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_Ico | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ↑e ⁻¹' Ico a b = Ico (↑(symm e) a) (↑(symm e) b)",
"tactic": "simp [← Ici_inter_Iio]"
}
] | [
56,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.restrict_singleton' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.710215\nγ : Type ?u.710218\nδ : Type ?u.710221\nι : Type ?u.710224\nR : Type ?u.710227\nR' : Type ?u.710230\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set... | [
3290,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3289,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.tendsto_nhds_nhds | [] | [
1049,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1047,
1
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Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.div_add_mod | [] | [
1048,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1047,
1
] |
Mathlib/Logic/Equiv/Fin.lean | finSuccEquivLast_castSucc | [] | [
274,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
] |
Mathlib/Data/Set/Prod.lean | Set.snd_image_prod_subset | [] | [
367,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
366,
1
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Mathlib/Analysis/Convex/Segment.lean | openSegment_subset_iff_segment_subset | [
{
"state_after": "no goals",
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163,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
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Mathlib/Topology/Basic.lean | Continuous.iterate | [] | [
1647,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1646,
1
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Mathlib/Order/Atoms.lean | Set.isAtom_singleton | [] | [
889,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
888,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ici_subset_Icc_union_Ici | [] | [
1334,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1333,
1
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Mathlib/Algebra/Hom/Iterate.lean | mul_right_iterate_apply_one | [
{
"state_after": "no goals",
"state_before": "M : Type ?u.67258\nN : Type ?u.67261\nG : Type u_1\nH : Type ?u.67267\ninst✝ : Monoid G\na : G\nn : ℕ\n⊢ ((fun x => x * a)^[n]) 1 = a ^ n",
"tactic": "simp [mul_right_iterate]"
}
] | [
199,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
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Mathlib/GroupTheory/Perm/Support.lean | Equiv.Perm.support_swap_mul_ge_support_diff | [
{
"state_after": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx y a✝ : α\n⊢ a✝ ∈ support f \\ {x, y} → a✝ ∈ support (swap x y * f)",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx y : α\n⊢ support f \\ {x, y} ≤ support (swap x y * f... | [
481,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
473,
1
] |
Mathlib/Data/Vector/Basic.lean | Vector.get_ofFn | [
{
"state_after": "case mk\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nf : Fin n → α\ni : ℕ\nhi : i < n\n⊢ get (ofFn f) { val := i, isLt := hi } = f { val := i, isLt := hi }",
"state_before": "n✝ : ℕ\nα : Type u_1\nn : ℕ\nf : Fin n → α\ni : Fin n\n⊢ get (ofFn f) i = f i",
"tactic": "cases' i with i hi"
},
{
"s... | [
144,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
140,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isLittleO_zero_right_iff | [] | [
1238,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1236,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean | closedBall_sub_ball | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.1341112\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y z : E\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 < δ\na b : E\n⊢ Metric.closedBall a ε - Metric.ball b δ = Metric.ball (a - ... | [
375,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
373,
1
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Mathlib/Order/CompleteLattice.lean | iSup_ne_bot_subtype | [
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.174146\nβ₂ : Type ?u.174149\nγ : Type ?u.174152\nι : Sort u_2\nι' : Sort ?u.174158\nκ : ι → Sort ?u.174163\nκ' : ι' → Sort ?u.174168\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : ι → α\nhtriv : ∀ (i : ι), f i = ⊥\n⊢ (⨆ (i : { i // f i ≠ ⊥ }), f ... | [
1603,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1595,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.lsub_le_sup_succ | [] | [
1614,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1612,
1
] |
Mathlib/Data/Multiset/Nodup.lean | Multiset.nodup_add | [] | [
119,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
118,
1
] |
Mathlib/Data/Finset/Option.lean | Finset.eraseNone_image_some | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.10620\ninst✝ : DecidableEq (Option α)\ns : Finset α\n⊢ ↑eraseNone (image some s) = s",
"tactic": "simpa only [map_eq_image] using eraseNone_map_some s"
}
] | [
111,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.cos_add_aux | [
{
"state_after": "no goals",
"state_before": "x y a b c d : ℂ\n⊢ (a + b) * (c + d) - (b - a) * (d - c) * -1 = 2 * (a * c + b * d)",
"tactic": "ring"
}
] | [
861,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
860,
9
] |
Mathlib/CategoryTheory/Abelian/Basic.lean | CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.hasImages | [] | [
186,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
185,
1
] |
Mathlib/Algebra/Module/Basic.lean | Module.ext' | [
{
"state_after": "case smul.h.h\nα : Type ?u.96211\nR✝ : Type ?u.96214\nk : Type ?u.96217\nS : Type ?u.96220\nM✝ : Type ?u.96223\nM₂ : Type ?u.96226\nM₃ : Type ?u.96229\nι : Type ?u.96232\nR : Type u_1\ninst✝¹ : Semiring R\nM : Type u_2\ninst✝ : AddCommMonoid M\nP Q : Module R M\nw : ∀ (r : R) (m : M), r • m = ... | [
293,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
] |
Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.get?_terminates_le | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\nm n : ℕ\nh : m ≤ n\n⊢ Terminates (get? s n) → Terminates (get? s m)",
"tactic": "induction' h with m' _ IH <;> [exact id;\n exact fun T => IH (@head_terminates_of_head_tail_terminates _ _ T)]"
}
] | [
897,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
894,
1
] |
Mathlib/LinearAlgebra/Eigenspace/Basic.lean | Module.End.mem_generalizedEigenspace | [] | [
265,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
264,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.inf_biUnion | [] | [
381,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
379,
1
] |
Mathlib/Topology/Instances/Sign.lean | continuousAt_sign_of_neg | [
{
"state_after": "α : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α\ninst✝² : PartialOrder α\ninst✝¹ : DecidableRel fun x x_1 => x < x_1\ninst✝ : OrderTopology α\na : α\nh : a < 0\n⊢ (fun x => -1) =ᶠ[nhds a] ↑SignType.sign",
"state_before": "α : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace ... | [
44,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.left_eq_union_iff_subset | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.151221\nγ : Type ?u.151224\ninst✝ : DecidableEq α\ns✝ s₁ s₂ t✝ t₁ t₂ u v : Finset α\na b : α\ns t : Finset α\n⊢ s = s ∪ t ↔ t ⊆ s",
"tactic": "rw [← union_eq_left_iff_subset, eq_comm]"
}
] | [
1471,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1470,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.count_le_card | [] | [
2363,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2362,
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] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.induce_mono_left | [] | [
1160,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1159,
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] |
Mathlib/LinearAlgebra/Dual.lean | Subspace.isCompl_dualAnnihilator | [
{
"state_after": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nh : W ⊓ W' = ⊥ ∧ W ⊔ W' = ⊤\n⊢ dualAnnihilator W ⊓ dualAnnihilator W' = ⊥ ∧ dualAnnihilator W ⊔ dualAnnihilator W' = ⊤... | [
1458,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1453,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.ConnectedComponent.ind | [] | [
1988,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1986,
11
] |
Std/Data/Array/Lemmas.lean | getElem?_pos | [] | [
24,
89
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
23,
1
] |
Mathlib/Algebra/Field/Basic.lean | inv_neg_one | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.37165\nβ : Type ?u.37168\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na b : K\n⊢ (-1)⁻¹ = -1",
"tactic": "rw [← neg_inv, inv_one]"
}
] | [
136,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
] |
Mathlib/Data/Set/NAry.lean | Set.image2_image2_image2_comm | [
{
"state_after": "case h\nα : Type u_4\nα' : Type ?u.42153\nβ : Type u_5\nβ' : Type ?u.42159\nγ : Type u_6\nγ' : Type ?u.42165\nδ : Type u_7\nδ' : Type ?u.42171\nε : Type u_2\nε' : Type u_8\nζ : Type u_3\nζ' : Type u_9\nν : Type u_1\nf✝ f'✝ : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' :... | [
342,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
334,
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Mathlib/Probability/CondCount.lean | ProbabilityTheory.condCount_of_univ | [] | [
137,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
] |
Mathlib/Topology/Basic.lean | interior_inter | [] | [
353,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
348,
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] |
Mathlib/Data/Polynomial/Basic.lean | Polynomial.coeff_zero | [] | [
680,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
679,
1
] |
Mathlib/Data/Set/Basic.lean | Set.diff_inter_self | [] | [
2030,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2029,
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] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean | tendstoUniformlyOn_iff_tendstoUniformlyOnFilter | [
{
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115,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | smul_mem_nhds | [
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\na : α\nx : β\nht : t ∈ 𝓝 x\nu : Set β\nut : u ⊆ t\nu_open : IsOpen u\nhu : x ∈ u\n⊢ a • t ∈ 𝓝 ... | [
1241,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1239,
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Mathlib/Topology/Constructions.lean | isOpen_range_inr | [] | [
930,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
929,
1
] |
Mathlib/FieldTheory/Subfield.lean | Subfield.mem_closure | [] | [
714,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
709,
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Mathlib/CategoryTheory/Elements.lean | CategoryTheory.CategoryOfElements.fromCostructuredArrow_obj_mk | [] | [
221,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
] |
Mathlib/LinearAlgebra/Alternating.lean | AlternatingMap.map_coord_zero | [] | [
220,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
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] |
Std/Data/Option/Lemmas.lean | Option.ext | [] | [
46,
32
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
43,
8
] |
Mathlib/Algebra/Lie/Nilpotent.lean | LieSubmodule.ucs_add | [] | [
373,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
372,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.sub_eq_sub_min | [
{
"state_after": "n m : Nat\n⊢ n - m = n - if n ≤ m then n else m",
"state_before": "n m : Nat\n⊢ n - m = n - min n m",
"tactic": "rw [Nat.min_def]"
},
{
"state_after": "case inl\nn m : Nat\nh✝ : n ≤ m\n⊢ n - m = n - n\n\ncase inr\nn m : Nat\nh✝ : ¬n ≤ m\n⊢ n - m = n - m",
"state_before": "n... | [
223,
8
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
220,
1
] |
Mathlib/Topology/MetricSpace/Contracting.lean | ContractingWith.toLipschitzWith | [] | [
51,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | Matrix.toLinearMap₂_basisFun | [
{
"state_after": "case h.h.h.h.h\nR : Type u_1\nR₁ : Type ?u.1404273\nR₂ : Type ?u.1404276\nM✝ : Type ?u.1404279\nM₁ : Type ?u.1404282\nM₂ : Type ?u.1404285\nM₁' : Type ?u.1404288\nM₂' : Type ?u.1404291\nn : Type u_2\nm : Type u_3\nn' : Type ?u.1404300\nm' : Type ?u.1404303\nι : Type ?u.1404306\ninst✝⁸ : CommRi... | [
411,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
407,
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] |
Mathlib/Topology/MetricSpace/Infsep.lean | Set.einfsep_insert_le | [
{
"state_after": "α : Type u_1\nβ : Type ?u.24010\ninst✝ : EDist α\nx y : α\ns t : Set α\n⊢ ∀ (i : α), i ∈ s → x ≠ i → einfsep (insert x s) ≤ edist x i",
"state_before": "α : Type u_1\nβ : Type ?u.24010\ninst✝ : EDist α\nx y : α\ns t : Set α\n⊢ einfsep (insert x s) ≤ ⨅ (y : α) (_ : y ∈ s) (_ : x ≠ y), edist... | [
150,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/LinearAlgebra/Basic.lean | Submodule.comap_iInf_map_of_injective | [] | [
970,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
968,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | Filter.HasBasis.mem_uniformity_iff | [
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.93034\ninst✝ : UniformSpace α\np : β → Prop\ns : β → Set (α × α)\nh : HasBasis (𝓤 α) p s\nt : Set (α × α)\n⊢ (∃ i, p i ∧ s i ⊆ t) ↔ ∃ i, p i ∧ ∀ (a b : α), (a, b) ∈ s i → (a, b) ∈ t",
"tactic":... | [
1042,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1039,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigO_iff_isBoundedUnder_le_div | [
{
"state_after": "α : Type u_1\nβ : Type ?u.313491\nE : Type u_3\nF : Type ?u.313497\nG : Type ?u.313500\nE' : Type ?u.313503\nF' : Type ?u.313506\nG' : Type ?u.313509\nE'' : Type ?u.313512\nF'' : Type u_2\nG'' : Type ?u.313518\nR : Type ?u.313521\nR' : Type ?u.313524\n𝕜 : Type ?u.313527\n𝕜' : Type ?u.313530\... | [
1404,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1399,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean | Cubic.monic_of_a_eq_one' | [] | [
250,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean | integral_pair | [] | [
1203,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1200,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.singleton_div | [] | [
625,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
624,
1
] |
Mathlib/Init/Classical.lean | Classical.eq_false_or_eq_true | [] | [
37,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
37,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | Real.arcsin_sin' | [
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ∈ Icc (-(π / 2)) (π / 2)\n⊢ sin (arcsin (sin x)) = sin x",
"tactic": "rw [sin_arcsin (neg_one_le_sin _) (sin_le_one _)]"
}
] | [
74,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Algebra/Associated.lean | Prime.not_square | [] | [
361,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
360,
1
] |
Mathlib/Data/Stream/Init.lean | Stream'.head_even | [] | [
466,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
465,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean | LinearMap.ker_lsmul | [] | [
424,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
422,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | Subring.closure_eq_of_le | [] | [
924,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
922,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | AffineEquiv.constVAdd_nsmul | [] | [
514,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
513,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.image_const_mul_uIcc | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a b c : α\n⊢ (fun x => a * x) '' [[b, c]] = [[a * b, a * c]]",
"tactic": "simpa only [mul_comm] using image_mul_const_uIcc a b c"
}
] | [
721,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
720,
1
] |
Mathlib/Data/List/Pairwise.lean | List.pairwise_and_iff | [
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.2465\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nx✝¹ : Pairwise R l✝ ∧ Pairwise S l✝\nhS✝ : Pairwise S l✝\na : α\nl : List α\nR1 : ∀ (a' : α), a' ∈ l → R a a'\nR2 : Pairwise R l\nIH : Pairwise R l ∧ Pairwise S l → Pairwise S l → Pairwise (fun a b => R a b ∧... | [
82,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.toNNReal_prod | [] | [
2228,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2226,
1
] |
Mathlib/CategoryTheory/GlueData.lean | CategoryTheory.GlueData.ι_jointly_surjective | [
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nC' : Type u₂\ninst✝⁴ : Category C'\nD : GlueData C\nF✝ : C ⥤ C'\nH : (i j k : D.J) → PreservesLimit (cospan (f D i j) (f D i k)) F✝\ninst✝³ : HasMulticoequalizer (diagram D)\ninst✝² : PreservesColimit (MultispanIndex.multispan (diagram D)) F✝\nF : C ⥤ Type v\n... | [
416,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.Term.realize_constantsToVars | [
{
"state_after": "case var\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.15741\nP : Type ?u.15744\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nv : β → M\n_a✝ : β\n⊢ ... | [
178,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean | Function.Surjective.lieModule_lcs_map_eq | [
{
"state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_2\nM₂ : Type u_1\ninst✝⁵ : LieRing ... | [
480,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
456,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | affineSpan_insert_eq_affineSpan | [
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.491936\np : P\nps : Set P\nh : p ∈ ↑(affineSpan k ps)\n⊢ affineSpan k (insert p ps) = affineSpan k ps",
"state_before": "k : Type u_2\nV : Type u_3\... | [
1393,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1390,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | TopCat.Presheaf.coveringOfPresieve_apply | [] | [
51,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
] |
Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | mem_parallelepiped_iff | [
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nι' : Type ?u.4165\nE : Type u_1\nF : Type ?u.4171\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : AddCommGroup F\ninst✝ : Module ℝ F\nv : ι → E\nx : E\n⊢ x ∈ parallelepiped v ↔ ∃ t _ht, x = ∑ i : ι, ... | [
53,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
] |
Mathlib/Topology/Order/Basic.lean | nhdsWithin_Iio_eq_bot_iff | [
{
"state_after": "case h.e'_2.h.e'_2.h.e'_2.h.h.a\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ne_1✝ : α = (fun x => αᵒᵈ) a\nx✝ : α\n⊢ x✝ ⋖ a ↔ ↑toDual a ⋖ x✝",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Topologic... | [
1723,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1720,
1
] |
Mathlib/RingTheory/Artinian.lean | isArtinian_of_fg_of_artinian' | [] | [
361,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
358,
1
] |
Mathlib/Order/SuccPred/Limit.lean | Order.not_isPredLimit_iff_exists_covby | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LT α\na✝ a : α\n⊢ ¬IsPredLimit a ↔ ∃ b, a ⋖ b",
"tactic": "simp [IsPredLimit]"
}
] | [
252,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
251,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.nfpBFamily_le_iff | [] | [
284,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Data/Sym/Sym2.lean | Sym2.eq_swap | [
{
"state_after": "α : Type u_1\nβ : Type ?u.4046\nγ : Type ?u.4049\na b : α\n⊢ (a, b) ≈ (b, a)",
"state_before": "α : Type u_1\nβ : Type ?u.4046\nγ : Type ?u.4049\na b : α\n⊢ Quotient.mk (Rel.setoid α) (a, b) = Quotient.mk (Rel.setoid α) (b, a)",
"tactic": "rw [Quotient.eq]"
},
{
"state_after": ... | [
145,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
1
] |
Std/Data/Int/Lemmas.lean | Int.negSucc_eq | [] | [
74,
64
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
74,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean | CategoryTheory.Limits.limMap_π | [] | [
212,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
] |
Mathlib/Analysis/Convex/Function.lean | LinearOrder.concaveOn_of_lt | [] | [
434,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
430,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean | Antitone.mul' | [] | [
1358,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1356,
1
] |
Mathlib/Data/Matrix/DMatrix.lean | DMatrix.map_sub | [
{
"state_after": "case a\nl : Type ?u.28777\nm : Type u_1\nn : Type u_2\no : Type ?u.28786\ninst✝⁵ : Fintype l\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype o\nα : m → n → Type v\ninst✝¹ : (i : m) → (j : n) → AddGroup (α i j)\nβ : m → n → Type w\ninst✝ : (i : m) → (j : n) → AddGroup (β i j)\nf : ⦃i ... | [
159,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/Algebra/Regular/Pow.lean | IsLeftRegular.pow_iff | [
{
"state_after": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nn0 : 0 < n\n⊢ IsLeftRegular (a ^ n) → IsLeftRegular a",
"state_before": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nn0 : 0 < n\n⊢ IsLeftRegular (a ^ n) ↔ IsLeftRegular a",
"tactic": "refine' ⟨_, IsLeftRegular.pow n⟩"
},
{
"st... | [
52,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
49,
1
] |
Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | AlgebraicTopology.DoldKan.HigherFacesVanish.on_Γ₀_summand_id | [
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK✝ K' : ChainComplex C ℕ\nf : K✝ ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\nj : Fin (n + 1)\na✝ : n + 1 ≤ ↑j + (n + 1)\n⊢ Splitting.ιSummand (Γ₀.splitting K) (Splitting.IndexSet.id [n ... | [
355,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
347,
1
] |
Mathlib/Topology/Basic.lean | DenseRange.exists_mem_open | [] | [
1868,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1866,
8
] |
Mathlib/Deprecated/Subgroup.lean | IsGroupHom.preimage | [
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type u_2\nA : Type ?u.90350\na a₁ a₂ b c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\ns : Set H\nhs : IsSubgroup s\n⊢ IsSubgroup (f ⁻¹' s)",
"tactic": "refine' { .. } <;>\n simp (config := { contextual := true }) [... | [
429,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
425,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean | CategoryTheory.Monad.Algebra.id_f | [] | [
129,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean | Real.tendsto_logb_atTop_of_base_lt_one | [
{
"state_after": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\n⊢ ∀ (b_1 : ℝ), ∃ i, ∀ (a : ℝ), i ≤ a → logb b a ≤ b_1",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\n⊢ Tendsto (logb b) atTop atBot",
"tactic": "rw [tendsto_atTop_atBot]"
},
{
"state_after": "b x y : ℝ\nb_pos : 0 < b\n... | [
351,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
342,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.restrict_univ | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.293051\nγ : Type ?u.293054\nδ : Type ?u.293057\nι : Type ?u.293060\nR : Type ?u.293063\nR' : Type ?u.293066\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t s : S... | [
1691,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1690,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean | mul_smul_one | [
{
"state_after": "no goals",
"state_before": "M✝ : Type ?u.27981\nN✝ : Type ?u.27984\nG : Type ?u.27987\nA : Type ?u.27990\nB : Type ?u.27993\nα : Type ?u.27996\nβ : Type ?u.27999\nγ : Type ?u.28002\nδ : Type ?u.28005\nM : Type u_1\nN : Type u_2\ninst✝² : MulOneClass N\ninst✝¹ : SMul M N\ninst✝ : SMulCommCl... | [
655,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
654,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean | PadicSeq.norm_nonarchimedean | [
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f + g ≈ 0\nthis : 0 ≤ max (norm f) (norm g)\n⊢ norm (f + g) ≤ max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f + g ≈ 0\n⊢ norm (f + g) ≤ max (norm f) (norm g)",
"tactic": "have : 0... | [
401,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
379,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | le_self_pow | [] | [
455,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
] |
Std/Data/Int/Lemmas.lean | Int.mul_nonpos_of_nonneg_of_nonpos | [
{
"state_after": "a b : Int\nha : 0 ≤ a\nhb : b ≤ 0\nh : a * b ≤ a * 0\n⊢ a * b ≤ 0",
"state_before": "a b : Int\nha : 0 ≤ a\nhb : b ≤ 0\n⊢ a * b ≤ 0",
"tactic": "have h : a * b ≤ a * 0 := Int.mul_le_mul_of_nonneg_left hb ha"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nha : 0... | [
1181,
26
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1178,
11
] |
Mathlib/SetTheory/ZFC/Basic.lean | PSet.eta | [] | [
147,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.normSq_mk | [] | [
594,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
593,
1
] |
Mathlib/Data/Nat/Choose/Multinomial.lean | Multiset.multinomial_filter_ne | [
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nm : Multiset α\n⊢ Finsupp.multinomial (↑toFinsupp m) =\n Nat.choose (↑card m) (count a m) * Finsupp.multinomial (↑toFinsupp (filter (fun x => a ≠ x) m))",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nm : Multiset α\n⊢ multinomi... | [
219,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
] |
Mathlib/Topology/Sheaves/Limits.lean | TopCat.isSheaf_of_isLimit | [
{
"state_after": "C : Type u\ninst✝² : Category C\nJ : Type v\ninst✝¹ : SmallCategory J\ninst✝ : HasLimits C\nX : TopCat\nF : J ⥤ Presheaf C X\nH : ∀ (j : J), Presheaf.IsSheaf (F.obj j)\nc : Cone F\nhc : IsLimit c\nF' : J ⥤ Sheaf C X :=\n CategoryTheory.Functor.mk\n { obj := fun j => { val := F.obj j, cond ... | [
52,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
44,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean | CategoryTheory.Sieve.functorPushforward_monotone | [] | [
676,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
674,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.leRecOn_succ' | [
{
"state_after": "no goals",
"state_before": "m n✝ k : ℕ\nC : ℕ → Sort u\nn : ℕ\nh : n ≤ n + 1\nnext : {k : ℕ} → C k → C (k + 1)\nx : C n\n⊢ leRecOn h (fun {k} => next) x = next x",
"tactic": "rw [leRecOn_succ (le_refl n), leRecOn_self]"
}
] | [
416,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
414,
1
] |
Mathlib/CategoryTheory/Arrow.lean | CategoryTheory.Arrow.inv_right | [
{
"state_after": "no goals",
"state_before": "T : Type u\ninst✝¹ : Category T\nf g : Arrow T\nsq : f ⟶ g\ninst✝ : IsIso sq\n⊢ sq.right ≫ (inv sq).right = 𝟙 f.right",
"tactic": "rw [← Comma.comp_right, IsIso.hom_inv_id, id_right]"
}
] | [
212,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
] |
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