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/Seq/WSeq.lean | Stream'.WSeq.head_congr | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\n⊢ ∀ {s t : WSeq α}, s ~ʷ t → ∀ {o : Option α}, o ∈ head s → o ∈ head t",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\n⊢ ∀ {s t : WSeq α}, s ~ʷ t → head s ~ head t",
"tactic": "suffices ∀ {s t : WSeq α}, s ~ʷ t → ∀ {o}, o ∈ head s → o ∈ head t... | [
1145,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1127,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean | YoungDiagram.exists_not_mem_row | [
{
"state_after": "case intro\nμ : YoungDiagram\ni j : ℕ\nhj :\n ¬j ∈\n Finset.preimage μ.cells (Prod.mk i)\n (_ :\n ∀ (x : ℕ),\n x ∈ Prod.mk i ⁻¹' ↑μ.cells → ∀ (x_2 : ℕ), x_2 ∈ Prod.mk i ⁻¹' ↑μ.cells → (i, x) = (i, x_2) → x = x_2)\n⊢ ∃ j, ¬(i, j) ∈ μ",
"state_before": "μ : Y... | [
303,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
296,
11
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.le_boundedBy | [
{
"state_after": "α : Type u_1\nm : Set α → ℝ≥0∞\nμ : OuterMeasure α\n⊢ ∀ (a : Set α), (↑μ a ≤ ⨆ (_ : Set.Nonempty a), m a) ↔ ↑μ a ≤ m a",
"state_before": "α : Type u_1\nm : Set α → ℝ≥0∞\nμ : OuterMeasure α\n⊢ μ ≤ boundedBy m ↔ ∀ (s : Set α), ↑μ s ≤ m s",
"tactic": "rw [boundedBy , le_ofFunction, forall... | [
875,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
873,
1
] |
Mathlib/Data/List/Basic.lean | List.nthLe_tail | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.45408\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\ni : ℕ\nh : i < length (tail l)\n⊢ i + 1 < length l",
"tactic": "simpa [← lt_tsub_iff_right] using h"
},
{
"state_after": "case nil\nι : Type ?u.45408\nα : Type u\nβ ... | [
951,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
945,
9
] |
Mathlib/Analysis/Calculus/Deriv/Star.lean | HasDerivAt.star | [] | [
50,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
48,
18
] |
Mathlib/Deprecated/Group.lean | Multiplicative.isMulHom | [] | [
451,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
449,
1
] |
Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean | CategoryTheory.Sieve.effectiveEpimorphic_family | [
{
"state_after": "case mp\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\n⊢ Presieve.EffectiveEpimorphic (Presieve.ofArrows X π) → EffectiveEpiFamily X π\n\ncase mpr\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\n⊢ EffectiveEpi... | [
434,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
] |
Mathlib/Order/UpperLower/Basic.lean | LowerSet.mem_mk | [] | [
473,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
472,
1
] |
Mathlib/Data/Seq/Seq.lean | Stream'.Seq.of_mem_append | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh this : a ∈ append s₁ s₂\n⊢ a ∈ s₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh : a ∈ append s₁ s₂\n⊢ a ∈ s₁ ∨ a ∈ s₂",
"tactic": "have := h"
},
{
"state_after": "α : Type u\nβ... | [
886,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
869,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean | Set.div_iInter_subset | [] | [
792,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
791,
1
] |
Mathlib/Init/IteSimp.lean | if_true_right_eq_or | [
{
"state_after": "no goals",
"state_before": "p : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then q else True) = (¬p ∨ q)",
"tactic": "by_cases p <;> simp [h]"
}
] | [
25,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
24,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.iUnion_prod_const | [
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.230579\nι : Sort u_3\nι' : Sort ?u.230585\nι₂ : Sort ?u.230588\nκ : ι → Sort ?u.230593\nκ₁ : ι → Sort ?u.230598\nκ₂ : ι → Sort ?u.230603\nκ' : ι' → Sort ?u.230608\ns : ι → Set α\nt : Set β\nx✝ : α × β\n⊢ x✝ ∈ (⋃ (i : ι), s i) ×ˢ t ↔ x✝ ∈ ⋃ (i : ι... | [
1783,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1781,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiff.snd | [] | [
802,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
801,
1
] |
Mathlib/Topology/ContinuousOn.lean | nhdsWithin_inter_of_mem' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18576\nγ : Type ?u.18579\nδ : Type ?u.18582\ninst✝ : TopologicalSpace α\na : α\ns t : Set α\nh : t ∈ 𝓝[s] a\n⊢ 𝓝[s ∩ t] a = 𝓝[s] a",
"tactic": "rw [inter_comm, nhdsWithin_inter_of_mem h]"
}
] | [
277,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
276,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | ContDiff.contDiffOn | [] | [
1438,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1437,
1
] |
Mathlib/Topology/Sets/Compacts.lean | TopologicalSpace.NonemptyCompacts.ext | [] | [
243,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
242,
11
] |
Mathlib/Data/Sym/Card.lean | Sym2.card_image_diag | [
{
"state_after": "α : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Set.InjOn Quotient.mk' ↑(Finset.diag s)",
"state_before": "α : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Finset.card (image Quotient.mk' (Finset.diag s)) = Finset.card s",
"tactic": "rw [c... | [
141,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
] |
Mathlib/Algebra/Lie/Classical.lean | LieAlgebra.Orthogonal.jd_transform | [
{
"state_after": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\nh : (PD l R)ᵀ ⬝ JD l R = fromBl... | [
279,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
1
] |
Mathlib/Data/Finset/Lattice.lean | iSup_eq_iSup_finset' | [
{
"state_after": "F : Type ?u.419112\nα : Type u_1\nβ : Type ?u.419118\nγ : Type ?u.419121\nι : Type ?u.419124\nκ : Type ?u.419127\nι' : Sort u_2\ninst✝ : CompleteLattice α\ns : ι' → α\n⊢ (⨆ (x : PLift ι'), s (↑Equiv.plift x)) = ⨆ (i : PLift ι'), s i.down",
"state_before": "F : Type ?u.419112\nα : Type u_1\... | [
1827,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1825,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean | Ordinal.cof_type | [] | [
160,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | inv_lt | [] | [
288,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
287,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | right_sub_midpoint | [] | [
157,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Complex.sin_two_pi_sub | [] | [
1165,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1164,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | zpow_lt_zpow | [] | [
355,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
354,
1
] |
Mathlib/RingTheory/Noetherian.lean | isNoetherian_of_range_eq_ker | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : IsNoetherian R P\n... | [
406,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
398,
1
] |
Mathlib/CategoryTheory/Sites/Pretopology.lean | CategoryTheory.Pretopology.toGrothendieck_bot | [] | [
218,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
] |
Mathlib/Algebra/Module/Equiv.lean | LinearEquiv.surjective | [] | [
563,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
562,
11
] |
Mathlib/NumberTheory/Multiplicity.lean | multiplicity.Nat.pow_sub_pow | [
{
"state_after": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n\n\ncase inr\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ ... | [
241,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
230,
1
] |
Mathlib/Topology/Basic.lean | isOpen_univ | [] | [
122,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
9
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigOWith.pow | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.494047\nE : Type ?u.494050\nF : Type ?u.494053\nG : Type ?u.494056\nE' : Type ?u.494059\nF' : Type ?u.494062\nG' : Type ?u.494065\nE'' : Type ?u.494068\nF'' : Type ?u.494071\nG'' : Type ?u.494074\nR : Type u_1\nR' : Type ?u.494080\n𝕜 :... | [
1623,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1620,
1
] |
Mathlib/Data/Real/Cardinality.lean | Cardinal.mk_Iic_real | [] | [
263,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
262,
1
] |
Mathlib/Order/Hom/Bounded.lean | BoundedOrderHom.ext | [] | [
600,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
599,
1
] |
Mathlib/Order/Ideal.lean | Order.Ideal.isIdeal | [] | [
141,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
140,
11
] |
Mathlib/Topology/FiberBundle/Basic.lean | FiberBundleCore.open_source' | [
{
"state_after": "case a\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\n⊢ (localTrivAsLocalEquiv Z i).source ∈\n ⋃ (i : ι) (s : Set (B × F)) (_ : IsOpen... | [
581,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
575,
1
] |
Mathlib/RingTheory/Localization/InvSubmonoid.lean | IsLocalization.finiteType_of_monoid_fg | [
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\nthis : Monoid.FG { x // x ∈ invSubmonoid M S }\n⊢ Algebra.FiniteType R S",
"s... | [
130,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
] |
Mathlib/CategoryTheory/Over.lean | CategoryTheory.Over.over_right | [
{
"state_after": "no goals",
"state_before": "T : Type u₁\ninst✝ : Category T\nX : T\nU : Over X\n⊢ U.right = { as := PUnit.unit }",
"tactic": "simp only"
}
] | [
70,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/Data/Rat/NNRat.lean | Rat.toNNRat_one | [] | [
344,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
344,
1
] |
Mathlib/Data/Fintype/Powerset.lean | Fintype.card_set | [] | [
77,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
] |
Mathlib/LinearAlgebra/Ray.lean | SameRay.add_right | [] | [
207,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
206,
1
] |
Mathlib/Logic/Function/Conjugate.lean | Function.Semiconj.comp_eq | [] | [
44,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
43,
11
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean | StieltjesFunction.outer_Ioc | [
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "f : StieltjesFunction\na b : ℝ\n⊢ ↑(StieltjesFunction.outer f) (Ioc a b) ... | [
447,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean | dist_le_of_le_geometric_two_of_tendsto₀ | [] | [
417,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
415,
1
] |
Std/Logic.lean | iff_true_right | [] | [
65,
83
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
65,
1
] |
Mathlib/Analysis/Calculus/Dslope.lean | dslope_of_ne | [] | [
46,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
] |
Mathlib/Algebra/Ring/Defs.lean | two_mul | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonAssocSemiring α\nn : α\n⊢ 1 * n + 1 * n = n + n",
"tactic": "rw [one_mul]"
}
] | [
181,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
180,
1
] |
Mathlib/Topology/Algebra/MulAction.lean | ContinuousAt.smul | [] | [
112,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.as_sum_range' | [] | [
425,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
] |
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.adjoin_univ | [] | [
352,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
350,
1
] |
Mathlib/Topology/Sets/Opens.lean | TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion | [
{
"state_after": "case hb\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ IsTopologicalBasis (rang... | [
337,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
330,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean | IsLocalMinOn.fderivWithin_eq_zero | [
{
"state_after": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nh : IsLocalMinOn f s a\ny : E\nhy : y ∈ posTangentConeAt s a\nhy' : -y ∈ posTangentConeAt s a\nhf : ¬DifferentiableWithinAt ℝ f s a\n⊢ ↑0 y = 0",
"state_before": "E : Type u\nin... | [
185,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
180,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | abs_pow | [] | [
707,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
706,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.mem_sSup_of_directedOn | [
{
"state_after": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nS : Set (Subsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x ≤ x_1) S\nx : R\nthis : Nonempty ↑S\n⊢ x ∈ sSup S ↔ ∃ s, s ∈ S ∧ x ∈ s",
... | [
1110,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1106,
1
] |
Mathlib/Data/Set/Image.lean | Set.Nontrivial.image | [] | [
1257,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1254,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.aleph0_le_aleph | [
{
"state_after": "o : Ordinal\n⊢ ω ≤ ω + o",
"state_before": "o : Ordinal\n⊢ ℵ₀ ≤ aleph o",
"tactic": "rw [aleph, aleph0_le_aleph']"
},
{
"state_after": "no goals",
"state_before": "o : Ordinal\n⊢ ω ≤ ω + o",
"tactic": "apply Ordinal.le_add_right"
}
] | [
294,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
292,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean | LinearPMap.graph_fst_eq_zero_snd | [] | [
828,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
826,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.Sized.node3L | [] | [
376,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
374,
1
] |
Mathlib/NumberTheory/Divisors.lean | Nat.prod_divisorsAntidiagonal' | [
{
"state_after": "n✝ : ℕ\nM : Type u_1\ninst✝ : CommMonoid M\nf : ℕ → ℕ → M\nn : ℕ\n⊢ ∏ x in divisorsAntidiagonal n,\n f (↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)) x).fst (↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)) x).snd =\n ∏ i in divisors n, f (n / i) i",
"state_before": "n✝ : ℕ\nM : Type u_1\nins... | [
480,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
477,
1
] |
Mathlib/Topology/LocalExtr.lean | IsLocalMaxOn.inter | [] | [
125,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean | ContDiffBump.normed_neg | [
{
"state_after": "no goals",
"state_before": "E : Type u_1\nX : Type ?u.1158619\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace ℝ X\ninst✝¹ : HasContDiffBump E\nc : E\nf✝ : ContDiffBump c\nx✝ : E\nn : ℕ∞\ninst✝ : MeasurableSpace E\nμ : MeasureThe... | [
509,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
508,
1
] |
Mathlib/RingTheory/JacobsonIdeal.lean | Ideal.radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot | [
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\n⊢ radical I = jacobson I ↔ radical ⊥ = jacobson ⊥",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\n⊢ radical I = jacobson... | [
305,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
292,
1
] |
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.AdjoinSimple.isIntegral_gen | [
{
"state_after": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\n⊢ IsIntegral F (gen F α) ↔ IsIntegral F (↑(algebraMap { x // x ∈ F⟮α⟯ } E) (gen F α))",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F ... | [
502,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
500,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean | ApproximatesLinearOn.surjective | [
{
"state_after": "case inl\n𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type... | [
401,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
11
] |
Mathlib/Data/Real/Irrational.lean | irrational_sqrt_two | [
{
"state_after": "no goals",
"state_before": "⊢ Irrational (Real.sqrt 2)",
"tactic": "simpa using Nat.prime_two.irrational_sqrt"
}
] | [
111,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | AEMeasurable.max | [] | [
798,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
795,
8
] |
Mathlib/Analysis/Convex/Join.lean | convexJoin_empty_left | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.16420\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns t✝ s₁ s₂ t₁ t₂ u : Set E\nx y : E\nt : Set E\n⊢ convexJoin 𝕜 ∅ t = ∅",
"tactic": "simp [convexJoin]"
}
] | [
62,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
62,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.nfp_mul_one | [
{
"state_after": "a : Ordinal\nha : 0 < a\n⊢ (fun a_1 => sup fun n => ((fun x => a * x)^[n]) a_1) 1 = sup fun n => a ^ ↑n\n\na : Ordinal\nha : 0 < a\n⊢ 0 < a",
"state_before": "a : Ordinal\nha : 0 < a\n⊢ nfp (fun x => a * x) 1 = a ^ ω",
"tactic": "rw [← sup_iterate_eq_nfp, ← sup_opow_nat]"
},
{
... | [
625,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
616,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | IsBoundedLinearMap.continuous | [] | [
178,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | lt_mul_of_one_lt_of_lt_of_nonneg | [] | [
910,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
908,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.Valid'.rotateL | [
{
"state_after": "case nil\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) nil o₂\nH1 : ¬size l + size nil ≤ 1\nH2 : delta * size l < size nil\nH3 : 2 * size nil ≤ 9 * size l + 5 ∨ size nil ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x nil) ... | [
1292,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1235,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.comapₗ_apply | [
{
"state_after": "case hc\nα : Type u_2\nβ✝ : Type ?u.221666\nγ : Type ?u.221669\nδ : Type ?u.221672\nι : Type ?u.221675\nR : Type ?u.221678\nR' : Type ?u.221681\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 α\nβ : Type u_1\nin... | [
1301,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1297,
1
] |
Mathlib/FieldTheory/RatFunc.lean | RatFunc.natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree | [
{
"state_after": "K : Type u\ninst✝ : Field K\nx : RatFunc K\nhx : x ≠ 0\ns : K[X]\nhs : s ≠ 0\n⊢ num x * s * denom x = num x * (s * denom x)",
"state_before": "K : Type u\ninst✝ : Field K\nx : RatFunc K\nhx : x ≠ 0\ns : K[X]\nhs : s ≠ 0\n⊢ ↑(natDegree (num x * s)) - ↑(natDegree (s * denom x)) = intDegree x... | [
1636,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1631,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean | CategoryTheory.Presieve.functorPullback_id | [] | [
186,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
185,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean | LieSubalgebra.mem_carrier | [] | [
153,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | Real.rpow_mul | [
{
"state_after": "no goals",
"state_before": "x✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\n⊢ x ^ (y * z) = (x ^ y) ^ z",
"tactic": "rw [← Complex.ofReal_inj, Complex.ofReal_cpow (rpow_nonneg_of_nonneg hx _),\n Complex.ofReal_cpow hx, Complex.ofReal_mul, Complex.cpow_mul, Complex.ofReal_cpow hx] <;>\n simp o... | [
324,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
320,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean | MeasureTheory.Measure.dirac_bind | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.40352\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nm : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(bind m dirac) s = ↑↑m s",
"state_before": "α : Type u_1\nβ : Type ?u.40352\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nm : Measur... | [
208,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean | BoxIntegral.Integrable.neg | [] | [
284,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
283,
1
] |
Mathlib/Topology/LocalExtr.lean | IsLocalMin.sub | [] | [
422,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
420,
8
] |
Mathlib/Data/Polynomial/EraseLead.lean | Polynomial.eraseLead_coeff_natDegree | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ coeff (eraseLead f) (natDegree f) = 0",
"tactic": "simp [eraseLead_coeff]"
}
] | [
54,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
1
] |
Mathlib/Order/Lattice.lean | sup_inf_left | [] | [
761,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
760,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | CategoryTheory.Limits.WidePullback.eq_lift_of_comp_eq | [
{
"state_after": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) a... | [
361,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
] |
Mathlib/RingTheory/Ideal/AssociatedPrime.lean | associatedPrimes.nonempty | [
{
"state_after": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nI J : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.65376\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsNoetherianRing R\ninst✝ : Nontrivial M\nx : M\nhx : x ≠ 0\n⊢ Set.Nonempty (as... | [
133,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Topology/DiscreteQuotient.lean | DiscreteQuotient.fiber_eq | [] | [
117,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | mul_singleton_mem_nhds | [
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\na b : α\nh : s ∈ 𝓝 b\n⊢ MulOpposite.op a • s ∈ 𝓝 (b * a)",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : Topological... | [
1318,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1316,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.eq_empty_of_forall_not_mem | [] | [
579,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
578,
1
] |
Mathlib/GroupTheory/Perm/List.lean | List.mem_of_formPerm_apply_ne | [] | [
87,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/Data/PNat/Factors.lean | PNat.factorMultiset_le_iff' | [
{
"state_after": "m : ℕ+\nv : PrimeMultiset\nh : factorMultiset m ≤ factorMultiset (PrimeMultiset.prod v) ↔ m ∣ PrimeMultiset.prod v := factorMultiset_le_iff\n⊢ factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v",
"state_before": "m : ℕ+\nv : PrimeMultiset\n⊢ factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v",
... | [
344,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | Complex.hasStrictDerivAt_tan | [
{
"state_after": "case h.e'_7\nx : ℂ\nh : cos x ≠ 0\n⊢ ↑1 / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2",
"state_before": "x : ℂ\nh : cos x ≠ 0\n⊢ HasStrictDerivAt tan (↑1 / cos x ^ 2) x",
"tactic": "convert (hasStrictDerivAt_sin x).div (hasStrictDerivAt_cos x) h using 1"
},
{
"stat... | [
33,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
30,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean | abs_le_abs_of_nonneg | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\nha : 0 ≤ a\nhab : a ≤ b\n⊢ abs a ≤ abs b",
"tactic": "rwa [abs_of_nonneg ha, abs_of_nonneg (ha.trans hab)]"
}
] | [
127,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | le_csSup_iff' | [] | [
1048,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1046,
1
] |
Mathlib/Topology/Bases.lean | TopologicalSpace.IsTopologicalBasis.isOpen_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nt : TopologicalSpace α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\n⊢ IsOpen s ↔ ∀ (a : α), a ∈ s → ∃ t, t ∈ b ∧ a ∈ t ∧ t ⊆ s",
"tactic": "simp [isOpen_iff_mem_nhds, hb.mem_nhds_iff]"
}
] | [
160,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
] |
Mathlib/CategoryTheory/Category/TwoP.lean | TwoP.coe_of | [] | [
52,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
] |
Mathlib/Algebra/Group/Basic.lean | mul_inv | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.29828\nG : Type ?u.29831\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ (a * b)⁻¹ = a⁻¹ * b⁻¹",
"tactic": "simp"
}
] | [
495,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
495,
1
] |
Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | FormalMultilinearSeries.removeZero_coeff_zero | [] | [
101,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean | WittVector.ghostFun_nat_cast | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\nS : Type ?u.681371\nT : Type ?u.681374\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.681389\nβ : Type ?u.681392\nx y : 𝕎 R\ni : ℕ\n⊢ WittVector.ghostFun (Nat.unaryCast i) = ↑i",
"tac... | [
201,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
9
] |
Mathlib/RingTheory/EuclideanDomain.lean | gcd_ne_zero_of_left | [] | [
40,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
] |
Mathlib/Control/Monad/Basic.lean | map_eq_bind_pure_comp | [] | [
54,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
] |
Mathlib/Topology/Sober.lean | IsGenericPoint.isIrreducible | [] | [
79,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
11
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | fderivWithin.log | [] | [
196,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean | MeasureTheory.unifIntegrable_subsingleton | [
{
"state_after": "α : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n ... | [
421,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
1
] |
Mathlib/Order/Filter/Prod.lean | Filter.prod_mem_prod | [] | [
65,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
] |
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