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/MetricSpace/Basic.lean | Metric.closedBall_eq_empty | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.34183\nι : Type ?u.34186\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\n⊢ closedBall x ε = ∅ ↔ ε < 0",
"tactic": "rw [← not_nonempty_iff_eq_empty, nonempty_closedBall, not_le]"
}
] | [
520,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
519,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.hasPushout_assoc_symm | [] | [
2582,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2581,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.map_add_left_Ico | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map ((fun x x_1 => x + x_1) c) (Ico a b) = Ico (c + a) (c + b)",
"tactic": "classical rw [Ico, Ico, ← Finset.image_add_left_Ico, Finset.... | [
296,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean | LocalRing.isUnit_or_isUnit_of_isUnit_add | [
{
"state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst✝¹ : CommSemiring R\ninst✝ : LocalRing R\na b : R\nu : Rˣ\nhu : ↑u = a + b\n⊢ IsUnit a ∨ IsUnit b",
"state_before": "R : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst✝¹ : CommSemiring R\ninst✝ : LocalRing R\na b : R\nh : ... | [
99,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Order/CompleteBooleanAlgebra.lean | sup_iInf₂_eq | [] | [
215,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean | Nat.dvd_mul | [
{
"state_after": "case mp\nx m n : ℕ\n⊢ x ∣ m * n → ∃ y z, y ∣ m ∧ z ∣ n ∧ y * z = x\n\ncase mpr\nx m n : ℕ\n⊢ (∃ y z, y ∣ m ∧ z ∣ n ∧ y * z = x) → x ∣ m * n",
"state_before": "x m n : ℕ\n⊢ x ∣ m * n ↔ ∃ y z, y ∣ m ∧ z ∣ n ∧ y * z = x",
"tactic": "constructor"
},
{
"state_after": "case mp\nx m n... | [
256,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.iUnion_eq_compl_iInter_compl | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.68971\nβ : Type u_1\nγ : Type ?u.68977\nι : Sort u_2\nι' : Sort ?u.68983\nι₂ : Sort ?u.68986\nκ : ι → Sort ?u.68991\nκ₁ : ι → Sort ?u.68996\nκ₂ : ι → Sort ?u.69001\nκ' : ι' → Sort ?u.69006\ns : ι → Set β\n⊢ (⋃ (i : ι), s i) = (⋂ (i : ι), s iᶜ)ᶜ",
... | [
513,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
512,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.set_biInter_option_toFinset | [] | [
2027,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2025,
1
] |
Mathlib/Algebra/Ring/Idempotents.lean | IsIdempotentElem.zero_compl | [] | [
136,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
] |
Mathlib/Topology/SubsetProperties.lean | Filter.cocompact_neBot_iff | [] | [
778,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
777,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.map_id | [] | [
1243,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1242,
1
] |
Mathlib/CategoryTheory/Generator.lean | CategoryTheory.isCoseparator_pi | [
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nβ : Type w\nf : β → C\ninst✝ : HasProduct f\nh : IsCoseparator (∏ f)\nX Y : C\nu v : X ⟶ Y\nhuv : ∀ (G : C), G ∈ Set.range f → ∀ (h : Y ⟶ G), u ≫ h = v ≫ h\n⊢ u = v\n\ncase refine'_... | [
620,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
1
] |
Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | Polynomial.cardPowDegree_anti_archimedean | [
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nx y z : Fq[X]\na : ℤ\nhxy : ↑cardPowDegree (x - y) < a\nhyz : ↑cardPowDegree (y - z) < a\nha : 0 < a\n⊢ ↑cardPowDegree (x - z) < a",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nx y z : Fq[X]\na : ℤ\nhxy : ↑car... | [
177,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean | RingHom.mem_rangeS_self | [] | [
617,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
616,
1
] |
Mathlib/Algebra/DirectSum/Module.lean | DirectSum.isInternal_submodule_iff_isCompl | [
{
"state_after": "R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nthis : ∀ (k : ι), k = i ∨ k = j\n⊢ IsInternal A ↔ IsCompl (A i) (A j)",
"state_before": "R : Type u... | [
421,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
416,
1
] |
Std/Data/String/Lemmas.lean | String.Iterator.Valid.remainingBytes_le | [
{
"state_after": "no goals",
"state_before": "x✝ : Iterator\nh✝ : Valid x✝\nl r : List Char\nh : ValidFor l r x✝\n⊢ remainingBytes x✝ ≤ utf8ByteSize x✝.s",
"tactic": "simp [h.remainingBytes, h.toString, Nat.le_add_left]"
}
] | [
632,
97
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
631,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean | bound'_sq | [
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\n⊢ Tendsto\n (fun n =>\n ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * ‖Polynomial.eval a (↑Polynom... | [
372,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
365,
9
] |
Mathlib/Data/List/Lex.lean | List.Lex.cons_iff | [
{
"state_after": "case cons\nα : Type u\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na : α\nl₁ l₂ : List α\nh : Lex r l₁ l₂\n⊢ Lex r l₁ l₂\n\ncase rel\nα : Type u\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na : α\nl₁ l₂ : List α\nh : r a a\n⊢ Lex r l₁ l₂",
"state_before": "α : Type u\nr : α → α → Prop\ninst✝ : IsI... | [
59,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
57,
1
] |
Mathlib/Tactic/Ring/Basic.lean | Mathlib.Tactic.Ring.single_pow | [
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na : R\nb : ℕ\nc : R\nx✝ : a ^ b = c\n⊢ (a + 0) ^ b = c + 0",
"tactic": "simp [*]"
}
] | [
765,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
765,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.real_smul_eq_coe_mul | [] | [
100,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.stronglyMeasurable_const' | [
{
"state_after": "case inl\nα✝ : Type ?u.15025\nβ✝ : Type ?u.15028\nγ : Type ?u.15031\nι : Type ?u.15034\ninst✝¹ : Countable ι\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : ∀ (x y : α), f x = f y\nh✝ : IsEmpty α\n⊢ StronglyMeasurable f\n\ncase inr\nα✝ : Type ?u.... | [
179,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
1
] |
Mathlib/Topology/Separation.lean | Embedding.t2Space | [] | [
1145,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1143,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.cycle_induction_on | [
{
"state_after": "case intro\nι : Type ?u.2779478\nα : Type ?u.2779481\nβ : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Finite β\nP : Perm β → Prop\nσ : Perm β\nbase_one : P 1\nbase_cycles : ∀ (σ : Perm β), IsCycle σ → P σ\ninduction_disjoint : ∀ (σ τ : Perm β), Disjoint σ τ → IsCycle σ → P σ → P τ → P (σ * τ)\nv... | [
1553,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1535,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.eval₂_mem | [
{
"state_after": "case hs\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Type u_1\nsubS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : SetLike subS S\ninst✝ : Subsemirin... | [
1675,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1652,
1
] |
Mathlib/Data/ULift.lean | ULift.forall | [] | [
127,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/Analysis/Convex/Join.lean | Convex.convexJoin | [
{
"state_after": "ι : Sort ?u.74403\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t u : Set E\nx y : E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\n⊢ ∀ ⦃x : E⦄,\n (∃ i h i_1 i_2, x ∈ [i-[𝕜]i_1]) →\n ∀ ⦃y : E⦄,\n (∃ i h i_1 i_2, y ∈ [i-[𝕜]i... | [
191,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
11
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.norm_ofNat | [] | [
703,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
702,
1
] |
Mathlib/LinearAlgebra/Basic.lean | Submodule.map_inf_comap_of_surjective | [] | [
921,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
919,
1
] |
Mathlib/Analysis/Convex/Join.lean | convexJoin_singleton_left | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.20440\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns t✝ s₁ s₂ t₁ t₂ u : Set E\nx✝ y : E\nt : Set E\nx : E\n⊢ convexJoin 𝕜 {x} t = ⋃ (y : E) (_ : y ∈ t), segment 𝕜 x y",
"tactic": "sim... | [
71,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/Data/Fintype/Card.lean | Function.RightInverse.leftInverse_of_card_le | [] | [
689,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
687,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearEquiv.ofLeftInverse_symm_apply | [] | [
2197,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2195,
1
] |
Mathlib/Analysis/Calculus/Deriv/Comp.lean | HasDerivAt.comp_hasDerivWithinAt | [] | [
190,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.bounded_iff_ediam_ne_top | [] | [
2649,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2646,
1
] |
Mathlib/RingTheory/NonZeroDivisors.lean | map_ne_zero_of_mem_nonZeroDivisors | [] | [
132,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Topology/Basic.lean | closure_iUnion | [
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι✝ : Sort w\na : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝¹ : TopologicalSpace α\nι : Type u_1\ninst✝ : Finite ι\nf : ι → Set α\nval✝ : Fintype ι\n⊢ closure (⋃ (i : ι), f i) = ⋃ (i : ι), closure (f i)",
"state_before": "α : Type u\nβ : Type v\nι✝ :... | [
538,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
535,
1
] |
Mathlib/Data/Multiset/Powerset.lean | Multiset.powersetLen_card_add | [] | [
293,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
291,
1
] |
Mathlib/Data/Nat/Parity.lean | Nat.odd_sub | [
{
"state_after": "no goals",
"state_before": "m n : ℕ\nh : n ≤ m\n⊢ Odd (m - n) ↔ (Odd m ↔ Even n)",
"tactic": "rw [odd_iff_not_even, even_sub h, not_iff, odd_iff_not_even]"
}
] | [
197,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
196,
17
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean | ContDiffBump.contDiff | [] | [
465,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
464,
11
] |
Mathlib/Data/Multiset/Bind.lean | Multiset.coe_join | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.238\nγ : Type ?u.241\nδ : Type ?u.244\nl : List α\nL : List (List α)\n⊢ join ↑(List.map ofList (l :: L)) = ↑(List.join (l :: L))",
"tactic": "simp only [join, List.map, coe_sum, List.sum_cons, List.join, ←coe_add, ←coe_join L]"
}
... | [
46,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/GroupTheory/SemidirectProduct.lean | SemidirectProduct.rightHom_inl | [
{
"state_after": "no goals",
"state_before": "N : Type u_2\nG : Type u_1\nH : Type ?u.113527\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nn : N\n⊢ ↑rightHom (↑inl n) = 1",
"tactic": "simp [rightHom]"
}
] | [
198,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
] |
Mathlib/Algebra/Order/Floor.lean | Int.image_fract | [
{
"state_after": "case h\nF : Type ?u.180856\nα : Type u_1\nβ : Type ?u.180862\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\ns : Set α\nx : α\n⊢ x ∈ fract '' s ↔ x ∈ ⋃ (m : ℤ), (fun x => x - ↑m) '' s ∩ Ico 0 1",
"state_before": "F : Type ?u.180856\nα : Type u_1\nβ : Type ?u.180862\ninst✝... | [
1008,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1001,
1
] |
Mathlib/Topology/Order.lean | continuous_induced_dom | [] | [
706,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
705,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean | gcd_zero_right | [] | [
385,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
384,
1
] |
Mathlib/Order/UpperLower/Basic.lean | UpperSet.prod_self_lt_prod_self | [] | [
1621,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1620,
1
] |
Mathlib/Data/Seq/Computation.lean | Computation.map_pure' | [] | [
883,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
882,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | Dense.borel_eq_generateFrom_Ioc_mem_aux | [
{
"state_after": "case h.e'_3.h.h.e'_2.h\nα✝ : Type ?u.661320\nβ : Type ?u.661323\nγ : Type ?u.661326\nγ₂ : Type ?u.661329\nδ : Type ?u.661332\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : Mea... | [
641,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
631,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.ConnectedComponent.connectedComponentMk_mem | [] | [
2163,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2162,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean | TendstoUniformlyOnFilter.uniformCauchySeqOnFilter | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nhF : TendstoUniformlyOnFilter F f p p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∀ᶠ (m : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', (F m.fst.fst m.snd, F... | [
430,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
422,
1
] |
Mathlib/RingTheory/IntegrallyClosed.lean | IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow | [] | [
98,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Data/Num/Lemmas.lean | Num.gcd_to_nat | [
{
"state_after": "this : ∀ (a b : Num), natSize (a * b) ≤ natSize a + natSize b\n⊢ ∀ (a b : Num), ↑(gcd a b) = Nat.gcd ↑a ↑b",
"state_before": "⊢ ∀ (a b : Num), ↑(gcd a b) = Nat.gcd ↑a ↑b",
"tactic": "have : ∀ a b : Num, (a * b).natSize ≤ a.natSize + b.natSize := by\n intros\n simp [natSize_to_nat]\n ... | [
1711,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1700,
1
] |
Mathlib/Data/Rat/NNRat.lean | NNRat.coe_inj | [] | [
82,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean | WittVector.constantCoeff_wittNSMul | [
{
"state_after": "p : ℕ\nR : Type ?u.168255\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nm n : ℕ\n⊢ ↑constantCoeff (m • X 0) = 0",
"state_before": "p : ℕ\nR : Type ?u.168255\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nm n : ℕ\n⊢ ↑constantCoeff (wittNSMul p m n) = 0",
"tactic": "apply constantCoeff_wi... | [
320,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean | Finsupp.coe_sum | [] | [
326,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
324,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean | Equiv.Perm.signAux_inv | [
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na b : Fin n\nhab : { fst := a, snd := b } ∈ finPairsLT n\nh : ↑f⁻¹ b < ↑f⁻¹ a\n⊢ 1 = if a ≤ b then -1 else 1",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (... | [
395,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
373,
1
] |
Mathlib/Analysis/Calculus/Deriv/Comp.lean | HasFDerivAt.comp_hasDerivAt | [] | [
259,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
] |
Mathlib/Data/Matrix/Block.lean | Matrix.fromBlocks_apply₂₁ | [] | [
68,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | mul_self_inj_of_nonneg | [
{
"state_after": "ι : Type ?u.197416\nα : Type u_1\nβ : Type ?u.197422\ninst✝ : LinearOrderedField α\nb c d : α\nn : ℤ\nb0 : 0 ≤ b\na0 : 0 ≤ -b\n⊢ -b = b",
"state_before": "ι : Type ?u.197416\nα : Type u_1\nβ : Type ?u.197422\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\na0 : 0 ≤ a\nb0 : 0 ≤ b\nh : a =... | [
989,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
984,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.tendsto_setToFun_approxOn_of_measurable | [] | [
1551,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1543,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean | MeasureTheory.IntegrableAtFilter.filter_mono | [] | [
409,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
406,
1
] |
Mathlib/Algebra/Group/Conj.lean | isConj_iff_conjugatesOf_eq | [
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : Monoid α\na b : α\nh : conjugatesOf a = conjugatesOf b\nha : b ∈ conjugatesOf b\n⊢ IsConj a b",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Monoid α\na b : α\nh : conjugatesOf a = conjugatesOf b\n⊢ IsConj a b",
"tactic": "have ha := @mem_conjugatesO... | [
300,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
297,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean | LieAlgebra.isNilpotent_ad_of_isNilpotent | [] | [
747,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
745,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean | Set.empty_smul | [] | [
111,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.comp_inf_eq_inf_comp | [] | [
447,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
445,
1
] |
Mathlib/Data/Finsupp/Pointwise.lean | Finsupp.support_mul | [
{
"state_after": "α : Type u₁\nβ : Type u₂\nγ : Type u₃\nδ : Type u₄\nι : Type u₅\ninst✝¹ : MulZeroClass β\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ β\na : α\nh : a ∈ (g₁ * g₂).support\n⊢ a ∈ g₁.support ∩ g₂.support",
"state_before": "α : Type u₁\nβ : Type u₂\nγ : Type u₃\nδ : Type u₄\nι : Type u₅\ninst✝¹ : MulZ... | [
63,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.left_not_mem_Ioo | [] | [
140,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.nat_lt_omega | [] | [
2409,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2408,
1
] |
Mathlib/Algebra/EuclideanDomain/Defs.lean | EuclideanDomain.lt_one | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\na : R\nthis : (p : Prop) → Decidable p\nh : ¬a = 0\n⊢ ¬a ≺ 1",
"tactic": "simpa only [one_mul] using mul_left_not_lt 1 h"
}
] | [
163,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | CategoryTheory.Limits.prodComparison_snd | [] | [
1265,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1264,
1
] |
Mathlib/Order/Filter/Curry.lean | Filter.curry_le_prod | [] | [
72,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Algebra/Quaternion.lean | QuaternionAlgebra.smul_imI | [] | [
368,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
9
] |
Mathlib/Analysis/Convex/Quasiconvex.lean | Convex.quasiconvexOn_of_convex_le | [] | [
85,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Data/List/Sigma.lean | List.kinsert_def | [] | [
571,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
570,
1
] |
Mathlib/Algebra/CharP/ExpChar.lean | char_prime_of_ne_zero | [
{
"state_after": "case inl\nR : Type u\ninst✝² : Semiring R\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors R\np : ℕ\nhp : CharP R p\np_ne_zero : p ≠ 0\nh : Nat.Prime p\n⊢ Nat.Prime p\n\ncase inr\nR : Type u\ninst✝² : Semiring R\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors R\np : ℕ\nhp : CharP R p\np_ne_zero :... | [
102,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean | prime_factors_unique | [
{
"state_after": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\nf : Multiset α\n⊢ ∀ {g : Multiset α},\n (∀ (x : α), x ∈ f → Prime x) →\n (∀ (x : α), x ∈ g → Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoidWithZe... | [
242,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean | StrictMonoOn.mul_const' | [] | [
1416,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1414,
1
] |
Mathlib/ModelTheory/Basic.lean | FirstOrder.Language.toEmbedding_embedding_empty | [] | [
993,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
992,
1
] |
Mathlib/ModelTheory/Syntax.lean | FirstOrder.Language.BoundedFormula.IsQF.isPrenex | [] | [
760,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
759,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean | Dfinsupp.zipWith_neLocus_eq_right | [
{
"state_after": "case a\nα : Type u_4\nN : α → Type u_3\ninst✝⁵ : DecidableEq α\nM : α → Type u_1\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 (M a)\ninst✝ : (a : α) → DecidableEq (P a)\nF : (a : α) → M a → N a → ... | [
110,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean | Submodule.mem_iSup_of_mem | [] | [
302,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
] |
Mathlib/CategoryTheory/CofilteredSystem.lean | CategoryTheory.Functor.IsMittagLeffler.subset_image_eventualRange | [
{
"state_after": "case intro.intro\nJ : Type u\ninst✝ : Category J\nF : J ⥤ Type v\ni j k✝ : J\ns : Set (F.obj i)\nh : IsMittagLeffler F\nf : j ⟶ i\nk : J\ng : k ⟶ j\nhg : eventualRange F j = range (F.map g)\n⊢ eventualRange F i ⊆ F.map f '' eventualRange F j",
"state_before": "J : Type u\ninst✝ : Category ... | [
164,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
] |
src/lean/Init/Prelude.lean | Nat.eq_or_lt_of_le | [] | [
1642,
42
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
1634,
11
] |
Mathlib/MeasureTheory/Measure/Regular.lean | IsOpen.exists_lt_isClosed | [] | [
566,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
564,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.eventually_map | [] | [
1820,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1819,
1
] |
Mathlib/Algebra/Group/Prod.lean | MonoidHom.coprod_apply | [] | [
638,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
637,
1
] |
Mathlib/GroupTheory/FreeGroup.lean | FreeGroup.invRev_involutive | [] | [
578,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
578,
1
] |
Mathlib/Logic/Relation.lean | Relation.transGen_eq_self | [
{
"state_after": "case single\nα : Type u_1\nβ : Type ?u.22591\nγ : Type ?u.22594\nδ : Type ?u.22597\nr : α → α → Prop\na✝¹ b✝¹ c d : α\ntrans : Transitive r\na b b✝ : α\na✝ : r a b✝\n⊢ r a b✝\n\ncase tail\nα : Type u_1\nβ : Type ?u.22591\nγ : Type ?u.22594\nδ : Type ?u.22597\nr : α → α → Prop\na✝² b✝¹ c d : α\... | [
461,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
456,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | Real.smul_map_volume_mul_right | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2207770\ninst✝ : Fintype ι\na : ℝ\nh : a ≠ 0\n⊢ ofReal (abs a) • Measure.map (fun x => x * a) volume = volume",
"tactic": "simpa only [mul_comm] using Real.smul_map_volume_mul_left h"
}
] | [
329,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
327,
1
] |
Mathlib/Data/List/Basic.lean | List.foldl_cons | [] | [
2401,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2399,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean | Nat.factorization_mul_support_of_coprime | [
{
"state_after": "a b : ℕ\nhab : coprime a b\n⊢ (factorization a + factorization b).support = (factorization a).support ∪ (factorization b).support",
"state_before": "a b : ℕ\nhab : coprime a b\n⊢ (factorization (a * b)).support = (factorization a).support ∪ (factorization b).support",
"tactic": "rw [fa... | [
791,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
788,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | IsBoundedLinearMap.neg | [
{
"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_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.41770\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\nhf : IsBo... | [
141,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Mathlib/Data/List/Basic.lean | List.nthLe_append_right | [] | [
1338,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1336,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | Real.sinh_sub_id_strictMono | [
{
"state_after": "x y z : ℝ\n⊢ StrictMonoOn (fun x => sinh x - x) (Ici 0)",
"state_before": "x y z : ℝ\n⊢ StrictMono fun x => sinh x - x",
"tactic": "refine' strictMono_of_odd_strictMonoOn_nonneg (fun x => by simp; abel) _"
},
{
"state_after": "case refine'_1\nx y z : ℝ\n⊢ ContinuousOn (fun x =>... | [
756,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
749,
1
] |
Mathlib/Logic/Basic.lean | xor_self | [
{
"state_after": "no goals",
"state_before": "a : Prop\n⊢ Xor' a a = False",
"tactic": "simp [Xor']"
}
] | [
295,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
295,
9
] |
Mathlib/Algebra/Algebra/Basic.lean | ULift.down_algebraMap | [] | [
500,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
499,
1
] |
Mathlib/Order/Grade.lean | grade_mono | [] | [
166,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
165,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.update_self | [
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni : ι\nf : Π₀ (i : ι), β i\nb : β i\nj i✝ : ι\n⊢ ↑(update i f (↑f i)) i✝ = ↑f i✝",
"state_before": "ι : Type u\nγ : Typ... | [
838,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
836,
1
] |
Mathlib/Combinatorics/Hall/Finite.lean | HallMarriageTheorem.hall_hard_inductive | [
{
"state_after": "case intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Finite ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nval✝ : Fintype ι\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\... | [
247,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
230,
1
] |
Mathlib/GroupTheory/Nilpotent.lean | is_decending_rev_series_of_is_ascending | [
{
"state_after": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊤\nh0 : H 0 = ⊥\nhH : ∀ (x : G) (n : ℕ), x ∈ H (n + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n\n⊢ IsDescendingCentralSeries fun m => H (n - m)",
"state_before": "G : Type u_1\... | [
246,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
233,
1
] |
Mathlib/Data/Nat/Parity.lean | Even.mod_even_iff | [] | [
353,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
351,
1
] |
Mathlib/RingTheory/Ideal/Quotient.lean | Ideal.map_pi | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nI : Ideal R\na b : R\nS ι✝ : Type v\nι : Type u_1\ninst✝ : Finite ι\nι' : Type w\nx : ι → R\nhi : ∀ (i : ι), x i ∈ I\nf : (ι → R) →ₗ[R] ι' → R\ni : ι'\n⊢ ↑f x i ∈ I",
"tactic": "classical\n cases nonempty_fintype ι\n rw [pi... | [
396,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
390,
1
] |
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