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/RingTheory/Coprime/Basic.lean | IsCoprime.add_mul_left_right | [
{
"state_after": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime (y + x * z) x",
"state_before": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (y + x * z)",
"tactic": "rw [isCoprime_comm]"
},
{
"state_after": "no goals",
"st... | [
295,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
293,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.pos_iff_ne_zero | [
{
"state_after": "no goals",
"state_before": "n m : ℕ\ninst✝ : NeZero n\na : Fin n\n⊢ 0 < a ↔ a ≠ 0",
"tactic": "rw [← val_fin_lt, val_zero, _root_.pos_iff_ne_zero, Ne.def, Ne.def, ext_iff, val_zero]"
}
] | [
424,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean | CategoryTheory.Presieve.FamilyOfElements.compPresheafMap_id | [] | [
355,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
] |
Mathlib/CategoryTheory/Functor/Category.lean | CategoryTheory.NatTrans.mono_of_mono_app | [
{
"state_after": "case w.h\nC : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G H I : C ⥤ D\nα : F ⟶ G\ninst✝ : ∀ (X : C), Mono (α.app X)\nZ✝ : C ⥤ D\ng h : Z✝ ⟶ F\neq : g ≫ α = h ≫ α\nX : C\n⊢ g.app X = h.app X",
"state_before": "C : Type u₁\ninst✝³ : C... | [
97,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Order/SupIndep.lean | CompleteLattice.setIndependent_iff_pairwiseDisjoint | [] | [
400,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
397,
1
] |
Mathlib/Data/Polynomial/Eval.lean | Polynomial.eval₂_at_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\n⊢ eval₂ f 0 p = ↑f (coeff p 0)",
"tactic": "simp (config := { contextual := true }) only [eval₂_eq_sum, zero_pow_eq, ... | [
65,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
62,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean | CategoryTheory.Monad.algebra_equiv_of_iso_monads_comp_forget | [] | [
315,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
313,
1
] |
Mathlib/Topology/Sheaves/Forget.lean | TopCat.Presheaf.isSheaf_iff_isSheaf_comp | [
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\n⊢ IsSheafEqualizerProducts F ↔ IsSheafEqualizerProducts (F ⋙ G)",
"state_befo... | [
224,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Topology/Order/Basic.lean | exists_Ioc_subset_of_mem_nhds' | [] | [
1232,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1228,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.two_zsmul_coe_div_two | [
{
"state_after": "no goals",
"state_before": "θ : ℝ\n⊢ 2 • ↑(θ / 2) = ↑θ",
"tactic": "rw [← coe_zsmul, two_zsmul, add_halves]"
}
] | [
147,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Measure.lean | BoxIntegral.Box.coe_ae_eq_Icc | [
{
"state_after": "ι : Type u_1\nI : Box ι\ninst✝ : Fintype ι\n⊢ (pi univ fun i => Ioc (lower I i) (upper I i)) =ᵐ[volume] ↑Box.Icc I",
"state_before": "ι : Type u_1\nI : Box ι\ninst✝ : Fintype ι\n⊢ ↑I =ᵐ[volume] ↑Box.Icc I",
"tactic": "rw [coe_eq_pi]"
},
{
"state_after": "no goals",
"state_b... | [
79,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Order/RelIso/Group.lean | RelIso.inv_apply_self | [] | [
48,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
] |
Mathlib/Algebra/Module/LinearMap.lean | LinearMap.map_smulₛₗ | [] | [
338,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
337,
11
] |
Std/Data/Nat/Lemmas.lean | Nat.min_le_left | [] | [
179,
97
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
179,
11
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.sSup_unit_ball_eq_norm | [
{
"state_after": "no goals",
"state_before": "𝕜✝ : Type ?u.819707\n𝕜₂✝ : Type ?u.819710\n𝕜₃ : Type ?u.819713\nE✝ : Type ?u.819716\nEₗ : Type ?u.819719\nF✝ : Type ?u.819722\nFₗ : Type ?u.819725\nG : Type ?u.819728\nGₗ : Type ?u.819731\n𝓕 : Type ?u.819734\ninst✝²³ : SeminormedAddCommGroup E✝\ninst✝²² : Se... | [
562,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
558,
1
] |
Mathlib/RingTheory/Polynomial/Vieta.lean | Multiset.prod_X_add_C_coeff' | [
{
"state_after": "case h\nR : Type u_2\ninst✝ : CommSemiring R\nσ : Type u_1\ns : Multiset σ\nr : σ → R\nk : ℕ\nh : k ≤ ↑card s\n⊢ k ≤ ↑card s",
"state_before": "R : Type u_2\ninst✝ : CommSemiring R\nσ : Type u_1\ns : Multiset σ\nr : σ → R\nk : ℕ\nh : k ≤ ↑card s\n⊢ coeff (prod (map (fun i => X + ↑C (r i)) ... | [
80,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean | LinearMap.map_sub₂ | [] | [
167,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
166,
1
] |
Mathlib/Data/List/Zip.lean | List.zipWith_distrib_take | [
{
"state_after": "case nil\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl'✝ : List β\nn✝ : ℕ\nl' : List β\nn : ℕ\n⊢ take n (zipWith f [] l') = zipWith f (take n []) (take n l')\n\ncase cons\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507... | [
481,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
474,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | contDiffAt_ring_inverse | [
{
"state_after": "case h0\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\nins... | [
1682,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1661,
1
] |
Std/Data/List/Lemmas.lean | List.find?_cons_of_neg | [
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\np : α✝ → Bool\na : α✝\nl : List α✝\nh : ¬p a = true\n⊢ find? p (a :: l) = find? p l",
"tactic": "simp [find?, h]"
}
] | [
1257,
21
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1256,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.minpoly_powerBasis_gen_of_monic | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\ng : R[X]\ninst✝ : Field K\nf : K[X]\nhf : Monic f\nhf' : optParam (f ≠ 0) (_ : f ≠ 0)\n⊢ minpoly K (powerBasis hf').gen = f",
"tactic": "rw [minpoly_powerBasis_gen hf', hf.leadingCoeff, inv_one, C.map_... | [
604,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
602,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.inv | [
{
"state_after": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Group G\ninst✝ : AddGroup A\ni : ℤ\nx : G\nhx : IsOfFinOrder x\nn : ℕ\nnpos : 0 < n\nhn : x ^ n = 1\n⊢ ∃ n, 0 < n ∧ x⁻¹ ^ n = 1",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝¹ : Gr... | [
544,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | DoubleCentralizer.zero_toProd | [] | [
223,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean | SimpleGraph.isEmpty_of_chromaticNumber_eq_zero | [
{
"state_after": "V : Type u\nG✝ : SimpleGraph V\nα : Type v\nC : Coloring G✝ α\nG : SimpleGraph V\ninst✝ : Finite V\nh : chromaticNumber G = 0\nh' : Colorable G (chromaticNumber G)\n⊢ IsEmpty V",
"state_before": "V : Type u\nG✝ : SimpleGraph V\nα : Type v\nC : Coloring G✝ α\nG : SimpleGraph V\ninst✝ : Fini... | [
301,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
297,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean | Bool.of_decide_false | [] | [
152,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/Analysis/Convex/Between.lean | affineSegment_const_vadd_image | [] | [
95,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/RingTheory/Algebraic.lean | exists_integral_multiple | [
{
"state_after": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"state_... | [
318,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
] |
Mathlib/Data/Polynomial/Derivative.lean | Polynomial.derivative_X_add_C_pow | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommSemiring R\nc : R\nm : ℕ\n⊢ ↑derivative ((X + ↑C c) ^ m) = ↑C ↑m * (X + ↑C c) ^ (m - 1)",
"tactic": "rw [derivative_pow, derivative_X_add_C, mul_one]"
}
] | [
526,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
524,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.two_zsmul_toReal_eq_two_mul | [
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ toReal (2 • θ) = 2 * toReal θ ↔ toReal θ ∈ Set.Ioc (-π / 2) (π / 2)",
"tactic": "rw [two_zsmul, ← two_nsmul, two_nsmul_toReal_eq_two_mul]"
}
] | [
678,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
676,
1
] |
Mathlib/Data/Set/Function.lean | Monotone.rangeFactorization | [] | [
1337,
4
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1336,
11
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.mem_comap | [] | [
1369,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1368,
1
] |
Std/Data/String/Lemmas.lean | Substring.ValidFor.nextn_stop | [
{
"state_after": "no goals",
"state_before": "l m r : List Char\nx✝ : Substring\nh : ValidFor l m r x✝\nn : Nat\n⊢ nextn x✝ (n + 1) { byteIdx := utf8Len m } = { byteIdx := utf8Len m }",
"tactic": "simp [Substring.nextn, h.next_stop, h.nextn_stop n]"
}
] | [
854,
72
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
852,
1
] |
Mathlib/GroupTheory/DoubleCoset.lean | Doset.bot_rel_eq_leftRel | [
{
"state_after": "case h.h.a\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ Setoid.Rel (setoid ↑⊥ ↑H) a b ↔ Setoid.Rel (QuotientGroup.leftRel H) a b",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : ... | [
103,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Analysis/Convex/Hull.lean | Set.Nonempty.convexHull | [] | [
122,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
11
] |
Mathlib/Combinatorics/Additive/Energy.lean | Finset.multiplicativeEnergy_univ_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : CommGroup α\ninst✝ : Fintype α\ns t : Finset α\n⊢ multiplicativeEnergy s univ = Fintype.card α * card s ^ 2",
"tactic": "rw [multiplicativeEnergy_comm, multiplicativeEnergy_univ_left]"
}
] | [
169,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean | CategoryTheory.CosimplicialObject.σ_naturality | [] | [
589,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
587,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearEquiv.coe_toHomeomorph | [] | [
1834,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1833,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.div_toGerm | [] | [
768,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
767,
1
] |
Mathlib/Order/UpperLower/Basic.lean | LowerSet.disjoint_coe | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.78350\nγ : Type ?u.78353\nι : Sort ?u.78356\nκ : ι → Sort ?u.78361\ninst✝ : LE α\nS : Set (LowerSet α)\ns t : LowerSet α\na : α\n⊢ Disjoint ↑s ↑t ↔ Disjoint s t",
"tactic": "simp [disjoint_iff, SetLike.ext'_iff]"
}
] | [
775,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
774,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | WithTop.isGLB_sInf' | [
{
"state_after": "case left\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ sInf s ∈ lowerBounds s\n\ncase right\nα : Type ?u.90006\nβ✝ : Ty... | [
1201,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1164,
1
] |
Mathlib/LinearAlgebra/Matrix/Basis.lean | Basis.toMatrix_eq_toMatrix_constr | [
{
"state_after": "case a.h\nι : Type u_1\nι' : Type ?u.55138\nκ : Type ?u.55141\nκ' : Type ?u.55144\nR : Type u_2\nM : Type u_3\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nR₂ : Type ?u.55337\nM₂ : Type ?u.55340\ninst✝⁴ : CommRing R₂\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R₂ M₂\n... | [
76,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Order/Filter/Pointwise.lean | Filter.inv_le_iff_le_inv | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.32470\nα : Type u_1\nβ : Type ?u.32476\nγ : Type ?u.32479\nδ : Type ?u.32482\nε : Type ?u.32485\ninst✝ : InvolutiveInv α\nf g : Filter α\ns : Set α\n⊢ f⁻¹ ≤ g ↔ f ≤ g⁻¹",
"tactic": "rw [← Filter.inv_le_inv_iff, inv_inv]"
}
] | [
262,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
262,
1
] |
Mathlib/Order/Antichain.lean | IsGreatest.antichain_iff | [] | [
249,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Data/Nat/PartENat.lean | PartENat.coe_le_iff | [
{
"state_after": "n : ℕ\nx : PartENat\n⊢ ↑n ≤ x ↔ ∀ (h : x.Dom), n ≤ Part.get x h",
"state_before": "n : ℕ\nx : PartENat\n⊢ ↑n ≤ x ↔ ∀ (h : x.Dom), n ≤ Part.get x h",
"tactic": "rw [← some_eq_natCast]"
},
{
"state_after": "n : ℕ\nx : PartENat\n⊢ (∀ (hy : x.Dom), Part.get ↑n (_ : (↑n).Dom) ≤ Part... | [
303,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.mul_self_le_self | [
{
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650,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
648,
1
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Mathlib/Topology/MetricSpace/Infsep.lean | Set.einfsep_pair | [
{
"state_after": "α : Type u_1\nβ : Type ?u.39983\ninst✝ : PseudoEMetricSpace α\nx y z : α\ns t : Set α\nhxy : x ≠ y\n⊢ einfsep {x, y} = min (edist x y) (edist x y)",
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220,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | Polynomial.cyclotomic_dvd_geom_sum_of_dvd | [
{
"state_after": "R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\n⊢ cyclotomic d ℤ ∣ ∑ i in range n, X ^ i",
"state_before": "R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\n⊢ cyclotomic d R ∣ ∑ i in range n, X ^ i",
"tactic": "suffices cyclotomic d ℤ ∣ ∑ i in Finset.rang... | [
431,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
422,
1
] |
Mathlib/Topology/CompactOpen.lean | Homeomorph.continuousMapOfUnique_symm_apply | [] | [
468,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
466,
1
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Mathlib/Algebra/Star/SelfAdjoint.lean | selfAdjoint.val_smul | [] | [
437,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
436,
1
] |
Mathlib/Topology/Constructions.lean | IsOpenMap.restrict | [] | [
1033,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1031,
1
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Mathlib/Data/Finset/Pointwise.lean | Finset.mem_one | [] | [
86,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
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Mathlib/RingTheory/JacobsonIdeal.lean | Ideal.isUnit_of_sub_one_mem_jacobson_bot | [
{
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258,
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254,
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Mathlib/FieldTheory/IntermediateField.lean | AlgHom.mem_fieldRange | [] | [
485,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
484,
1
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Mathlib/ModelTheory/FinitelyGenerated.lean | FirstOrder.Language.Substructure.FG.cg | [
{
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"state_before": "L : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\nh : FG N\n⊢ CG N... | [
116,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
114,
1
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Mathlib/Order/LiminfLimsup.lean | Filter.blimsup_eq_iInf_biSup | [
{
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741,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
727,
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Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_eq_zero' | [
{
"state_after": "no goals",
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"tactic": "cla... | [
373,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
367,
1
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Mathlib/Algebra/GroupPower/Order.lean | sq_le_one_iff_abs_le_one | [
{
"state_after": "no goals",
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"tactic": "simpa only [one_pow, abs_one] using @sq_le_sq _ _ x 1"
}
] | [
731,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
730,
1
] |
Mathlib/Topology/Connected.lean | isTotallyDisconnected_of_clopen_set | [
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\n⊢ Set.Subsingleton S",
"state_before": "α... | [
1291,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1280,
1
] |
Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints | [
{
"state_after": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∃ᶠ (x : C(X, ℝ)) in 𝓝 f, x ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : C... | [
315,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
] |
Mathlib/Analysis/Complex/Basic.lean | Complex.nnnorm_int | [] | [
196,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
195,
1
] |
Mathlib/RingTheory/Localization/NumDen.lean | IsFractionRing.num_mul_den_eq_num_iff_eq | [
{
"state_after": "no goals",
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84,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
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Mathlib/Data/Finset/Lattice.lean | Finset.sup_mul_le_mul_sup_of_nonneg | [] | [
1893,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1889,
1
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Mathlib/Order/CompleteLattice.lean | le_iInf_const | [] | [
1040,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1039,
1
] |
Mathlib/Analysis/NormedSpace/CompactOperator.lean | IsCompactOperator.clm_comp | [] | [
276,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.sup_closed_of_sup_closed | [] | [
998,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
996,
1
] |
Mathlib/Data/Seq/Seq.lean | Stream'.Seq.eq_or_mem_of_mem_cons | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na b : α\nf : Stream' (Option α)\nal : IsSeq f\nh✝ : a ∈ cons b { val := f, property := al }\nh : some a = some b\n⊢ a = b",
"tactic": "injection h"
}
] | [
199,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.coe_filterMap | [] | [
2110,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2109,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean | Polynomial.aeval_eq_sum_range | [
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1271761\nB' : Type ?u.1271764\na b : R\nn : ℕ\ninst✝⁸ : CommSemiring A'\ninst✝⁷ : Semiring B'\ninst✝⁶ : CommSemiring R\np✝ q : R[X]\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB : Type ?u.1271977\ninst✝³ : Semiring B\ninst✝² : Algebr... | [
358,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
355,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_insert | [] | [
1451,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1449,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean | CategoryTheory.MonoOver.bot_arrow | [] | [
103,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean | NormedSpace.isBounded_iff_subset_smul_closedBall | [
{
"state_after": "case mp\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ Bornology.IsBounded s → ∃ a, s ⊆ a • Metric.closedBall 0 1\n\n... | [
338,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
] |
Mathlib/MeasureTheory/Measure/Haar/Basic.lean | MeasureTheory.Measure.haar.index_union_eq | [
{
"state_after": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V",
"state... | [
269,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
239,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean | UniqueFactorizationMonoid.normalizedFactors_pos | [
{
"state_after": "case mp\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\n⊢ 0 < normalizedFactors x → ¬IsUnit x\n\ncase mpr\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq... | [
789,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
778,
1
] |
Mathlib/Algebra/Algebra/Spectrum.lean | spectrum.preimage_units_mul_eq_swap_mul | [] | [
250,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.coe_map | [] | [
318,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean | Dfinsupp.zipWith_neLocus_eq_left | [
{
"state_after": "case a\nα : Type u_4\nN : α → Type u_1\ninst✝⁵ : DecidableEq α\nM : α → Type u_3\nP : α → Type u_2\ninst✝⁴ : (a : α) → Zero (N a)\ninst✝³ : (a : α) → Zero (M a)\ninst✝² : (a : α) → Zero (P a)\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → DecidableEq (P a)\nF : (a : α) → M a → N a → ... | [
102,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
] |
Mathlib/RingTheory/TensorProduct.lean | TensorProduct.AlgebraTensorModule.lift_apply | [] | [
151,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
149,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.card_set_walk_length_eq | [
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu v : V\nn : ℕ\np : Walk G u v\n⊢ p ∈ finsetWalkLength G n u v ↔ p ∈ {p | Walk.length p = n}",
"tactic": "r... | [
2391,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2388,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.mem_top | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.57666\nι : Sort x\nf g : Filter α\ns✝ t s : Set α\n⊢ s ∈ ⊤ ↔ s = univ",
"tactic": "rw [mem_top_iff_forall, eq_univ_iff_forall]"
}
] | [
461,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
460,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.Ici_iSup | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.295226\nγ : Type ?u.295229\nι : Sort u_2\nι' : Sort ?u.295235\nι₂ : Sort ?u.295238\nκ : ι → Sort ?u.295243\nκ₁ : ι → Sort ?u.295248\nκ₂ : ι → Sort ?u.295253\nκ' : ι' → Sort ?u.295258\ninst✝ : CompleteLattice α\nf : ι → α\nx✝ : α\n⊢ x✝ ∈... | [
2123,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2122,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.eventuallyEq_of_toReal_eventuallyEq | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.27073\nγ : Type ?u.27076\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nl : Filter α\nf g : α → ℝ≥0∞\nhfi : ∀ᶠ (x : α) in l, f x ≠ ⊤\nhgi : ∀ᶠ (x : α) in l, g x ≠ ⊤\nhfg : (fun x => ENNReal.toReal (f x)) =ᶠ[l] fun x => ENNReal.toReal (g x)\n... | [
127,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | zpow_strictMono_right | [
{
"state_after": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : OrderedCommGroup α\nm✝ n✝ : ℤ\na b : α\nha : 1 < a\nm n : ℤ\nh : m < n\n⊢ a ^ (m + (n - m)) = a ^ n",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH... | [
334,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
330,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean | OrderRingHom.id_apply | [] | [
283,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Data/Bool/AllAny.lean | List.any_iff_exists | [
{
"state_after": "case nil\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl : List α\na : α\np : α → Bool\n⊢ any [] p = true ↔ ∃ a, a ∈ [] ∧ p a = true\n\ncase cons\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl✝ : List α\na✝ : α\np : α → Bool\na : α\nl : List α\nih : any l p = true ↔ ∃ a, a... | [
48,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
] |
Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearEquiv.coe_toContinuousLinearEquiv | [] | [
370,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
369,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean | PiLp.nnnorm_eq_sum | [
{
"state_after": "case a\np✝ : ℝ≥0∞\n𝕜 : Type ?u.229074\n𝕜' : Type ?u.229077\nι : Type u_2\nα : ι → Type ?u.229085\nβ✝ : ι → Type ?u.229090\ninst✝³ : Fintype ι\ninst✝² : Fact (1 ≤ p✝)\np : ℝ≥0∞\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\nhp : p ≠ ⊤\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nf : PiLp p β\n... | [
564,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
560,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.replicate_add | [] | [
898,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
897,
1
] |
Mathlib/Data/Sym/Sym2.lean | Sym2.mem_iff | [] | [
339,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | pow_four_le_pow_two_of_pow_two_le | [] | [
750,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
749,
1
] |
Mathlib/Order/Antisymmetrization.lean | ofAntisymmetrization_lt_ofAntisymmetrization_iff | [] | [
203,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/RingTheory/EuclideanDomain.lean | EuclideanDomain.isCoprime_of_dvd | [] | [
96,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Topology/UrysohnsLemma.lean | Urysohns.CU.continuous_lim | [
{
"state_after": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\n⊢ Continuous (CU.lim c)",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\n⊢ Continuous (CU.lim c)",
... | [
292,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | Monotone.measurable | [] | [
1171,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1168,
11
] |
Mathlib/Data/Int/Bitwise.lean | Int.lnot_bit | [
{
"state_after": "no goals",
"state_before": "b : Bool\nn : ℕ\n⊢ lnot (bit b ↑n) = bit (!b) (lnot ↑n)",
"tactic": "simp [lnot]"
},
{
"state_after": "no goals",
"state_before": "b : Bool\nn : ℕ\n⊢ lnot (bit b -[n+1]) = bit (!b) (lnot -[n+1])",
"tactic": "simp [lnot]"
}
] | [
331,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
329,
1
] |
Mathlib/Topology/Order/Basic.lean | isOpen_iff_generate_intervals | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\ns : Set α\n⊢ IsOpen s ↔ GenerateOpen {s | ∃ a, s = Ioi a ∨ s = Iio a} s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : Orde... | [
900,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
898,
1
] |
Mathlib/Order/CompleteLattice.lean | iInf_lt_iff | [] | [
614,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
613,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.mapRange_single | [
{
"state_after": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), ... | [
1190,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1184,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean | PadicInt.coe_mul | [] | [
129,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean | exists_norm_eq | [
{
"state_after": "case intro\nα : Type ?u.296964\nβ : Type ?u.296967\nγ : Type ?u.296970\nι : Type ?u.296973\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.297066\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst... | [
357,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
] |
Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | CategoryTheory.MonoidalCategory.pentagon_hom_inv | [
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nW X Y Z : C\n⊢ (α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) = (α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (X ⊗ Y) Z).hom",
"tactic": "coherence"
}
] | [
83,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
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