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/HausdorffDistance.lean | IsClosed.cthickening_eq_biUnion_closedBall | [
{
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Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | MeasureTheory.Measure.add_haar_closedBall_center | [
{
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"state_before": "E✝ : Type ?u.1969525\ninst✝¹² : NormedAddCommGroup E✝\ninst✝¹¹ : NormedSpace ℝ E✝\ninst✝¹⁰ : MeasurableSpace E✝\ninst✝⁹ : BorelSpace E✝\ninst✝⁸ : FiniteDimensional ℝ E✝\nμ✝ : Measure E✝\ninst✝⁷ : IsAddHaarMeasure μ✝\nF : Type ?u.1970194\ninst✝⁶ : NormedAddCommGr... | [
413,
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409,
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Mathlib/Data/Multiset/FinsetOps.lean | Multiset.coe_ndunion | [] | [
149,
6
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148,
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Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.inl_comp_pushoutSymmetry_inv | [
{
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"tactic": "simp [Iso.comp_inv_eq]"
}
] | [
1582,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1581,
1
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Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.BoundedFormula.realize_iff | [
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.70178\nP : Type ?u.70181\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nx... | [
356,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
355,
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Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le | [
{
"state_after": "α : Type u_2\nβ✝ : Type ?u.23759\nγ : Type ?u.23762\nι : Type ?u.23765\ninst✝³ : Countable ι\nf✝ g : α → β✝\ninst✝² : TopologicalSpace β✝\nβ : Type u_1\nf : α → β\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nhf : StronglyMeasurable f\nc : ℝ\nx : α\nhfx : ‖f x... | [
247,
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Mathlib/MeasureTheory/Function/UniformIntegrable.lean | MeasureTheory.tendstoInMeasure_iff_tendsto_Lp | [] | [
633,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
626,
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Mathlib/Algebra/IsPrimePow.lean | IsUnit.not_isPrimePow | [] | [
53,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
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Mathlib/Data/Set/Intervals/WithBotTop.lean | WithTop.preimage_coe_Ioi | [] | [
41,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
40,
1
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Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | vsub_mem_vectorSpan_pair | [] | [
1256,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1255,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | Commute.cast_int_mul_right | [] | [
1154,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1153,
1
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Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | ENNReal.rpow_lt_rpow | [] | [
570,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
569,
1
] |
Std/Data/Array/Lemmas.lean | Array.get_set_ne | [
{
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"state_before": "α : Type ?u.15121\na : Array α\ni : Fin (size a)\nj : Nat\nv : α\nhj : j < size a\nh : i.val ≠ j\n⊢ j < size (set a i v)",
"tactic": "simp [*]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : Array α\ni : Fin (size a)\nj : Nat\... | [
93,
58
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
91,
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Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_of_controlled | [
{
"state_after": "α : Type u\nβ : Type v\ninst✝² : UniformSpace α\ninst✝¹ : SemilatticeSup β\ninst✝ : Nonempty β\nU : β → Set (α × α)\nhU : ∀ (s : Set (α × α)), s ∈ 𝓤 α → ∃ n, U n ⊆ s\nf : β → α\nhf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N\ns : Set (α × α)\nhs : s ∈ 𝓤 α\n⊢ s ∈ map (Prod.map f f) atTo... | [
316,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
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Mathlib/MeasureTheory/Group/Integration.lean | MeasureTheory.integral_mul_right_eq_self | [
{
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114,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
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Mathlib/Order/MinMax.lean | Antitone.map_min | [] | [
269,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
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Std/Logic.lean | iff_of_eq | [] | [
47,
49
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
47,
1
] |
Mathlib/RingTheory/IntegralClosure.lean | le_integralClosure_iff_isIntegral | [] | [
579,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
574,
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Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineIsometry.comp_assoc | [] | [
256,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | AEMeasurable.prod_swap | [
{
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696,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
693,
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Mathlib/Data/List/Basic.lean | List.get?_eq_some' | [] | [
1253,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1253,
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Mathlib/GroupTheory/QuotientGroup.lean | QuotientGroup.lift_quot_mk | [] | [
220,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
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Mathlib/Data/PNat/Xgcd.lean | PNat.gcd_rel_left | [] | [
543,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
542,
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Mathlib/Init/Algebra/Order.lean | le_antisymm | [] | [
193,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
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Mathlib/Algebra/Lie/Subalgebra.lean | LieSubalgebra.top_coe_submodule | [] | [
460,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
459,
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Mathlib/Data/Real/Basic.lean | Real.ext_cauchy_iff | [
{
"state_after": "no goals",
"state_before": "x y : ℝ\na b : Cauchy abs\n⊢ { cauchy := a } = { cauchy := b } ↔ { cauchy := a }.cauchy = { cauchy := b }.cauchy",
"tactic": "rw [ofCauchy.injEq]"
}
] | [
62,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
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Mathlib/CategoryTheory/Preadditive/Biproducts.lean | CategoryTheory.Biprod.ofComponents_snd | [
{
"state_after": "no goals",
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"tactic":... | [
700,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
698,
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Mathlib/Data/Set/Lattice.lean | Set.sUnion_union | [] | [
1131,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1130,
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Mathlib/GroupTheory/Torsion.lean | IsTorsion.not_torsion_free | [
{
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"state_before": "G : Type u_1\nH : Type ?u.210692\ninst✝ : Group G\nhN : Nontrivial G\ntG : IsTorsion G\n⊢ ∃ g, g ≠ 1 ∧ IsOfFinOrder g",
... | [
387,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
384,
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Mathlib/MeasureTheory/Integral/CircleIntegral.lean | circleIntegral.norm_integral_le_of_norm_le_const' | [
{
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401,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
391,
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Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.maxDegree_lt_card_verts | [
{
"state_after": "case intro\nι : Sort ?u.296737\n𝕜 : Type ?u.296740\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nv : V\nhv : maxDegree G = degree G v\n⊢ maxDegree G < Fintype.card ... | [
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1590,
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Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.Integrable.bdd_mul' | [
{
"state_after": "α : Type u_2\nβ : Type ?u.1127420\nγ : Type ?u.1127423\nδ : Type ?u.1127426\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\n𝕜 : Type u_1\ninst✝ : NormedRing 𝕜\nf✝ f g : α → 𝕜\nc : ℝ\nhg : Integrable g\nhf : A... | [
1113,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1107,
1
] |
Mathlib/Topology/Bases.lean | TopologicalSpace.IsTopologicalBasis.mem_nhds_iff | [
{
"state_after": "α : Type u\nt : TopologicalSpace α\na : α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\n⊢ s ∈ (𝓝 a).sets ↔ ∃ t, t ∈ b ∧ a ∈ t ∧ t ⊆ s",
"state_before": "α : Type u\nt : TopologicalSpace α\na : α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\n⊢ s ∈ 𝓝 a ↔ ∃ t, t ∈ b ∧ a ... | [
156,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
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] |
Mathlib/Data/Real/EReal.lean | EReal.neg_mul | [
{
"state_after": "no goals",
"state_before": "x y : EReal\n⊢ -x * y = -(x * y)",
"tactic": "induction x, y using induction₂_neg_left with\n| top_zero | zero_top | zero_bot => simp only [zero_mul, mul_zero, neg_zero]\n| top_top | top_bot => rfl\n| neg_left h => rw [h, neg_neg, neg_neg]\n| coe_coe => norm... | [
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1025,
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] |
Mathlib/Topology/Order/Basic.lean | isClosed_ge' | [] | [
143,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
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Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.coe_zsmul | [] | [
983,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
983,
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Mathlib/Analysis/Calculus/LHopital.lean | HasDerivAt.lhopital_zero_nhds_left | [
{
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Order/Heyting/Basic.lean | ofDual_compl | [] | [
1137,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1136,
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Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.measure_diff' | [
{
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Mathlib/Data/Multiset/Basic.lean | Multiset.Rel.add | [
{
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2705,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2702,
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Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | AffineEquiv.constVAdd_zero | [] | [
491,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
490,
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] |
Mathlib/SetTheory/Cardinal/Cofinality.lean | Cardinal.bsup_lt_ord_lift_of_isRegular | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.159715\nr : α → α → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (card o) < c\n⊢ lift (card o) < Ordinal.cof (ord c)",
"tactic": "rwa [hc.cof_eq]"
}
] | [
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40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1092,
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Mathlib/Algebra/Lie/Abelian.lean | LieAlgebra.ad_ker_eq_self_module_ker | [] | [
267,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
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Mathlib/Data/Multiset/FinsetOps.lean | Multiset.dedup_cons | [
{
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"tactic": "by_cases h : a ∈ s <;> simp [h]"
}
] | [
87,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
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Mathlib/Algebra/Order/LatticeGroup.lean | LatticeOrderedCommGroup.abs_div_sup_mul_abs_div_inf | [
{
"state_after": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\nthis : DistribLattice α := latticeOrderedCommGroupToDistribLattice α\n⊢ abs ((a ⊔ c) / (b ⊔ c)) * abs ((a ⊓ c) / (b ⊓ c)) = abs (a / b)",
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459,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
442,
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Mathlib/Algebra/Group/Units.lean | Units.inv_mul_of_eq | [
{
"state_after": "no goals",
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"tactic": "rw [← h, u.inv_mul]"
}
] | [
274,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
1
] |
Mathlib/Order/Filter/Pointwise.lean | Filter.NeBot.vsub | [] | [
1117,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1116,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.Integrable.const_mul | [] | [
1079,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1077,
1
] |
Mathlib/Data/Int/Cast/Lemmas.lean | RingHom.eq_intCast' | [] | [
326,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
325,
1
] |
Mathlib/Algebra/BigOperators/Order.lean | Finset.le_sum_card | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.109273\nα : Type u_1\nβ : Type ?u.109279\nM : Type ?u.109282\nN : Type ?u.109285\nG : Type ?u.109288\nk : Type ?u.109291\nR : Type ?u.109294\ninst✝¹ : DecidableEq α\ns : Finset α\nB : Finset (Finset α)\nn : ℕ\ninst✝ : Fintype α\nh : ∀ (a : α), n ≤ ca... | [
352,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
348,
1
] |
Mathlib/Algebra/Order/CompleteField.lean | LinearOrderedField.coe_mem_cutMap_iff | [] | [
121,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/Data/List/Infix.lean | List.length_tails | [
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.48588\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\n⊢ length (tails []) = length [] + 1\n\ncase cons\nα : Type u_1\nβ : Type ?u.48588\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nx : α\nl : List α\nIH : length (tails l) = length l + 1\n⊢ length (tails (x :: l)) = leng... | [
429,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
426,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | Submodule.isClosed_orthogonal | [
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.53341\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ IsClosed ↑(⨅ (v : { x // x ∈ K }), LinearMap.ker (↑(innerSL 𝕜) ↑v))",
"s... | [
137,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
] |
Mathlib/Data/Nat/Digits.lean | Nat.NormDigits.digits_succ | [
{
"state_after": "case intro.intro\nn✝ b n m r : ℕ\nl : List ℕ\ne : r + b * m = n\nhr : r < b\nh : digits b m = l\nb2 : 1 < b\nm0 : 0 < m\n⊢ digits b n = r :: l ∧ 1 < b ∧ 0 < n",
"state_before": "n✝ b n m r : ℕ\nl : List ℕ\ne : r + b * m = n\nhr : r < b\nh : digits b m = l ∧ 1 < b ∧ 0 < m\n⊢ digits b n = r ... | [
639,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
632,
1
] |
Mathlib/CategoryTheory/Bicategory/Coherence.lean | CategoryTheory.FreeBicategory.normalizeAux_nil_comp | [
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Quiver B\na b c : B\nf : Hom a b\ng : Hom b c\n⊢ normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g)",
"tactic": "induction g generalizing a with\n| id => rfl\n| of => rfl\n| comp g _ ihf ihg => erw [ihg (f.co... | [
196,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
] |
Mathlib/Algebra/Tropical/Basic.lean | Tropical.add_pow | [
{
"state_after": "case inl\nR : Type u\ninst✝³ : LinearOrder R\ninst✝² : AddMonoid R\ninst✝¹ : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass R R (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nx y : Tropical R\nn : ℕ\nh : x ≤ y\n⊢ (x + y) ^ n = x ^ n + y ^ n\n... | [
550,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
545,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Ioc_subset_Iic_self | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.54007\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\ninst✝ : LocallyFiniteOrderBot α\n⊢ Ioc a b ⊆ Iic b",
"tactic": "simpa [← coe_subset] using Set.Ioc_subset_Iic_self"
}
] | [
431,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
430,
1
] |
Mathlib/Topology/Order/Basic.lean | nhdsWithin_Iic_self_neBot | [] | [
264,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
263,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.coeff_monomial_mul | [
{
"state_after": "σ : Type u_2\nR : Type u_1\ninst✝ : Semiring R\nm n : σ →₀ ℕ\nφ ψ : MvPowerSeries σ R\na : R\nthis :\n ∀ (p : (σ →₀ ℕ) × (σ →₀ ℕ)),\n p ∈ antidiagonal m → ↑(coeff R p.fst) (↑(monomial R n) a) * ↑(coeff R p.snd) φ ≠ 0 → p.fst = n\n⊢ ↑(coeff R m) (↑(monomial R n) a * φ) = if n ≤ m then a * ↑... | [
234,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
227,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean | Nat.bodd_mul | [
{
"state_after": "case zero\nm : ℕ\n⊢ bodd (m * zero) = (bodd m && bodd zero)\n\ncase succ\nm n : ℕ\nIH : bodd (m * n) = (bodd m && bodd n)\n⊢ bodd (m * succ n) = (bodd m && bodd (succ n))",
"state_before": "m n : ℕ\n⊢ bodd (m * n) = (bodd m && bodd n)",
"tactic": "induction' n with n IH"
},
{
"... | [
88,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.ofReal_exp_ofReal_re | [
{
"state_after": "no goals",
"state_before": "x✝ y : ℂ\nx : ℝ\n⊢ ↑(starRingEnd ℂ) (exp ↑x) = exp ↑x",
"tactic": "rw [← exp_conj, conj_ofReal]"
}
] | [
576,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
575,
1
] |
Mathlib/Algebra/Group/Basic.lean | mul_div_mul_comm | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.47654\nG : Type ?u.47657\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a * b / (c * d) = a / c * (b / d)",
"tactic": "simp"
}
] | [
600,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
600,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean | neg_of_mul_neg_left | [] | [
821,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
820,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean | DifferentiableOn.norm_sq | [] | [
278,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/Topology/Maps.lean | embedding_of_embedding_compose | [] | [
221,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean | Finsupp.single_finset_sum | [
{
"state_after": "α : Type u_3\nι : Type u_2\nγ : Type ?u.271042\nA : Type ?u.271045\nB : Type ?u.271048\nC : Type ?u.271051\ninst✝³ : AddCommMonoid A\ninst✝² : AddCommMonoid B\ninst✝¹ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns✝ : Finset... | [
287,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Data/Set/Countable.lean | Set.Countable.mono | [] | [
89,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
] |
Mathlib/Topology/Sets/Opens.lean | TopologicalSpace.Opens.comap_id | [] | [
368,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
367,
1
] |
Mathlib/Data/PFun.lean | PFun.bind_apply | [] | [
218,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
] |
Mathlib/Algebra/Group/Defs.lean | inv_mul_self | [] | [
1085,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1084,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean | CategoryTheory.Limits.hasFiniteCoproducts_of_opposite | [] | [
383,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
382,
1
] |
Mathlib/Data/Nat/ModEq.lean | Nat.ModEq.dvd_iff | [
{
"state_after": "m n a b c d : ℕ\nh : a ≡ b [MOD m]\nhdm : d ∣ m\n⊢ a ≡ 0 [MOD d] ↔ b ≡ 0 [MOD d]",
"state_before": "m n a b c d : ℕ\nh : a ≡ b [MOD m]\nhdm : d ∣ m\n⊢ d ∣ a ↔ d ∣ b",
"tactic": "simp only [← modEq_zero_iff_dvd]"
},
{
"state_after": "m n a b c d : ℕ\nhdm : d ∣ m\nh : a ≡ b [MOD ... | [
252,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
] |
Mathlib/SetTheory/Cardinal/Continuum.lean | Cardinal.aleph_one_le_continuum | [
{
"state_after": "⊢ Order.succ ℵ₀ ≤ 𝔠",
"state_before": "⊢ aleph 1 ≤ 𝔠",
"tactic": "rw [← succ_aleph0]"
},
{
"state_after": "no goals",
"state_before": "⊢ Order.succ ℵ₀ ≤ 𝔠",
"tactic": "exact Order.succ_le_of_lt aleph0_lt_continuum"
}
] | [
107,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/Data/Part.lean | Part.mem_toOption | [
{
"state_after": "α : Type u_1\nβ : Type ?u.7431\nγ : Type ?u.7434\no : Part α\ninst✝ : Decidable o.Dom\na : α\n⊢ (a ∈ if h : o.Dom then Option.some (get o h) else Option.none) ↔ a ∈ o",
"state_before": "α : Type u_1\nβ : Type ?u.7431\nγ : Type ?u.7434\no : Part α\ninst✝ : Decidable o.Dom\na : α\n⊢ a ∈ toOp... | [
296,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
292,
1
] |
Mathlib/LinearAlgebra/Dimension.lean | rank_zero_iff_forall_zero | [
{
"state_after": "case refine'_1\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.245408\nR : Type u\nM : Type v\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : NoZeroSMulDivisors R M\ninst✝ : Nontrivial R\nh : M... | [
512,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
503,
1
] |
Mathlib/Algebra/CharZero/Lemmas.lean | add_halves' | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\na : R\n⊢ a / 2 + a / 2 = a",
"tactic": "rw [← add_div, half_add_self]"
}
] | [
167,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean | MeasureTheory.Lp.nnnorm_toLp | [] | [
283,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
281,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | le_of_forall_sub_le | [
{
"state_after": "ι : Type ?u.195915\nα : Type u_1\nβ : Type ?u.195921\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nh : a < b\n⊢ ∃ ε, ε > 0 ∧ a < b - ε",
"state_before": "ι : Type ?u.195915\nα : Type u_1\nβ : Type ?u.195921\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nh : ∀ (ε : α), ε > 0 → b - ... | [
981,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
978,
1
] |
Mathlib/FieldTheory/Subfield.lean | Subfield.coe_map | [] | [
517,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
516,
1
] |
Mathlib/Data/Set/Ncard.lean | Set.ncard_diff_singleton_of_mem | [] | [
194,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
9
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean | MultilinearMap.map_nonempty | [] | [
1642,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1640,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.map_eq_iff | [] | [
1003,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1002,
1
] |
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | gramSchmidt_of_orthogonal | [
{
"state_after": "case h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nhf : Pairwise fun i j => inner (f i) (f j) = 0... | [
202,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
184,
1
] |
Mathlib/Data/Bundle.lean | Bundle.TotalSpace.eta | [] | [
84,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean | CategoryTheory.GrothendieckTopology.Cover.condition | [] | [
456,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
455,
1
] |
Mathlib/Order/Interval.lean | Interval.exists | [] | [
364,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
363,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.eval₂_zero'_apply | [] | [
1508,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1506,
1
] |
Mathlib/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular.neg | [] | [
92,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
11
] |
Mathlib/Order/UpperLower/Basic.lean | UpperSet.inf_prod | [] | [
1587,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1586,
1
] |
Mathlib/Topology/Instances/Matrix.lean | Summable.matrix_transpose | [] | [
301,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
299,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | uniformContinuous_norm' | [] | [
1117,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1116,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | Filter.Tendsto.eventually_intervalIntegrable | [] | [
435,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
431,
1
] |
Mathlib/Order/Antisymmetrization.lean | toAntisymmetrization_ofAntisymmetrization | [] | [
129,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.IsCycleOn.zpow_apply_eq_zpow_apply | [
{
"state_after": "ι : Type ?u.1791500\nα : Type u_1\nβ : Type ?u.1791506\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nm n : ℤ\n⊢ ↑(f ^ m) a = ↑(f ^ n) a ↔ ↑(f ^ (n - m)) a = a",
"state_before": "ι : Type ?u.1791500\nα : Type u_1\nβ : Type ?u.1791506\nf g : Perm α\... | [
889,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
886,
1
] |
Mathlib/FieldTheory/Finiteness.lean | IsNoetherian.iff_rank_lt_aleph0 | [
{
"state_after": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nb : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\n⊢ IsNoetherian K V ↔ Module.rank K V < ℵ₀",
"state_before": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : A... | [
44,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
33,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean | Nat.coprime_mul_left_add_right | [
{
"state_after": "no goals",
"state_before": "m n k : ℕ\n⊢ coprime m (m * k + n) ↔ coprime m n",
"tactic": "rw [coprime, coprime, gcd_mul_left_add_right]"
}
] | [
169,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | isConformalMap_id | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\nN : Type ?u.8223\nG : Type ?u.8226\nM' : Type ?u.8229\ninst✝⁸ : NormedField R\ninst✝⁷ : SeminormedAddCommGroup M\ninst✝⁶ : SeminormedAddCommGroup N\ninst✝⁵ : SeminormedAddCommGroup G\ninst✝⁴ : NormedSpace R M\ninst✝³ : NormedSpace R N... | [
62,
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61,
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Mathlib/Data/MvPolynomial/Comap.lean | MvPolynomial.comap_rename | [
{
"state_after": "case h\nσ : Type u_1\nτ : Type u_3\nυ : Type ?u.158664\nR : Type u_2\ninst✝ : CommSemiring R\nf : σ → τ\nx : τ → R\nx✝ : σ\n⊢ comap (rename f) x x✝ = (x ∘ f) x✝",
"state_before": "σ : Type u_1\nτ : Type u_3\nυ : Type ?u.158664\nR : Type u_2\ninst✝ : CommSemiring R\nf : σ → τ\nx : τ → R\n⊢ ... | [
95,
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93,
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Mathlib/Algebra/Order/Floor.lean | Nat.lt_of_ceil_lt | [] | [
340,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Monad.lean | Pmf.bindOnSupport_bindOnSupport | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np✝ : Pmf α\nf✝ : (a : α) → a ∈ support p✝ → Pmf β\np : Pmf α\nf : (a : α) → a ∈ support p → Pmf β\ng : (b : β) → b ∈ support (bindOnSupport p f) → Pmf γ\na : γ\n⊢ ↑(bindOnSupport (bindOnSupport p f) g) a =\n ↑(bindOnSupport p fun a ha => bindOnSuppo... | [
294,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
278,
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