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/Topology/Basic.lean | IsOpen.inter | [] | [
126,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | Real.cos_arcsin | [
{
"state_after": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "x : ℝ\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "by_cases hx₁ : -1 ≤ x"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ ... | [
318,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
306,
1
] |
Mathlib/Algebra/Homology/ImageToKernel.lean | imageToKernel_arrow | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.5291\nV : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ninst✝¹ : HasImage f\ng : B ⟶ C\ninst✝ : HasKernel g\nw : f ≫ g = 0\n⊢ imageToKernel f g w ≫ Subobject.arrow (kernelSubobject g) = Subobject.arrow (imageSubobjec... | [
67,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean | UniqueDiffWithinAt.mono_nhds | [
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.112500\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.112590\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : S... | [
285,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
281,
1
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Mathlib/Data/Set/Intervals/Monotone.lean | Antitone.Ici | [] | [
49,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
48,
11
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Mathlib/GroupTheory/Abelianization.lean | rank_commutator_le_card | [
{
"state_after": "G : Type u\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\n⊢ Group.rank { x // x ∈ Subgroup.closure (commutatorSet G) } ≤ Nat.card ↑(commutatorSet G)",
"state_before": "G : Type u\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\n⊢ Group.rank { x // x ∈ commutator G } ≤ Nat.card ... | [
69,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
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Mathlib/Data/Real/ENNReal.lean | ENNReal.toNNReal_inv | [
{
"state_after": "case top\nα : Type ?u.836153\nβ : Type ?u.836156\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ ENNReal.toNNReal ⊤⁻¹ = (ENNReal.toNNReal ⊤)⁻¹\n\ncase coe\nα : Type ?u.836153\nβ : Type ?u.836156\na✝ b c d : ℝ≥0∞\nr p q a : ℝ≥0\n⊢ ENNReal.toNNReal (↑a)⁻¹ = (ENNReal.toNNReal ↑a)⁻¹",
"state_before": "α : Typ... | [
2314,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2311,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.pred_eq_of_eq_succ | [
{
"state_after": "no goals",
"state_before": "m✝ n✝ k m n : ℕ\nH : m = succ n\n⊢ pred m = n",
"tactic": "simp [H]"
}
] | [
311,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
1
] |
Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | SL_reduction_mod_hom_val | [] | [
45,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
43,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | ZFSet.sUnion_lem | [
{
"state_after": "α β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nhb : PSet.Equiv (A a) (B b)\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"state_before": "... | [
1029,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1016,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean | Nat.factorial_mul_ascFactorial | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ n ! * ascFactorial n 0 = (n + 0)!",
"tactic": "rw [ascFactorial, add_zero, mul_one]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ n ! * ascFactorial n (k + 1) = (n + (k + 1))!",
"tactic": "rw [ascFactorial_succ, mul_le... | [
267,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
263,
1
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Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearMap.coe_comp | [] | [
783,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
781,
1
] |
Mathlib/Data/Set/Prod.lean | Set.prod_inter | [
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.13035\nδ : Type ?u.13038\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ (t₁ ∩ t₂) ↔ (x, y) ∈ s ×ˢ t₁ ∩ s ×ˢ t₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.13035\nδ : Type ?u.13038\ns s₁ s₂ : S... | [
168,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
166,
1
] |
Mathlib/Algebra/Module/Projective.lean | Module.Projective.of_lifting_property | [] | [
202,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Mathlib/ModelTheory/Satisfiability.lean | FirstOrder.Language.completeTheory.isMaximal | [] | [
509,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
508,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Real.sin_eq_sqrt_one_sub_cos_sq | [
{
"state_after": "no goals",
"state_before": "x : ℝ\nhl : 0 ≤ x\nhu : x ≤ π\n⊢ sin x = sqrt (1 - cos x ^ 2)",
"tactic": "rw [← abs_sin_eq_sqrt_one_sub_cos_sq, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi hl hu)]"
}
] | [
489,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
487,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean | UniqueFactorizationMonoid.normalizedFactors_prod_of_prime | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero α\ninst✝⁴ : DecidableEq α\ninst✝³ : NormalizationMonoid α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : Nontrivial α\ninst✝ : Unique αˣ\nm : Multiset α\nh : ∀ (p : α), p ∈ m → Prime p\n⊢ normalizedFactors (Multis... | [
767,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
763,
1
] |
Mathlib/Topology/Covering.lean | IsEvenlyCovered.continuousAt | [] | [
69,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
11
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigO_zero | [] | [
1208,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1207,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.copy_eq | [] | [
102,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
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Mathlib/Algebra/Periodic.lean | Function.Antiperiodic.smul | [
{
"state_after": "no goals",
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"tactic": "simp_all"
}
] | [
455,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Group.subset_conjugatesOfSet | [] | [
2419,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2418,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.power_one | [] | [
514,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
513,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean | Matrix.IsDiag.fromBlocks_of_isSymm | [
{
"state_after": "α : Type u_1\nβ : Type ?u.26056\nR : Type ?u.26059\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\nh : IsSymm (Matrix.fromBlocks A 0 C D)\nha : IsDiag A\nhd : IsDiag D\n⊢ IsDiag (Matrix.fromBlocks A 0 0ᵀ D)",
"state_before": "α : Type u_1\... | [
191,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean | Zsqrtd.le_antisymm | [
{
"state_after": "no goals",
"state_before": "d : ℕ\ndnsq : Nonsquare d\na b : ℤ√↑d\nab : a ≤ b\nba : b ≤ a\n⊢ Nonneg (-(a - b))",
"tactic": "rwa [neg_sub]"
}
] | [
947,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
946,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | CategoryTheory.Limits.diagonalObjPullbackFstIso_inv_snd_fst | [
{
"state_after": "C : Type u_2\ninst✝⁵ : Category C\nX✝ Y✝ Z✝ : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf✝ : X✝ ⟶ T\ng✝ : Y✝ ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f✝ g✝\ninst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)\ninst✝ :\n HasPullback (diagonal i)\n (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S)... | [
346,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
] |
Mathlib/CategoryTheory/Sites/Sheafification.lean | CategoryTheory.Meq.refine_apply | [] | [
85,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.span_pair_mul_span_pair | [
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.306186\ninst✝ : CommSemiring R\nI J K L : Ideal R\nw x y z : R\n⊢ span {w, x} * span {y, z} = span {w * y, w * z, x * y, x * z}",
"tactic": "simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc]"
}
] | [
828,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
826,
1
] |
Mathlib/Data/Part.lean | Part.none_toOption | [] | [
249,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean | pow_eq_zero_iff' | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.25956\nS : Type ?u.25959\nM : Type u_1\ninst✝² : MonoidWithZero M\ninst✝¹ : NoZeroDivisors M\ninst✝ : Nontrivial M\na : M\nn : ℕ\n⊢ a ^ n = 0 ↔ a = 0 ∧ n ≠ 0",
"tactic": "cases (zero_le n).eq_or_gt <;> simp [*, ne_of_gt]"
}
] | [
73,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
72,
1
] |
Mathlib/Algebra/Algebra/Bilinear.lean | LinearMap.mulRight_mul | [
{
"state_after": "case h\nR : Type u_2\nA : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : SMulCommClass R A A\ninst✝ : IsScalarTower R A A\na b x✝ : A\n⊢ ↑(mulRight R (a * b)) x✝ = ↑(comp (mulRight R b) (mulRight R a)) x✝",
"state_before": "R : Type u_2\nA : ... | [
152,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/Analysis/Convex/Combination.lean | Finset.inf_le_centerMass | [
{
"state_after": "R : Type ?u.76609\nE : Type ?u.76612\nF : Type ?u.76615\nι : Type ?u.76618\nι' : Type ?u.76621\nα : Type ?u.76624\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R ... | [
147,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Data/Set/Image.lean | Set.eq_preimage_iff_image_eq | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nf✝ f : α → β\nhf : Bijective f\ns : Set α\nt : Set β\n⊢ s = f ⁻¹' t ↔ f '' s = t",
"tactic": "rw [← image_eq_image hf.1, hf.2.image_preimage]"
}
] | [
1567,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1566,
1
] |
Mathlib/Algebra/Algebra/Tower.lean | IsScalarTower.of_algebraMap_eq' | [] | [
94,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
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] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | ContinuousLinearMap.map_add₂ | [
{
"state_after": "no goals",
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297,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
296,
1
] |
Mathlib/Order/Filter/Archimedean.lean | Rat.comap_cast_atTop | [
{
"state_after": "no goals",
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"tactic": "simpa"
}
] | [
78,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean | is_const_of_deriv_eq_zero | [
{
"state_after": "case h\nE : Type ?u.290841\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.290937\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_1\nG : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\... | [
696,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
694,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean | EuclideanDomain.gcd_val | [
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ (fun b =>\n if a0 : a = 0 then b\n else\n let_fun x := (_ : EuclideanDomain.r (b % a) a);\n gcd (b % a) a)\n b =\n gcd (b % a) a",
"state_before": "R : Type u\ninst✝¹ : Euclide... | [
146,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
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Mathlib/MeasureTheory/Integral/Average.lean | MeasureTheory.set_average_const | [
{
"state_after": "no goals",
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207,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean | TrivSqZeroExt.inl_pow | [
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝³ : Monoid R\ninst✝² : AddMonoid M\ninst✝¹ : DistribMulAction R M\ninst✝ : DistribMulAction Rᵐᵒᵖ M\nr : R\nn : ℕ\n⊢ snd (inl r ^ n) = snd (inl (r ^ n))",
"tactic": "simp [snd_pow_eq_sum]"
}
] | [
640,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
638,
1
] |
src/lean/Init/SimpLemmas.lean | eq_false' | [] | [
20,
60
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
20,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean | MeasureTheory.IsFundamentalDomain.set_integral_eq_tsum | [
{
"state_after": "no goals",
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439,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
433,
1
] |
Std/Data/Array/Lemmas.lean | Array.fin_cast_val | [
{
"state_after": "case refl\nn : Nat\ni : Fin n\n⊢ (_ : n = n) ▸ i = { val := i.val, isLt := (_ : i.val < n) }",
"state_before": "n n' : Nat\ne : n = n'\ni : Fin n\n⊢ e ▸ i = { val := i.val, isLt := (_ : i.val < n') }",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "cas... | [
113,
98
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
113,
9
] |
Mathlib/Topology/Sheaves/Functors.lean | TopCat.Presheaf.SheafConditionPairwiseIntersections.pushforward_sheaf_of_sheaf | [
{
"state_after": "case h.e'_1.h.e'_5\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\n⊢ Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefuncto... | [
73,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
] |
Mathlib/Logic/Basic.lean | imp_iff_or_not | [] | [
375,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
375,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | ContinuousLinearEquiv.differentiableOn | [] | [
94,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | deriv_csinh | [] | [
288,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
286,
1
] |
Mathlib/RingTheory/MatrixAlgebra.lean | matrixEquivTensor_apply | [] | [
166,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.smul_finset_inter | [] | [
1992,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1991,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | CategoryTheory.Limits.kernelIsoOfEq_inv_comp_ι | [
{
"state_after": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nf : X ⟶ Y\ninst✝¹ inst✝ : HasKernel f\n⊢ (kernelIsoOfEq (_ : f = f)).inv ≫ kernel.ι f = kernel.ι f",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphi... | [
365,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
363,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean | GroupSeminorm.mul_bddBelow_range_add | [
{
"state_after": "case intro\nι : Type ?u.89294\nR : Type ?u.89297\nR' : Type ?u.89300\nE : Type u_1\nF : Type ?u.89306\nG : Type ?u.89309\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝¹ y : E\np q : GroupSeminorm E\nx✝ x : E\n⊢ 0 ≤ (fun y => ↑p y + ↑q (x✝ / y)) x",
"state_before": ... | [
402,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
397,
1
] |
Mathlib/Order/Lattice.lean | le_inf | [] | [
408,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
407,
1
] |
Mathlib/Algebra/DirectSum/Ring.lean | DirectSum.toSemiring_coe_addMonoidHom | [] | [
640,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
638,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean | CategoryTheory.Limits.biproduct.map_eq | [
{
"state_after": "case w.w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nJ : Type\ninst✝² : Fintype J\nf✝ : J → C\ninst✝¹ : HasBiproduct f✝\ninst✝ : HasFiniteBiproducts C\nf g : J → C\nh : (j : J) → f j ⟶ g j\nj✝¹ j✝ : J\n⊢ (ι (fun b => f b) j✝¹ ≫ map h) ≫ π (fun b => g b) j✝ =\n (ι (fun b => f b... | [
238,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.max'_image | [
{
"state_after": "F : Type ?u.364381\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.364390\nι : Type ?u.364393\nκ : Type ?u.364396\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\ny : β\nhy : y ∈ im... | [
1503,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1498,
1
] |
Mathlib/Data/Nat/Prime.lean | Nat.Prime.dvd_iff_eq | [
{
"state_after": "p a : ℕ\nhp : Prime p\na1 : a ≠ 1\n⊢ a ∣ p → p = a",
"state_before": "p a : ℕ\nhp : Prime p\na1 : a ≠ 1\n⊢ a ∣ p ↔ p = a",
"tactic": "refine'\n ⟨_, by\n rintro rfl\n rfl⟩"
},
{
"state_after": "case intro\na : ℕ\na1 : a ≠ 1\nj : ℕ\nhp : Prime (a * j)\n⊢ a * j = a",
"s... | [
241,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
233,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | edist_le_tsum_of_edist_le_of_tendsto | [
{
"state_after": "α : Type u_1\nβ : Type ?u.567007\nγ : Type ?u.567010\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ m ∈ {x | (fun c => edist (f n) (f c) ≤ ∑' (m : ℕ), d (n + m)) x}",
"s... | [
1581,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1575,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean | NormedAddGroupHom.isClosed_ker | [] | [
776,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
774,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.tendsto_atTop_add_left_of_le' | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.145656\nι' : Type ?u.145659\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.145668\ninst✝ : OrderedAddCommGroup β\nl : Filter α\nf g : α → β\nC : β\nhf : ∀ᶠ (x : α) in l, C ≤ f x\nhg : Tendsto g l atTop\n⊢ ∀ᶠ (x : α) in l, (fun x => -f x) x ≤ -C",
"tact... | [
770,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
767,
1
] |
Mathlib/CategoryTheory/Sites/Sheaf.lean | CategoryTheory.Presheaf.isSheaf_of_isTerminal | [] | [
255,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
253,
1
] |
Mathlib/RingTheory/Noetherian.lean | fg_of_ker_bot | [] | [
198,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
195,
1
] |
Std/Data/Array/Init/Lemmas.lean | Array.foldl_eq_foldl_data | [] | [
49,
53
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
47,
1
] |
Mathlib/Topology/Inseparable.lean | specializes_iff_forall_open | [] | [
116,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean | Set.Nonempty.of_div_left | [] | [
653,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
652,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean | PrincipalIdealRing.irreducible_iff_prime | [] | [
267,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
265,
1
] |
Mathlib/Order/Circular.lean | sbtw_irrefl_left | [] | [
260,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
260,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.coeff_inv_aux | [
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Ring R\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\na : R\nφ : MvPowerSeries σ R\n⊢ (if n = 0 then a\n else -a * ∑ x in antidiagonal n, if x_1 : x.snd < n then ↑(coeff R x.fst) φ * inv.aux a φ x.snd else 0) =\n if n = 0 then a\n else -a * ∑ x in antidiagona... | [
835,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
826,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.one_lt_of_mem_cycleType | [] | [
104,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/Order/BoundedOrder.lean | Pi.top_def | [] | [
650,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
649,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | BoxIntegral.Prepartition.mem_biUnionIndex | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nhJ : J ∈ biUnion π πi\n⊢ J ∈ πi (biUnionIndex π πi J)",
"tactic": "convert (π.mem_biUnion.1 hJ).choose_spec.2 <;> exact dif_pos hJ"
}
] | [
375,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
374,
1
] |
src/lean/Init/Control/Lawful.lean | ExceptT.run_throw | [] | [
112,
99
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
112,
9
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.mem_map_iff_mem_of_injective | [] | [
1531,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1529,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.mul_ne_mul_right | [] | [
350,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
349,
1
] |
Mathlib/Order/RelIso/Group.lean | RelIso.coe_mul | [] | [
39,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
38,
1
] |
Mathlib/Data/Analysis/Filter.lean | Filter.Realizer.ofEquiv_F | [] | [
151,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.comap_iInf | [] | [
1481,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1480,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean | expNegInvGlue.pos_of_pos | [
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : 0 < x\n⊢ 0 < expNegInvGlue x",
"tactic": "simp [expNegInvGlue, not_le.2 hx, exp_pos]"
}
] | [
75,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.castSucc_fin_succ | [
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\nj : Fin n\n⊢ ↑castSucc (succ j) = succ (↑castSucc j)",
"tactic": "simp [Fin.ext_iff]"
}
] | [
1299,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1298,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean | Bool.decide_false | [] | [
148,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
147,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | ModelWithCorners.unique_diff | [] | [
258,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
11
] |
Mathlib/Order/SuccPred/Basic.lean | Order.Ioo_succ_right_of_not_isMax | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : SuccOrder α\na b : α\nhb : ¬IsMax b\n⊢ Ioo a (succ b) = Ioc a b",
"tactic": "rw [← Ioi_inter_Iio, Iio_succ_of_not_isMax hb, Ioi_inter_Iic]"
}
] | [
316,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | Class.eq_univ_of_forall | [] | [
1528,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1527,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | AffineMap.map_vadd | [] | [
132,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.lift_aleph0 | [] | [
1251,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1250,
1
] |
Mathlib/GroupTheory/Perm/List.lean | List.formPerm_eq_formPerm_iff | [
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\n⊢ formPerm [] = formPerm l' ↔ [] ~r l' ∨ length [] ≤ 1 ∧ length l' ≤ 1\n\ncase cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α... | [
444,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
426,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.mem_range | [] | [
2985,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2984,
1
] |
Mathlib/Data/Set/NAry.lean | Set.image2_empty_left | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.16359\nβ : Type u_3\nβ' : Type ?u.16365\nγ : Type u_1\nγ' : Type ?u.16371\nδ : Type ?u.16374\nδ' : Type ?u.16377\nε : Type ?u.16380\nε' : Type ?u.16383\nζ : Type ?u.16386\nζ' : Type ?u.16389\nν : Type ?u.16392\nf f' : α → β → γ\ng g' :... | [
149,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean | Isometry.mapsTo_emetric_ball | [] | [
167,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
165,
1
] |
Mathlib/Topology/PathConnected.lean | pathComponentIn_univ | [
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.633769\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.633784\nF : Set X\nx : X\n⊢ pathComponentIn x univ = pathComponent x",
"tactic": "simp [pathComponentIn, pathComponent, JoinedIn, Joined, exist... | [
943,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
942,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.lintegral_zero | [] | [
859,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
858,
8
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | Continuous.rpow_const | [] | [
327,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
325,
1
] |
Mathlib/Data/Multiset/Fintype.lean | Multiset.image_toEnumFinset_fst | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nm✝ m : Multiset α\n⊢ Finset.image Prod.fst (toEnumFinset m) = toFinset m",
"tactic": "rw [Finset.image, Multiset.map_toEnumFinset_fst]"
}
] | [
230,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
228,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | inv_lt_one | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.50280\nα : Type u_1\nβ : Type ?u.50286\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 1 < a\n⊢ a⁻¹ < 1",
"tactic": "rwa [inv_lt (zero_lt_one.trans ha) zero_lt_one, inv_one]"
}
] | [
300,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
299,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_mul_const | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.944622\nγ : Type ?u.944625\nδ : Type ?u.944628\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ (∫⁻ (a : α), f a * r ∂μ) = (∫⁻ (a : α), f a ∂μ) * r",
"tactic": "simp_rw [mul_comm, lintegral_const... | [
734,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
733,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean | StieltjesFunction.borel_le_measurable | [
{
"state_after": "f : StieltjesFunction\n⊢ MeasurableSpace.generateFrom (range Ioi) ≤ OuterMeasure.caratheodory (StieltjesFunction.outer f)",
"state_before": "f : StieltjesFunction\n⊢ borel ℝ ≤ OuterMeasure.caratheodory (StieltjesFunction.outer f)",
"tactic": "rw [borel_eq_generateFrom_Ioi]"
},
{
... | [
499,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
496,
1
] |
Mathlib/Topology/LocalExtr.lean | IsLocalExtrOn.neg | [] | [
417,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
416,
8
] |
Mathlib/GroupTheory/FreeGroup.lean | FreeGroup.of_injective | [
{
"state_after": "α : Type u\nL L₁✝ L₂ L₃ L₄ : List (α × Bool)\nx✝¹ x✝ : α\nH : of x✝¹ = of x✝\nL₁ : List (α × Bool)\nhx : Red [(x✝¹, true)] L₁\nhy : Red [(x✝, true)] L₁\n⊢ x✝¹ = x✝",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx✝¹ x✝ : α\nH : of x✝¹ = of x✝\n⊢ x✝¹ = x✝",
"tactic": "le... | [
676,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
674,
1
] |
src/lean/Init/Data/Fin/Basic.lean | Fin.val_lt_of_le | [] | [
116,
30
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
115,
1
] |
Mathlib/Logic/Function/Conjugate.lean | Function.Semiconj.option_map | [] | [
77,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
] |
Mathlib/LinearAlgebra/Pi.lean | LinearMap.pi_eq_zero | [
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nφ : ι → Type i\ninst✝¹ : (i : ι)... | [
69,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
67,
1
] |
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