀ {x y z : β}, r x (op y z) ↔ r x y ∨ r x z\nc : β\n⊢ r c (fold op b f ∅) ↔ r c b ∨ ∃ x, x ∈ ∅ ∧ r c (f x)\... | [
198,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
185,
1
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Mathlib/Algebra/Homology/HomotopyCategory.lean | HomotopyCategory.homologyFunctor_map_factors | [] | [
174,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
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Mathlib/Init/Algebra/Classes.lean | symm | [] | [
267,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
] |
Mathlib/Data/PFun.lean | PFun.preimage_mono | [] | [
443,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
442,
1
] |
Mathlib/Analysis/Seminorm.lean | Seminorm.coe_add | [] | [
192,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.norm_eq_zero_of_empty | [] | [
816,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
815,
1
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Mathlib/Topology/Maps.lean | IsOpenMap.range_mem_nhds | [] | [
364,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
363,
1
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Mathlib/Combinatorics/Configuration.lean | Configuration.ProjectivePlane.Dual.order | [] | [
443,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
442,
1
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Mathlib/LinearAlgebra/Multilinear/Basic.lean | MultilinearMap.mk_coe | [
{
"state_after": "case H\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMo... | [
166,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
1
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Mathlib/Algebra/Algebra/Prod.lean | AlgHom.fst_prod | [
{
"state_after": "case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B\ng : A →ₐ[R] C\nx✝ : A\n⊢ ↑(comp (fst R B C) (prod f g)) x✝ = ↑... | [
90,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
90,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | HomogeneousLocalization.one_eq | [] | [
397,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
396,
1
] |
Mathlib/Data/Matrix/Block.lean | Matrix.blockDiagonal'_mul | [
{
"state_after": "case a.mk.h.mk\nl : Type ?u.225429\nm : Type ?u.225432\nn : Type ?u.225435\no : Type u_3\np : Type ?u.225441\nq : Type ?u.225444\nm' : o → Type u_4\nn' : o → Type u_2\np' : o → Type u_5\nR : Type ?u.225462\nS : Type ?u.225465\nα : Type u_1\nβ : Type ?u.225471\ninst✝³ : DecidableEq o\ninst✝² : ... | [
764,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
755,
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Mathlib/ModelTheory/Order.lean | FirstOrder.Language.Term.realize_lt | [
{
"state_after": "no goals",
"state_before": "L : Language\nα : Type w\nM : Type w'\nn : ℕ\ninst✝³ : IsOrdered L\ninst✝² : Structure L M\ninst✝¹ : Preorder M\ninst✝ : OrderedStructure L M\nt₁ t₂ : Term L (α ⊕ Fin n)\nv : α → M\nxs : Fin n → M\n⊢ BoundedFormula.Realize (lt t₁ t₂) v xs ↔ realize (Sum.elim v x... | [
225,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
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Mathlib/Data/Multiset/FinsetOps.lean | Multiset.ndinter_eq_inter | [] | [
278,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
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Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Term.realize_mem | [
{
"state_after": "case var\nL : Language\nM : Type w\nN : Type ?u.14398\nP : Type ?u.14401\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nα : Type u_1\nxs : α → M\nh : ∀ (a : α), xs a ∈ S\na : α\n⊢ realize xs (var a) ∈ S\n\ncase func\nL : Language\nM : Type w\nN : ... | [
148,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/FieldTheory/Subfield.lean | Subfield.inv_mem | [] | [
280,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
279,
11
] |
Mathlib/CategoryTheory/Sites/Plus.lean | CategoryTheory.GrothendieckTopology.plus_hom_ext | [
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη γ : plusObj J P ⟶ Q\nhQ : P... | [
339,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
332,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.sup_mul_eq_of_coprime_left | [
{
"state_after": "R : Type u\nι : Type ?u.266258\ninst✝ : CommSemiring R\nI J K L : Ideal R\nh : 1 ∈ I ⊔ J\ni : R\nhi : i ∈ I ⊔ K\n⊢ i ∈ I ⊔ J * K",
"state_before": "R : Type u\nι : Type ?u.266258\ninst✝ : CommSemiring R\nI J K L : Ideal R\nh : I ⊔ J = ⊤\ni : R\nhi : i ∈ I ⊔ K\n⊢ i ∈ I ⊔ J * K",
"tactic... | [
685,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
680,
1
] |
Mathlib/CategoryTheory/Category/Preorder.lean | CategoryTheory.homOfLE_leOfHom | [] | [
107,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.IsPullback.of_hasBinaryProduct' | [] | [
269,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
] |
Mathlib/Topology/SubsetProperties.lean | Embedding.isCompact_iff_isCompact_image | [] | [
891,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
889,
1
] |
Mathlib/Topology/Algebra/Monoid.lean | continuous_multiset_prod | [
{
"state_after": "case mk\nι : Type u_1\nα : Type ?u.421560\nX : Type u_2\nM : Type u_3\nN : Type ?u.421569\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\ns : Multiset ι\nl : List ι\n⊢ (∀ (i : ι), i ∈ Quot.mk Setoid.r l → Continuous (f i... | [
750,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
747,
1
] |
Mathlib/Data/List/Zip.lean | List.prod_mul_prod_eq_prod_zipWith_of_length_eq | [
{
"state_after": "α : Type u\nβ : Type ?u.290919\nγ : Type ?u.290922\nδ : Type ?u.290925\nε : Type ?u.290928\ninst✝ : CommMonoid α\nL L' : List α\nh : length L = length L'\n⊢ prod (zipWith (fun x x_1 => x * x_1) L L') * prod (drop (length L') L) * prod (drop (length L) L') =\n prod (zipWith (fun x x_1 => x *... | [
548,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
545,
1
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Mathlib/Topology/SubsetProperties.lean | exists_compact_subset | [
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.147163\nπ : ι → Type ?u.147168\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace α\nx : α\nU : Set α\nhU : IsOpen U\nhx : x ∈ U\nK : Set α\nh1K : K ∈ 𝓝 x\nh2K : K ⊆ U\nh3K : IsCompact... | [
1156,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1153,
1
] |
Mathlib/LinearAlgebra/Dual.lean | LinearMap.dualMap_surjective_iff | [
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nf : V₁ →ₗ[K] V₂\n⊢ Function.Surjective ↑(dualMap f) ↔ Function.Injective ↑f",
"tactic": "rw [← LinearMap... | [
1372,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1369,
1
] |
Mathlib/RingTheory/Localization/Basic.lean | IsLocalization.isLocalization_of_algEquiv | [
{
"state_after": "case map_units'\nR : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_2\ninst✝² : CommSemiring P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization M S\nh : S ≃ₐ[R] P\n⊢ ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap R P) ↑y)\... | [
764,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
750,
1
] |
Mathlib/Algebra/DirectLimit.lean | Module.DirectLimit.of_f | [] | [
120,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
119,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | Subring.coe_comap | [] | [
575,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
574,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean | mul_ball_one | [
{
"state_after": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ s * ball 1 δ = ⋃ (x : E) (_ : x ∈ s), ball x δ",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ s * ball 1 δ = thickening δ s",
"tactic": "rw [thickening_eq_biU... | [
212,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
207,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.IsPullback.of_is_product | [
{
"state_after": "case mk.right\nC : Type u₁\ninst✝ : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nc : BinaryFan X Y\nh : IsLimit c\nt : IsTerminal Z\n⊢ BinaryFan.snd c = 𝟙 c.pt ≫ BinaryFan.snd c",
"state_before": "case mk.right\nC : Type u₁\ninst✝ : Category C\nP X Y Z : C\nfst... | [
256,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
247,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean | Equiv.swap_mul_eq_iff | [
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\ni j : α\nσ : Perm α\nh : swap i j * σ = σ\nswap_id : swap i j = 1\n⊢ i = j",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\ni j : α\nσ : Perm α\nh : swap i j * σ = σ\n⊢ i = j",
"tactic": "have swap_id : swap i j = 1 := mul... | [
578,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
572,
1
] |
Mathlib/Data/Stream/Init.lean | Stream'.interleave_tail_tail | [
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\n⊢ tail s₁ ⋈ tail s₂ = tail (tail (head s₁ :: head s₂ :: (tail s₁ ⋈ tail s₂)))",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\n⊢ tail s₁ ⋈ tail s₂ = tail (tail (s₁ ⋈ s₂))",
"tactic": "rw [interleave_eq s₁ s... | [
428,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
427,
1
] |
Mathlib/Algebra/AddTorsor.lean | vsub_left_cancel | [
{
"state_after": "no goals",
"state_before": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np1 p2 p : P\nh : p1 -ᵥ p = p2 -ᵥ p\n⊢ p1 = p2",
"tactic": "rwa [← sub_eq_zero, vsub_sub_vsub_cancel_right, vsub_eq_zero_iff_eq] at h"
}
] | [
211,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean | RingHom.map_adjugate | [
{
"state_after": "case a.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : CommRing α\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nM : Matrix n n R\ni k : n\n⊢ ↑(RingHom.mapMatrix f) (adju... | [
368,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
] |
Mathlib/Data/Nat/Size.lean | Nat.size_bit1 | [] | [
98,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
] |
Mathlib/Logic/Basic.lean | exists_unique_eq | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.16326\nα : Sort u_1\nκ : ι → Sort ?u.16328\np q : α → Prop\na' : α\n⊢ ∃! a, a = a'",
"tactic": "simp only [eq_comm, ExistsUnique, and_self, forall_eq', exists_eq']"
}
] | [
766,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
765,
9
] |
Mathlib/Algebra/Hom/Ring.lean | NonUnitalRingHom.one_def | [] | [
312,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
1
] |
Mathlib/Algebra/Parity.lean | IsSquare.div | [
{
"state_after": "F : Type ?u.49647\nα : Type u_1\nβ : Type ?u.49653\nR : Type ?u.49656\ninst✝ : DivisionCommMonoid α\na b : α\nha : IsSquare a\nhb : IsSquare b\n⊢ IsSquare (a * b⁻¹)",
"state_before": "F : Type ?u.49647\nα : Type u_1\nβ : Type ?u.49653\nR : Type ?u.49656\ninst✝ : DivisionCommMonoid α\na b :... | [
211,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
] |
Mathlib/Algebra/Order/WithZero.lean | Units.zero_lt | [] | [
172,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
1
] |
Mathlib/Algebra/Associated.lean | Prime.irreducible | [
{
"state_after": "α : Type u_1\nβ : Type ?u.97319\nγ : Type ?u.97322\nδ : Type ?u.97325\ninst✝ : CancelCommMonoidWithZero α\na✝ p : α\nhp : ¬p = 0 ∧ ¬IsUnit p ∧ ∀ (a b : α), p ∣ a * b → p ∣ a ∨ p ∣ b\na b : α\nhab : p = a * b\nx✝ : a * b ∣ a\nx : α\nhx : a = a * b * x\nh : a = 0\n⊢ False",
"state_before": "... | [
345,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
11
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | CategoryTheory.Functor.biproductComparison'_comp_biproductComparison | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nJ : Type w₁\nF : C ⥤ D\nf : J → C\ninst✝² : HasBiproduct f\ninst✝¹ : HasBiproduct (F.obj ∘ f)\ninst✝ : PreservesZeroMorphisms F\n⊢ bipro... | [
265,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
] |
Mathlib/GroupTheory/Complement.lean | Subgroup.transferTransversal_apply | [] | [
631,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
629,
1
] |
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.sInf_toSubfield | [
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set (IntermediateField F E)\n⊢ ↑(toSubfield (sInf S)) = ↑(sInf (toSubfield '' S))",
"tactic": "simp [Set.sUnion_image]"
}
] | [
159,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Topology/Inseparable.lean | Specializes.map_of_continuousAt | [] | [
189,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean | Ordinal.enum_le_enum' | [] | [
1162,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1160,
1
] |
Mathlib/LinearAlgebra/Quotient.lean | Submodule.linearMap_qext | [] | [
344,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean | linearIndependent_singleton | [] | [
1129,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1127,
1
] |
Mathlib/LinearAlgebra/Pi.lean | Submodule.pi_mono | [] | [
297,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
296,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean | lipschitzWith_max | [] | [
427,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
425,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal | [] | [
297,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
295,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_and' | [] | [
1330,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1328,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.lift_root | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : CommRing S\ni : R →+* S\na : S\nh : eval₂ i a f = 0\n⊢ ↑(lift i a h) (root f) = a",
"tactic": "rw [root, lift_mk, eval₂_X]"
}
] | [
290,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
290,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | norm_sub_norm_le' | [] | [
566,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
565,
1
] |
Mathlib/Algebra/Order/AbsoluteValue.lean | AbsoluteValue.coe_mk | [] | [
77,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean | Cubic.ne_zero | [
{
"state_after": "R : Type u_1\nS : Type ?u.553084\nF : Type ?u.553087\nK : Type ?u.553090\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nh0 : toPoly P = 0\n⊢ P.a = 0 ∧ P.b = 0 ∧ P.c = 0 ∧ P.d = 0",
"state_before": "R : Type u_1\nS : Type ?u.553084\nF : Type ?u.553087\nK : Type ?u.553090\nP Q ... | [
184,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
9
] |
Mathlib/Analysis/Fourier/FourierTransform.lean | Fourier.fourierIntegral_def | [] | [
207,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
] |
Mathlib/Algebra/Algebra/Pi.lean | Pi.constRingHom_eq_algebraMap | [] | [
90,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | CategoryTheory.Limits.IsInitial.epi_to | [
{
"state_after": "C : Type u₁\ninst✝ : Category C\nX Y : C\nt : IsInitial X\nf : Y ⟶ X\nthis : IsSplitEpi f\n⊢ Epi f",
"state_before": "C : Type u₁\ninst✝ : Category C\nX Y : C\nt : IsInitial X\nf : Y ⟶ X\n⊢ Epi f",
"tactic": "haveI := t.isSplitEpi_to f"
},
{
"state_after": "no goals",
"stat... | [
189,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean | isConnected_Ioo | [] | [
486,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
485,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | GromovHausdorff.HD_candidatesBDist_le | [
{
"state_after": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nx : X\n⊢ (⨅ (y : Y), ↑(candidatesBDist X Y) (inl x, inr y))... | [
346,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
326,
1
] |
Mathlib/Algebra/Hom/Aut.lean | MulAut.inv_def | [] | [
88,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Std/Data/List/Lemmas.lean | List.eraseP_subset | [] | [
987,
81
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
987,
1
] |
Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | tendsto_pow_div_pow_atTop_atTop | [
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\nhpq : q < p\n⊢ Tendsto (fun x => x ^ (↑p - ↑q)) atTop atTop",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\nhpq : q < p\n⊢ Tendsto (fun x => x ^ p / x ^ q) atTop atTop",
"tactic": "rw [tendsto_congr' pow_d... | [
69,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean | Matrix.Commute.zpow_self | [] | [
233,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
232,
1
] |
Mathlib/MeasureTheory/PiSystem.lean | isPiSystem_generatePiSystem | [] | [
232,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snormEssSup_lt_top_of_ae_nnnorm_bound | [] | [
433,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
431,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean | Nat.mono_cast | [
{
"state_after": "α : Type u_1\nβ : Type ?u.8929\ninst✝ : OrderedSemiring α\nn : ℕ\n⊢ ↑n ≤ ↑n + 1",
"state_before": "α : Type u_1\nβ : Type ?u.8929\ninst✝ : OrderedSemiring α\nn : ℕ\n⊢ ↑n ≤ ↑(n + 1)",
"tactic": "rw [Nat.cast_succ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_... | [
96,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean | Matrix.TransvectionStruct.toMatrix_mk | [] | [
179,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Real.dist_0_eq_abs | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.194421\nι : Type ?u.194424\ninst✝ : PseudoMetricSpace α\nx : ℝ\n⊢ dist x 0 = abs x",
"tactic": "simp [Real.dist_eq]"
}
] | [
1363,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1363,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean | Set.inv_singleton | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.6817\nα : Type u_1\nβ : Type ?u.6823\nγ : Type ?u.6826\ninst✝ : InvolutiveInv α\ns t : Set α\na✝ a : α\n⊢ {a}⁻¹ = {a⁻¹}",
"tactic": "rw [← image_inv, image_singleton]"
}
] | [
280,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
280,
1
] |
Mathlib/Data/Vector.lean | Vector.toList_take | [
{
"state_after": "case mk\nα : Type u\nβ : Type v\nφ : Type w\nn✝ n m : ℕ\nval✝ : List α\nproperty✝ : List.length val✝ = m\n⊢ toList (take n { val := val✝, property := property✝ }) = List.take n (toList { val := val✝, property := property✝ })",
"state_before": "α : Type u\nβ : Type v\nφ : Type w\nn✝ n m : ℕ... | [
258,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Data/Nat/Order/Lemmas.lean | Nat.succ_div | [
{
"state_after": "no goals",
"state_before": "a✝ b m n k a : ℕ\n⊢ (a + 1) / 0 = a / 0 + if 0 ∣ a + 1 then 1 else 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "a b m n k : ℕ\n⊢ (0 + 1) / 1 = 0 / 1 + if 1 ∣ 0 + 1 then 1 else 0",
"tactic": "simp"
},
{
"state_... | [
156,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Algebra/Algebra/Hom.lean | AlgHom.toLinearMap_id | [] | [
387,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
386,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean | HasFDerivAtFilter.prod | [] | [
73,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/Order/Iterate.lean | StrictMono.strictMono_iterate_of_lt_map | [
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nf : α → α\nx : α\nhf : StrictMono f\nhx : x < f x\nn : ℕ\n⊢ (f^[n]) x < (f^[n]) (f x)",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nf : α → α\nx : α\nhf : StrictMono f\nhx : x < f x\nn : ℕ\n⊢ (f^[n]) x < (f^[n + 1]) x",
"tactic": "rw [iterate_su... | [
259,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
255,
1
] |
Mathlib/Topology/Basic.lean | Function.Surjective.denseRange | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.174230\nβ : Type u_2\nγ : Type ?u.174236\nδ : Type ?u.174239\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type u_1\nι : Type ?u.174254\nf : κ → β\ng : β → γ\nhf : Surjective f\nx : β\n⊢ x ∈ closure (range... | [
1799,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1798,
1
] |
Mathlib/Data/Nat/Pairing.lean | Nat.pair_unpair' | [
{
"state_after": "no goals",
"state_before": "n a b : ℕ\nH : unpair n = (a, b)\n⊢ pair a b = n",
"tactic": "simpa [H] using pair_unpair n"
}
] | [
64,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.find_eq_iff | [
{
"state_after": "case mp\nm n k : ℕ\np q : ℕ → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\nh : ∃ n, p n\n⊢ Nat.find h = m → p m ∧ ∀ (n : ℕ), n < m → ¬p n\n\ncase mpr\nm n k : ℕ\np q : ℕ → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\nh : ∃ n, p n\n⊢ (p m ∧ ∀ (n : ℕ), n < m → ¬p n) → Nat... | [
824,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
819,
1
] |
Mathlib/Algebra/Quandle.lean | Rack.self_invAct_act_eq | [
{
"state_after": "R : Type u_1\ninst✝ : Rack R\nx y : R\nh : (op x ◃ op x) ◃⁻¹ op y = op x ◃⁻¹ op y\n⊢ (x ◃⁻¹ x) ◃ y = x ◃ y",
"state_before": "R : Type u_1\ninst✝ : Rack R\nx y : R\n⊢ (x ◃⁻¹ x) ◃ y = x ◃ y",
"tactic": "have h := @self_act_invAct_eq _ _ (op x) (op y)"
},
{
"state_after": "no goa... | [
307,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean | HasStrictDerivAt.smul | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : F... | [
76,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
8
] |
Mathlib/Algebra/Regular/Basic.lean | not_isRightRegular_zero | [] | [
265,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
264,
1
] |
Mathlib/Algebra/IndicatorFunction.lean | Set.prod_mulIndicator_subset | [] | [
617,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
615,
1
] |
Mathlib/Data/Complex/Exponential.lean | Real.tan_div_sqrt_one_add_tan_sq | [
{
"state_after": "no goals",
"state_before": "x✝ y x : ℝ\nhx : 0 < cos x\n⊢ tan x / sqrt (1 + tan x ^ 2) = sin x",
"tactic": "rw [← tan_mul_cos hx.ne', ← inv_sqrt_one_add_tan_sq hx, div_eq_mul_inv]"
}
] | [
1329,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1327,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean | fderivWithin_smul | [] | [
221,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean | Finset.card_biUnion_le | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.862077\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq β\ns : Finset α\nt : α → Finset β\nthis : DecidableEq α\n⊢ card (Finset.biUnion ∅ t) ≤ ∑ a in ∅, card (t a)",
"tactic": "simp"
},
{
"... | [
1880,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1873,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean | Subsemigroup.comap_map_comap | [] | [
325,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.closedComplemented_bot | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : Ring R\nM : Type u_2\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nM₂ : Type ?u.2064642\ninst✝² : TopologicalSpace M₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\nx : { x // x ∈ ⊥ }\n⊢ ↑0 ↑x = x",
"tact... | [
2584,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2583,
1
] |
Mathlib/Logic/Equiv/Set.lean | Set.preimage_equiv_eq_image_symm | [] | [
61,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
1
] |
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