file_path stringlengths 11 79 | full_name stringlengths 2 100 | traced_tactics list | end list | commit stringclasses 4
values | url stringclasses 4
values | start list |
|---|---|---|---|---|---|---|
Mathlib/Data/Set/Intervals/OrderIso.lean | OrderIso.preimage_Ici | [
{
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... | [
34,
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32,
1
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Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | CategoryTheory.SemiadditiveOfBinaryBiproducts.isUnital_leftAdd | [
{
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71,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
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Std/Data/String/Lemmas.lean | Substring.Valid.next | [
{
"state_after": "m₁ : List Char\nc : Char\nm₂ : List Char\nx✝ : Substring\nh✝ : Valid x✝\ne : (toString x✝).data = m₁ ++ c :: m₂\nl m r : List Char\nh : ValidFor l m r x✝\n⊢ Substring.next x✝ { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + csize c }",
"state_before": "m₁ : List Char\nc : Char\nm₂ : L... | [
990,
51
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986,
1
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Mathlib/Order/SuccPred/Basic.lean | Order.Ico_succ_right | [] | [
377,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
376,
1
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Mathlib/Data/Real/CauSeq.lean | CauSeq.lt_irrefl | [
{
"state_after": "no goals",
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"tactic": "simp [zero_limZero]"
}
] | [
745,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
744,
1
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Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | LiouvilleWith.nat_sub_iff | [
{
"state_after": "no goals",
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"tactic": "simp [sub_eq_add_neg]"
}
] | [
307,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
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Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.compMeasurable_toGerm | [
{
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399,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
395,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | affineSpan_singleton_union_vadd_eq_top_of_span_eq_top | [
{
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1238,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1227,
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Mathlib/Data/Real/CauSeq.lean | CauSeq.const_limZero | [] | [
464,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
458,
1
] |
Mathlib/Topology/MetricSpace/Kuratowski.lean | KuratowskiEmbedding.exists_isometric_embedding | [
{
"state_after": "case inl\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : univ = ∅\n⊢ ∃ f, Isometry f\n\ncase inr\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g ... | [
108,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
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Mathlib/Algebra/Module/Submodule/Lattice.lean | Submodule.toAddSubmonoid_toNatSubmodule | [] | [
376,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
374,
1
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Std/Data/List/Lemmas.lean | List.get?_modifyNth_eq | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α\nn : Nat\nl : List α\n⊢ get? (modifyNth f n l) n = f <$> get? l n",
"tactic": "simp only [get?_modifyNth, if_pos]"
}
] | [
758,
37
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
756,
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Mathlib/Tactic/LinearCombination.lean | Mathlib.Tactic.LinearCombination.add_pf | [] | [
39,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.filter_sub | [] | [
480,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
478,
1
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Mathlib/Analysis/Calculus/FDeriv/Comp.lean | Differentiable.comp | [] | [
188,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
186,
1
] |
Mathlib/FieldTheory/RatFunc.lean | RatFunc.liftOn_ofFractionRing_mk | [
{
"state_after": "K : Type u\ninst✝ : CommRing K\nP : Sort v\nn : K[X]\nd : { x // x ∈ K[X]⁰ }\nf : K[X] → K[X] → P\nH : ∀ {p q p' q' : K[X]}, q ∈ K[X]⁰ → q' ∈ K[X]⁰ → q' * p = q * p' → f p q = f p' q'\n⊢ Localization.liftOn { toFractionRing := Localization.mk n d }.toFractionRing (fun p q => f p ↑q)\n (_ ... | [
172,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
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Mathlib/Logic/Function/Basic.lean | eq_rec_on_bijective | [] | [
997,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
995,
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Mathlib/RingTheory/Ideal/Operations.lean | Ideal.map_of_equiv | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nF : Type ?u.1386892\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf✝ : F\nI✝ I : Ideal R\nf : R ≃+* S\n⊢ map (↑(RingEquiv.symm f)) (map (↑f) I) = I",
"tactic": "simp [← RingEquiv.toRingHom_eq_coe, map_map]"
}
] | [
1713,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1711,
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Mathlib/Topology/MetricSpace/Lipschitz.lean | LipschitzWith.eval | [
{
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219,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/LinearAlgebra/Alternating.lean | AlternatingMap.map_swap_add | [
{
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694,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
689,
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Mathlib/Data/List/Intervals.lean | List.Ico.succ_singleton | [
{
"state_after": "n : ℕ\n⊢ range' n (n + 1 - n) = [n]",
"state_before": "n : ℕ\n⊢ Ico n (n + 1) = [n]",
"tactic": "dsimp [Ico]"
},
{
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"tactic": "simp [range', add_tsub_cancel_left]"
}
] | [
124,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
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Mathlib/LinearAlgebra/Determinant.lean | LinearMap.associated_det_of_eq_comp | [
{
"state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M),... | [
496,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
489,
1
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Mathlib/Algebra/GroupPower/Basic.lean | zpow_two | [
{
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"state_before": "α : Type ?u.55852\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR :... | [
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293,
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Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.map_units | [] | [
577,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
576,
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Mathlib/Analysis/NormedSpace/Multilinear.lean | ContinuousMultilinearMap.uncurry_curryRight | [
{
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1508,
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Mathlib/Algebra/DirectSum/Module.lean | DirectSum.lequivCongrLeft_apply | [] | [
245,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
243,
1
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Mathlib/Data/Finset/Pointwise.lean | Finset.inv_empty | [] | [
229,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
228,
1
] |
Mathlib/Algebra/Hom/Iterate.lean | Commute.function_commute_mul_right | [] | [
260,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
1
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Mathlib/Topology/MetricSpace/HausdorffDistance.lean | Metric.thickening_thickening_subset | [
{
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1472,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1463,
1
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Mathlib/Analysis/SpecificLimits/Normed.lean | NormedAddCommGroup.cauchy_series_of_le_geometric' | [] | [
427,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
425,
1
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Mathlib/Combinatorics/Composition.lean | Composition.eq_ones_iff_length | [
{
"state_after": "case mp\nn : ℕ\nc✝ c : Composition n\n⊢ c = ones n → length c = n\n\ncase mpr\nn : ℕ\nc✝ c : Composition n\n⊢ length c = n → c = ones n",
"state_before": "n : ℕ\nc✝ c : Composition n\n⊢ c = ones n ↔ length c = n",
"tactic": "constructor"
},
{
"state_after": "case mp\nn : ℕ\nc :... | [
562,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
545,
1
] |
Mathlib/Topology/Basic.lean | IsOpen.preimage | [] | [
1572,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1570,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean | Nat.pos_of_mem_factorization | [] | [
145,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Data/Real/ENNReal.lean | Set.OrdConnected.preimage_coe_nnreal_ennreal | [] | [
2508,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2507,
1
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Mathlib/Data/Prod/Basic.lean | Prod.fst_eq_iff | [
{
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"tactic": "simp"
}
] | [
218,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
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Mathlib/Analysis/Convex/Star.lean | StarConvex.add_smul_mem | [
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.100004\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 x s\nhy : x + y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\nh : x + t • y = (1 - t) • x + ... | [
340,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
335,
1
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Mathlib/Data/Num/Lemmas.lean | PosNum.cast_to_num | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.355613\nn : PosNum\n⊢ ↑n = pos n",
"tactic": "rw [← cast_to_nat, ← of_to_nat n]"
}
] | [
655,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
655,
1
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Mathlib/RingTheory/Multiplicity.lean | multiplicity.ne_top_iff_finite | [
{
"state_after": "no goals",
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"tactic": "rw [Ne.def, eq_top_iff_not_finite, Classical.not_not]"
}
] | [
218,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
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Mathlib/Data/List/Range.lean | List.length_finRange | [
{
"state_after": "no goals",
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"tactic": "rw [finRange, length_pmap, length_range]"
}
] | [
152,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
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Mathlib/Analysis/InnerProductSpace/Basic.lean | norm_inner_eq_norm_tfae | [
{
"state_after": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x... | [
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14
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1589,
1
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Mathlib/Algebra/GroupWithZero/Basic.lean | mul_self_mul_inv | [
{
"state_after": "case pos\nα : Type ?u.24169\nM₀ : Type ?u.24172\nG₀ : Type u_1\nM₀' : Type ?u.24178\nG₀' : Type ?u.24181\nF : Type ?u.24184\nF' : Type ?u.24187\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : a = 0\n⊢ a * a * a⁻¹ = a\n\ncase neg\nα : Type ?u.24169\nM₀ : Type ?u.24172\nG₀ : Type u_1\nM₀' : Type ?... | [
348,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
345,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean | Metric.hausdorffDist_empty' | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.75311\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ hausdorffDist ∅ s = 0",
"tactic": "simp [hausdorffDist_comm]"
}
] | [
740,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
740,
1
] |
Mathlib/Order/Heyting/Basic.lean | inf_sdiff_left | [] | [
549,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
548,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.NullMeasurableSet.mono_ac | [] | [
2665,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2663,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | CategoryTheory.Limits.Cofork.ofCocone_ι | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf g : X ⟶ Y\nF : WalkingParallelPair ⥤ C\nt : Cocone F\nj : WalkingParallelPair\n⊢ (parallelPair (F.map left) (F.map right)).obj j = F.obj j",
"tactic": "aesop"
}
] | [
659,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
658,
1
] |
Mathlib/Logic/Basic.lean | exists_unique_const | [
{
"state_after": "no goals",
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"tactic": "simp"
}
] | [
743,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
742,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.dvd_refl | [
{
"state_after": "no goals",
"state_before": "a : Nat\n⊢ a = a * 1",
"tactic": "simp"
}
] | [
651,
61
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
651,
11
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Mathlib/Computability/Partrec.lean | Computable.vector_head | [] | [
382,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
381,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean | EMetric.tendsto_nhds | [] | [
725,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
723,
1
] |
Mathlib/Data/Num/Lemmas.lean | Num.mul_to_nat | [] | [
321,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.sign_pi_sub | [
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ sign (↑π - θ) = sign θ",
"tactic": "simp [sign_antiperiodic.sub_eq']"
}
] | [
890,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
889,
1
] |
Mathlib/Topology/Separation.lean | t2_iff_ultrafilter | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : TopologicalSpace α\n⊢ (∀ {x y : α}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y) ↔ ∀ {x y : α} (f : Ultrafilter α), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y",
"tactic": "simp only [← exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp]"
}
] | [
948,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
946,
1
] |
Mathlib/Algebra/BigOperators/Pi.lean | prod_mk_prod | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns : Finset γ\nf : γ → α\ng : γ → β\nthis : DecidableEq γ\n⊢ ∀ ⦃a : γ⦄ {s : Finset γ},\n ¬a ∈ s →\n (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) →\n (∏ ... | [
70,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
67,
1
] |
Std/Data/Int/Lemmas.lean | Int.le_of_neg_le_neg | [
{
"state_after": "a b : Int\nh : -b ≤ -a\nthis : a ≤ b\n⊢ a ≤ b",
"state_before": "a b : Int\nh : -b ≤ -a\nthis : - -a ≤ - -b\n⊢ a ≤ b",
"tactic": "simp [Int.neg_neg] at this"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh : -b ≤ -a\nthis : a ≤ b\n⊢ a ≤ b",
"tactic": "assu... | [
863,
19
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
861,
11
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.toNNReal_sInf | [
{
"state_after": "α : Type ?u.839566\nβ : Type ?u.839569\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nι : Sort ?u.839586\nf g : ι → ℝ≥0∞\ns : Set ℝ≥0∞\nhs : ∀ (r : ℝ≥0∞), r ∈ s → r ≠ ⊤\nhf : ∀ (i : { x // x ∈ s }), ↑i ≠ ⊤\n⊢ ENNReal.toNNReal (sInf s) = sInf (ENNReal.toNNReal '' s)",
"state_before": "α : Type ?u.839566\nβ ... | [
2360,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2355,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | csSup_insert | [] | [
708,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
707,
1
] |
Mathlib/MeasureTheory/Constructions/Polish.lean | IsClosed.analyticSet | [
{
"state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type ?u.62127\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed s\nthis : PolishSpace ↑s\n⊢ AnalyticSet s",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type ?u.62127\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed s\n⊢ Anal... | [
216,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
212,
1
] |
Mathlib/Algebra/Support.lean | Function.mulSupport_eq_preimage | [] | [
52,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tsum_neg | [
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.442177\nδ : Type ?u.442180\ninst✝³ : AddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\ninst✝ : T2Space α\nhf : Summable f\n⊢ (∑' (b : β), -f b) = -∑' (b : β), f b\n\ncase neg\nα : Type u_1\nβ... | [
907,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
904,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean | Nat.pow_succ_le_ascFactorial | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (n + 1) ^ 0 ≤ ascFactorial n 0",
"tactic": "rw [ascFactorial_zero, pow_zero]"
},
{
"state_after": "n k : ℕ\n⊢ (n + 1) * (n + 1) ^ k ≤ ascFactorial n (k + 1)",
"state_before": "n k : ℕ\n⊢ (n + 1) ^ (k + 1) ≤ ascFactorial n (k + 1)",
"... | [
289,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
285,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean | Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.28643\ninst✝ : Semiring R\nn : ℕ\np : R[X]\nhp : p ∈ degreeLT R n\n⊢ ↑(degreeLTEquiv R n) { val := p, property := hp } = 0 ↔ p = 0",
"tactic": "rw [LinearEquiv.map_eq_zero_iff, Submodule.mk_eq_zero]"
}
] | [
177,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
] |
Mathlib/Algebra/Invertible.lean | invOf_mul_eq_iff_eq_mul_left | [
{
"state_after": "no goals",
"state_before": "α : Type u\nc a b : α\ninst✝¹ : Monoid α\ninst✝ : Invertible c\n⊢ ⅟c * a = b ↔ a = c * b",
"tactic": "rw [← mul_left_inj_of_invertible (c := c), mul_invOf_self_assoc]"
}
] | [
309,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.lf_of_lt_of_lf | [] | [
583,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
582,
1
] |
Mathlib/Data/List/Perm.lean | List.exists_perm_sublist | [
{
"state_after": "no goals",
"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' l₁ : List α\ns : l₁ <+ []\n⊢ ∃ l₁' x, l₁' <+ []",
"tactic": "exact ⟨[], eq_nil_of_sublist_nil s ▸ Perm.refl _, nil_sublist _⟩"
},
{
"state_after": "case cons.cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹... | [
313,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
290,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | WithBot.coe_sSup' | [] | [
142,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
140,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean | IsometryEquiv.eq_symm_apply | [] | [
441,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
440,
1
] |
Mathlib/Computability/TuringMachine.lean | Turing.TM2to1.tr_respects | [
{
"state_after": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nc₁ : Cfg₂\nc₂ : TM1.Cfg Γ' Λ' σ\nh : TrCfg c₁ c₂\n⊢ match TM2.step M c₁ with\n | some b₁ => ∃ b₂, TrCfg b₁ b₂ ∧ Reaches₁ (TM1.step (tr M)) c₂ b₂\n | n... | [
2723,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2704,
1
] |
Mathlib/Topology/Order.lean | continuous_iSup_rng | [] | [
769,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
767,
1
] |
Mathlib/MeasureTheory/Function/L2Space.lean | MeasureTheory.L2.inner_indicatorConstLp_eq_inner_set_integral | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.159069\n𝕜 : Type u_3\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ns : Set α\ninst✝¹ : CompleteSpace E\ninst... | [
270,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean | Ordinal.type_ne_zero_of_nonempty | [] | [
239,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
238,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_mul_const_Icc | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a b c : α\nh : 0 < c\n⊢ (fun x => x * c) ⁻¹' Icc a b = Icc (a / c) (b / c)",
"tactic": "simp [← Ici_inter_Iic, h]"
}
] | [
551,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
550,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean | PadicInt.zmod_congr_of_sub_mem_span | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑p ^ n}\nhb : x - ↑b ∈ Ideal.span {↑p ^ n}\n⊢ ↑a = ↑b",
"tactic": "simpa using zmod_congr_of_sub_mem_span_aux n x a b ha hb"
}
] | [
158,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.inter_empty_of_inter_sUnion_empty | [
{
"state_after": "α : Type u_1\nβ : Type ?u.148065\nγ : Type ?u.148068\nι : Sort ?u.148071\nι' : Sort ?u.148074\nι₂ : Sort ?u.148077\nκ : ι → Sort ?u.148082\nκ₁ : ι → Sort ?u.148087\nκ₂ : ι → Sort ?u.148092\nκ' : ι' → Sort ?u.148097\ns t : Set α\nS : Set (Set α)\nhs : t ∈ S\nh : s ∩ ⋃₀ S = ∅\n⊢ s ∩ t ⊆ s ∩ ⋃₀ S... | [
1260,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1257,
1
] |
Mathlib/Data/Real/Sqrt.lean | Real.sqrt_eq_zero' | [
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ sqrt x = 0 ↔ x ≤ 0",
"tactic": "rw [sqrt, NNReal.coe_eq_zero, NNReal.sqrt_eq_zero, Real.toNNReal_eq_zero]"
}
] | [
341,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.subtypeDomain_zero | [] | [
498,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
496,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Real.sin_add_pi | [] | [
239,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
238,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean | Submodule.isHilbertSumOrthogonal | [
{
"state_after": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[... | [
399,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
390,
1
] |
Mathlib/Algebra/GroupWithZero/Divisibility.lean | eq_of_forall_dvd' | [] | [
140,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean | sInfHom.top_apply | [] | [
521,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
520,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean | tsub_pos_iff_not_le | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\n⊢ 0 < a - b ↔ ¬a ≤ b",
"tactic": "rw [pos_iff_ne_zero, Ne.def, tsub_eq_zero_iff_le]"
}
] | [
353,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
] |
Mathlib/Topology/Order/Basic.lean | eventually_gt_of_tendsto_gt | [] | [
367,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
365,
1
] |
Mathlib/RingTheory/IsTensorProduct.lean | IsTensorProduct.map_eq | [
{
"state_after": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.102460\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R... | [
122,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
119,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean | Ideal.span_singleton_one | [] | [
163,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
] |
Mathlib/Data/Set/Function.lean | Set.piecewise_compl | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.83376\nγ : Type ?u.83379\nι : Sort ?u.83382\nπ : α → Type ?u.83387\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : (i : α) → Decidable (i ∈ sᶜ)\nx : α\nhx : x ∈ s\n⊢ piecewise (sᶜ) f g x ... | [
1434,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1433,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | InnerProductGeometry.angle_neg_right | [
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\n⊢ arccos (inner x (-y) / (‖x‖ * ‖-y‖)) = π - arccos (inner x y / (‖x‖ * ‖y‖))",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\n⊢ angle x (... | [
110,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/Algebra/Order/Kleene.lean | Prod.kstar_def | [] | [
312,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean | Measurable.nullMeasurable | [] | [
422,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
421,
11
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean | Matrix.Pivot.listTransvecCol_mul_last_row_drop | [
{
"state_after": "case refine'_1\nn : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\n⊢ ∀ (k_1 : ℕ),\n k_1 < r →\n ... | [
369,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
359,
1
] |
Mathlib/Order/BoundedOrder.lean | max_bot_right | [] | [
855,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
854,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | Matrix.isUnit_diagonal | [
{
"state_after": "no goals",
"state_before": "l : Type ?u.338640\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nv : n → α\n⊢ IsUnit (diagonal v) ↔ IsUnit v",
"tactic": "simp only [← nonempty_invertible_iff_isUnit,\n (diagonalInv... | [
565,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
563,
1
] |
Mathlib/Algebra/Ring/Commute.lean | Commute.mul_self_sub_mul_self_eq' | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonUnitalNonAssocRing R\na b : R\nh : Commute a b\n⊢ a * a - b * b = (a - b) * (a + b)",
"tactic": "rw [mul_add, sub_mul, sub_mul, h.eq, sub_add_sub_cancel]"
}
] | [
81,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
] |
Mathlib/Topology/Algebra/Monoid.lean | continuous_one | [] | [
36,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
35,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.eq_inv_of_mul_eq_one_left | [
{
"state_after": "α : Type ?u.297877\nβ : Type ?u.297880\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a * b = 1\n⊢ b ≠ ⊤",
"state_before": "α : Type ?u.297877\nβ : Type ?u.297880\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a * b = 1\n⊢ a = b⁻¹",
"tactic": "rw [← mul_one a, ← ENNReal.mul_inv_cancel (right_ne_zero_of_mul_eq_o... | [
1653,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1649,
11
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean | Matrix.TransvectionStruct.det_toMatrix_prod | [
{
"state_after": "case nil\nn : Type u_1\np : Type ?u.23948\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\n⊢ det (List.prod (List.map toMatrix [])) = 1\n\ncase cons\nn : Type u_1\np : Type ?u.23948\nR : Type u₂\n𝕜... | [
192,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/Tactic/Linarith/Lemmas.lean | Linarith.lt_of_eq_of_lt | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedSemiring α\na b : α\nha : a = 0\nhb : b < 0\n⊢ a + b < 0",
"tactic": "simp [*]"
}
] | [
34,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
33,
1
] |
Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.get?_add | [] | [
747,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
746,
1
] |
Mathlib/Logic/Lemmas.lean | ite_dite_distrib_right | [] | [
53,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean | HolderWith.ediam_image_le | [] | [
227,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
224,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.X_eq | [] | [
1425,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1424,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | Matrix.det_smul_inv_mulVec_eq_cramer | [
{
"state_after": "no goals",
"state_before": "l : Type ?u.360583\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\nb : n → α\nh : IsUnit (det A)\n⊢ det A • mulVec A⁻¹ b = ↑(cramer A) b",
"tactic": "rw [cramer_eq_adjugate_mulVec, ... | [
607,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
604,
1
] |
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