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/Dynamics/FixedPoints/Basic.lean | Function.mem_fixedPoints_iff | [
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nf✝ fa g : α✝ → α✝\nx✝ y : α✝\nfb : β → β\nm n k : ℕ\ne : Perm α✝\nα : Type u_1\nf : α → α\nx : α\n⊢ x ∈ fixedPoints f ↔ f x = x",
"tactic": "rfl"
}
] | [
146,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
1
] |
Mathlib/Topology/UniformSpace/CompactConvergence.lean | ContinuousMap.compactConvNhd_mono | [] | [
111,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
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Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.mem_closure_iff_exists_term | [
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type w\nN : Type ?u.33484\nP : Type ?u.33487\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\ns : Set M\nx : M\n⊢ x ∈ LowerAdjoint.toFun (closure L) s ↔ ∃ t, Term.realize Subtype.val t = x",
"tac... | [
320,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
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Mathlib/GroupTheory/Subsemigroup/Membership.lean | Subsemigroup.mem_sSup_of_mem | [
{
"state_after": "ι : Sort ?u.8999\nM : Type u_1\nA : Type ?u.9005\nB : Type ?u.9008\ninst✝ : Mul M\nS : Set (Subsemigroup M)\ns : Subsemigroup M\nhs : s ∈ S\nthis : s ≤ sSup S\n⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S",
"state_before": "ι : Sort ?u.8999\nM : Type u_1\nA : Type ?u.9005\nB : Type ?u.9008\ninst✝ : Mul... | [
113,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
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Mathlib/LinearAlgebra/Ray.lean | SameRay.rfl | [] | [
86,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
11
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Mathlib/Analysis/Calculus/Deriv/Mul.lean | derivWithin_smul_const | [] | [
111,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.map_id | [] | [
593,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
592,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | MeasureTheory.IntegrableOn.mul_continuousOn | [] | [
401,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
399,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.blsub_succ_of_mono | [] | [
1879,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1877,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean | Ordinal.ord_cof_eq | [
{
"state_after": "α : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nhS : Unbounded r S\ne : (#↑S) = cof (type r)\n⊢ ∃ S, Unbounded r S ∧ type (Subrel r S) = ord (cof (type r))",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ ∃ S, Unbounded r S ∧ type (Sub... | [
215,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.ker_codRestrict | [
{
"state_after": "R : Type u_2\nR₁ : Type ?u.1232780\nR₂ : Type u_1\nR₃ : Type ?u.1232786\nR₄ : Type ?u.1232789\nS : Type ?u.1232792\nK : Type ?u.1232795\nK₂ : Type ?u.1232798\nM : Type u_3\nM' : Type ?u.1232804\nM₁ : Type ?u.1232807\nM₂ : Type u_4\nM₃ : Type ?u.1232813\nM₄ : Type ?u.1232816\nN : Type ?u.123281... | [
1372,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1371,
1
] |
Mathlib/CategoryTheory/Sites/Types.lean | CategoryTheory.eval_app | [] | [
159,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/LinearAlgebra/Determinant.lean | Basis.det_unitsSMul_self | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type ?u.2918948\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nι : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\ne : Basis ι R M\nA : Type ?u.2919968\ni... | [
677,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
676,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/UV.lean | UV.sup_sdiff_mem_of_mem_compression | [
{
"state_after": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ s ∧ (a ⊔ u) \\ v ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\nhva : v ≤ a\nhua : Disjoint u a\n⊢ (a ⊔ u) \\ v ∈ s",
"state_b... | [
277,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Additive.lean | BoxIntegral.BoxAdditiveMap.coe_inj | [] | [
81,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/NumberTheory/VonMangoldt.lean | Nat.ArithmeticFunction.log_apply | [] | [
56,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.join_think | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nS : WSeq (WSeq α)\n⊢ Seq.join\n ((fun o =>\n match o with\n | none => Seq1.ret none\n | some s => (none, s)) <$>\n Seq.cons none S) =\n Seq.cons none\n (Seq.join\n ((fun o =>\n match o with\n ... | [
763,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
760,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.ofReal_zero | [] | [
146,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Group.conjugatesOfSet_subset | [] | [
2435,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2433,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | AffineMap.id_apply | [] | [
398,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
397,
1
] |
Mathlib/Topology/SubsetProperties.lean | isCompact_univ_pi | [
{
"state_after": "case h.e'_3\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nh : ∀ (i : ι), IsCompact (s i)\n⊢ Set.pi univ s = {x | ∀ (i : ι), x i ∈ s i}",
"st... | [
1042,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1039,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.eventually_bind | [] | [
2719,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2717,
1
] |
Mathlib/Algebra/Order/Chebyshev.lean | Antivary.card_smul_sum_le_sum_smul_sum | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : LinearOrderedAddCommGroup β\ninst✝² : Module α β\ninst✝¹ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\ninst✝ : Fintype ι\nhfg : Antivary f g\n⊢ Fintype.card ι •... | [
93,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
] |
Mathlib/MeasureTheory/CardMeasurableSpace.lean | MeasurableSpace.cardinal_generateMeasurable_le | [
{
"state_after": "α : Type u\ns : Set (Set α)\n⊢ (#↑(⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i)) ≤ max (#↑s) 2 ^ ℵ₀",
"state_before": "α : Type u\ns : Set (Set α)\n⊢ (#↑{t | GenerateMeasurable s t}) ≤ max (#↑s) 2 ^ ℵ₀",
"tactic": "rw [generateMeasurable_eq_rec]"
},
{
"s... | [
167,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
] |
Mathlib/Data/Nat/ModEq.lean | Nat.odd_of_mod_four_eq_three | [
{
"state_after": "no goals",
"state_before": "m n✝ a b c d n : ℕ\n⊢ n % 4 = 3 → n % 2 = 1",
"tactic": "simpa [ModEq, show 2 * 2 = 4 by norm_num, show 3 % 4 = 3 by norm_num] using\n @ModEq.of_mul_left 2 n 3 2"
},
{
"state_after": "no goals",
"state_before": "m n✝ a b c d n : ℕ\n⊢ 2 * 2 = 4",... | [
518,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
516,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean | deriv_const_smul | [] | [
153,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.FinMeasSupp.map_iff | [] | [
1220,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1218,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | lt_of_pow_lt_pow' | [] | [
289,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
288,
1
] |
Mathlib/Data/Nat/Prime.lean | Nat.Prime.ne_one | [] | [
88,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/Analysis/SpecialFunctions/Stirling.lean | Stirling.log_stirlingSeq_bounded_by_constant | [
{
"state_after": "case intro\nd : ℝ\nh : ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (succ n)) ≤ d\n⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (succ n))",
"state_before": "⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (succ n))",
"tactic": "obtain ⟨d, h⟩ := log_stirlingSeq_bounded_aux"
... | [
183,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.continuousWithinAt_setToFun_of_dominated | [] | [
1795,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1789,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | InnerProductSpace.Core.inner_self_im | [
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.327167\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx : F\n⊢ I * (↑(starRingEnd 𝕜) (inner x x) - inner x x) / 2 = ↑0",
"state_before": "𝕜 : Type u_1\nE : Type ?u.327167\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹... | [
218,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
216,
1
] |
Mathlib/Order/SymmDiff.lean | symmDiff_le | [] | [
150,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
149,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.iInter₂_subset | [] | [
340,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
] |
Mathlib/Order/WellFoundedSet.lean | Set.WellFoundedOn.insert | [] | [
512,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
511,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean | Equiv.Perm.subtypeCongrHom_injective | [
{
"state_after": "case mk.mk\nα : Type u\nβ : Type v\np : α → Prop\ninst✝ : DecidablePred p\nfst✝¹ : Perm { a // p a }\nsnd✝¹ : Perm { a // ¬p a }\nfst✝ : Perm { a // p a }\nsnd✝ : Perm { a // ¬p a }\nh : ↑(subtypeCongrHom p) (fst✝¹, snd✝¹) = ↑(subtypeCongrHom p) (fst✝, snd✝)\n⊢ (fst✝¹, snd✝¹) = (fst✝, snd✝)",
... | [
291,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
287,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.dense_iff_topologicalClosure_eq_top | [
{
"state_after": "R R' : Type u\nM M' : Type v\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Ring R'\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : TopologicalSpace M'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M\ninst✝³ : ContinuousConstSMul R M\ninst✝² : Module R' M'\ninst✝¹ : ContinuousConstSMul R' M'\... | [
191,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.coe_prod | [] | [
1711,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1709,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | tendsto_log_div_rpow_nhds_zero | [] | [
330,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
328,
1
] |
Mathlib/Algebra/Algebra/Operations.lean | Submodule.pow_induction_on_left' | [
{
"state_after": "case zero\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n✝ : A\nC : (n : ℕ) → (x : A) → x ∈ M ^ n → Prop\nhr : ∀ (r : R), C 0 (↑(algebraMap R A) r) (_ : ↑(algebraMap R A) r ∈ 1)\nhadd : ∀ (x y : ... | [
450,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
437,
11
] |
Mathlib/ModelTheory/ElementaryMaps.lean | FirstOrder.Language.ElementaryEmbedding.injective | [
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.306353\nQ : Type ?u.306356\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nφ : M ↪ₑ[L] N\nx y : M\n⊢ ↑φ x = ↑φ y → x = y",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Ty... | [
134,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean | Convex.image_sub_le_mul_sub_of_deriv_le | [
{
"state_after": "E : Type ?u.408885\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.408981\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nC : (fun x => ℝ) 1\nle_hf' : ∀ (x : ℝ)... | [
876,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
869,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | CategoryTheory.Limits.hasPushout_of_preservesPushout | [] | [
214,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/Algebra/CovariantAndContravariant.lean | Covariant.flip | [] | [
140,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean | Nat.Prime.pow_dvd_iff_dvd_ord_proj | [
{
"state_after": "no goals",
"state_before": "p k n : ℕ\npp : Prime p\nhn : n ≠ 0\n⊢ p ^ k ∣ n ↔ p ^ k ∣ p ^ ↑(Nat.factorization n) p",
"tactic": "rw [pow_dvd_pow_iff_le_right pp.one_lt, pp.pow_dvd_iff_le_factorization hn]"
}
] | [
497,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
495,
1
] |
Mathlib/Data/Real/Irrational.lean | irrational_nat_div_iff | [
{
"state_after": "no goals",
"state_before": "q : ℚ\nm : ℤ\nn : ℕ\nx : ℝ\n⊢ Irrational (↑n / x) ↔ n ≠ 0 ∧ Irrational x",
"tactic": "simp [div_eq_mul_inv]"
}
] | [
654,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
653,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean | Polynomial.natTrailingDegree_le_of_mem_supp | [] | [
319,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
Std/Data/RBMap/WF.lean | Std.RBNode.RedRed.balance2 | [
{
"state_after": "α : Type u_1\nc : RBColor\nn : Nat\np : Prop\nl : RBNode α\nv : α\nr : RBNode α\nhl : Balanced l c n\nhr : RedRed p r n\n⊢ ∃ c,\n Balanced\n (match l, v, r with\n | a, x, node red (node red b y c) z d => node red (node black a x b) y (node black c z d)\n | a, x, node red b y ... | [
175,
66
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
166,
11
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | Liouville.liouvilleWith | [
{
"state_after": "x : ℝ\nhx : Liouville x\np : ℝ\nthis : LiouvilleWith (↑⌈p⌉₊) x\n⊢ LiouvilleWith p x\n\ncase this\nx : ℝ\nhx : Liouville x\np : ℝ\n⊢ LiouvilleWith (↑⌈p⌉₊) x",
"state_before": "x : ℝ\nhx : Liouville x\np : ℝ\n⊢ LiouvilleWith p x",
"tactic": "suffices : LiouvilleWith ⌈p⌉₊ x"
},
{
... | [
380,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
375,
11
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean | MeasureTheory.VectorMeasure.AbsolutelyContinuous.trans | [] | [
1107,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1105,
1
] |
Mathlib/RingTheory/WittVector/StructurePolynomial.lean | wittStructureRat_existsUnique | [
{
"state_after": "case refine'_1\np : ℕ\nR : Type ?u.216558\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℚ\n⊢ (fun φ => ∀ (n : ℕ), ↑(bind₁ φ) (W_ ℚ n) = ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) Φ) (wittStructureRat p Φ)\n\ncase refine'_2\np : ℕ\nR : Type ?u.216558\n... | [
164,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
154,
1
] |
Mathlib/Analysis/Convex/Gauge.lean | gauge_lt_one_of_mem_of_open | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.199119\nE : Type u_1\nF : Type ?u.199125\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\nhs₁ : Convex ℝ s\nhs₀ : 0 ∈ s\nhs₂ : IsOpen s\nx : E\nhx : x ∈ s\n⊢ gauge s x < 1",
... | [
362,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
] |
Mathlib/Data/Bool/Basic.lean | Bool.le_and | [
{
"state_after": "no goals",
"state_before": "⊢ ∀ {x y z : Bool}, x ≤ y → x ≤ z → x ≤ (y && z)",
"tactic": "decide"
}
] | [
361,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
] |
Mathlib/Data/QPF/Multivariate/Basic.lean | MvQPF.has_good_supp_iff | [
{
"state_after": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\n⊢ (∀ (p : (i : Fin2 n) → α i → Prop), LiftP p x ↔ ∀ (i : Fin2 n) (u : α i), u ∈ supp x i → p i u) →\n ∃ a f,\n abs { fst := a, snd := f } = x ∧\n ∀ (i : Fin2 n) (a' : (P F).A) (f'... | [
210,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
] |
Mathlib/Data/PNat/Basic.lean | PNat.dvd_antisymm | [] | [
439,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
438,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearMap.exists_ne_zero | [
{
"state_after": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.448146\ninst✝¹⁶ : Semiring R₁\ninst✝¹⁵ : Semiring R₂\ninst✝¹⁴ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_3\ninst✝¹³ : TopologicalSpace M₁\ninst✝¹² : AddCommMonoid M₁\nM'₁ : Type ?u.448233\ninst✝¹¹ : TopologicalSpace ... | [
654,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
652,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | intervalIntegral.integral_of_le | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.9584325\n𝕜 : Type ?u.9584328\nE : Type u_1\nF : Type ?u.9584334\nA : Type ?u.9584337\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\nh : a ≤ b\n⊢ (∫ (x : ℝ) in a..... | [
468,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
467,
1
] |
Mathlib/NumberTheory/PythagoreanTriples.lean | PythagoreanTriple.coprime_classification | [
{
"state_after": "x y z : ℤ\n⊢ PythagoreanTriple x y z ∧ Int.gcd x y = 1 ↔\n ∃ m n,\n (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧\n (z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)) ∧ Int.gcd m n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)",
"state_before": "x... | [
622,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
596,
1
] |
Mathlib/GroupTheory/DoubleCoset.lean | Doset.mk_out'_eq_mul | [
{
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147,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
1
] |
Mathlib/Analysis/Quaternion.lean | Quaternion.norm_coe | [
{
"state_after": "no goals",
"state_before": "a : ℝ\n⊢ ‖↑a‖ = ‖a‖",
"tactic": "rw [norm_eq_sqrt_real_inner, inner_self, normSq_coe, Real.sqrt_sq_eq_abs, Real.norm_eq_abs]"
}
] | [
78,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.prod_inf_prod | [] | [
1139,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1137,
1
] |
Mathlib/CategoryTheory/ConcreteCategory/Basic.lean | CategoryTheory.ConcreteCategory.hasCoeToFun_Type | [] | [
196,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
196,
1
] |
Mathlib/Probability/Independence/Basic.lean | ProbabilityTheory.IndepFun.ae_eq | [
{
"state_after": "case intro.intro.intro.intro\nΩ : Type u_2\nι : Type ?u.3257019\nβ : Type u_1\nβ' : Type ?u.3257025\nγ : Type ?u.3257028\nγ' : Type ?u.3257031\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β\ng✝ : Ω → β'\nmβ : MeasurableSpace β\nf g f' g' : Ω → β\nhfg : IndepFun f g\nhf : f =ᵐ... | [
719,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
713,
1
] |
Mathlib/RingTheory/PowerSeries/WellKnown.lean | PowerSeries.exp_mul_exp_eq_exp_add | [
{
"state_after": "case h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\na b : A\nn : ℕ\n⊢ ↑(coeff A n) (↑(rescale a) (exp A) * ↑(rescale b) (exp A)) = ↑(coeff A n) (↑(rescale (a + b)) (exp A))",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\na b : A\n⊢ ↑(rescale a) (exp A... | [
183,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
] |
Mathlib/Topology/Instances/Irrational.lean | eventually_residual_irrational | [] | [
56,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.lintegral_mono | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.899767\nγ : Type ?u.899770\nδ : Type ?u.899773\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\nhfg : f ≤ g\nhμν : μ ≤ ν\n⊢ lintegral (f ⊔ g) μ = lintegral g μ",
"tactic": "rw [sup_of_le_right hfg]"
}
] | [
1129,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1122,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.enumOrd_univ | [
{
"state_after": "S : Set Ordinal\n⊢ enumOrd (range id) = id",
"state_before": "S : Set Ordinal\n⊢ enumOrd Set.univ = id",
"tactic": "rw [← range_id]"
},
{
"state_after": "no goals",
"state_before": "S : Set Ordinal\n⊢ enumOrd (range id) = id",
"tactic": "exact enumOrd_range strictMono_i... | [
2223,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2221,
1
] |
Mathlib/Order/Heyting/Basic.lean | LE.le.codisjoint_hnot_left | [] | [
1045,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1044,
1
] |
Std/Data/List/Lemmas.lean | List.append_ne_nil_of_left_ne_nil | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b : List α\nh0 : a ≠ []\n⊢ a ++ b ≠ []",
"tactic": "simp [*]"
}
] | [
120,
95
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
120,
1
] |
Mathlib/LinearAlgebra/Basic.lean | Submodule.map_subtype_range_ofLe | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.1550580\nR₂ : Type ?u.1550583\nR₃ : Type ?u.1550586\nR₄ : Type ?u.1550589\nS : Type ?u.1550592\nK : Type ?u.1550595\nK₂ : Type ?u.1550598\nM : Type u_2\nM' : Type ?u.1550604\nM₁ : Type ?u.1550607\nM₂ : Type ?u.1550610\nM₃ : Type ?u.155... | [
1656,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1655,
1
] |
Mathlib/Algebra/Hom/Iterate.lean | RingHom.iterate_map_one | [] | [
132,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/Analysis/Normed/Group/InfiniteSum.lean | cauchySeq_finset_of_summable_norm | [] | [
96,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Order/Basic.lean | lt_of_lt_of_eq | [] | [
183,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean | NormedAddGroupHom.lift_unique | [
{
"state_after": "case H\nM : Type u_2\nN✝ : Type ?u.463092\ninst✝² : SeminormedAddCommGroup M\ninst✝¹ : SeminormedAddCommGroup N✝\nN : Type u_1\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : ∀ (s : M), s ∈ S → ↑f s = 0\ng : NormedAddGroupHom (M ⧸ S) N\nh : NormedAddGroupH... | [
373,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
366,
1
] |
Mathlib/Order/UpperLower/Basic.lean | IsUpperSet.image | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.18624\nι : Sort ?u.18627\nκ : ι → Sort ?u.18632\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set α\np : α → Prop\na : α\nhs : IsUpperSet s\nf : α ≃o β\n⊢ IsUpperSet (↑↑f '' s)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.18624\nι : Sor... | [
256,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
253,
1
] |
Std/Data/Int/Lemmas.lean | Int.natAbs_mul_self' | [
{
"state_after": "no goals",
"state_before": "a : Int\n⊢ ↑(natAbs a) * ↑(natAbs a) = a * a",
"tactic": "rw [← Int.ofNat_mul, natAbs_mul_self]"
}
] | [
179,
40
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
178,
9
] |
Mathlib/Computability/TuringMachine.lean | Turing.frespects_eq | [
{
"state_after": "σ₁ : Type u_1\nσ₂ : Type u_2\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂\na₂ b₂ : σ₂\nh : f₂ a₂ = f₂ b₂\n⊢ (match none with\n | some b₁ => Reaches₁ f₂ a₂ (tr b₁)\n | none => f₂ a₂ = none) ↔\n match none with\n | some b₁ => Reaches₁ f₂ b₂ (tr b₁)\n | none => f₂ b₂ = none",
"state_befo... | [
968,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
965,
1
] |
Mathlib/Order/WellFoundedSet.lean | Set.isWf_iff_no_descending_seq | [] | [
223,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
220,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | Real.neg_pi_div_two_lt_arcsin | [
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ sin (-(π / 2)) < x ↔ -1 < x",
"tactic": "rw [sin_neg, sin_pi_div_two]"
}
] | [
240,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
238,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean | Subalgebra.star_mem_star_iff | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.406937\nR : Type u_1\nA : Type u_2\nB : Type ?u.406946\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRin... | [
363,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
362,
1
] |
Mathlib/NumberTheory/LSeries.lean | Nat.ArithmeticFunction.LSeries_add | [
{
"state_after": "f g : ArithmeticFunction ℂ\nz : ℂ\nhf : LSeriesSummable f z\nhg : LSeriesSummable g z\n⊢ (∑' (n : ℕ), (↑f n + ↑g n) / ↑n ^ z) = (∑' (n : ℕ), ↑f n / ↑n ^ z) + ∑' (n : ℕ), ↑g n / ↑n ^ z",
"state_before": "f g : ArithmeticFunction ℂ\nz : ℂ\nhf : LSeriesSummable f z\nhg : LSeriesSummable g z\n... | [
132,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/LinearAlgebra/Prod.lean | LinearMap.coprod_map_prod | [
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.90184\nM₆ : Type ?u.90187\nS✝ : Type ?u.90190\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring S✝\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝... | [
268,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
263,
1
] |
Mathlib/Logic/Equiv/Basic.lean | Equiv.optionEquivSumPUnit_some | [] | [
430,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
429,
1
] |
Mathlib/Data/Int/Div.lean | Int.eq_zero_of_dvd_of_natAbs_lt_natAbs | [
{
"state_after": "a b : ℤ\nw : natAbs a ∣ natAbs b\nh : natAbs b < natAbs a\n⊢ b = 0",
"state_before": "a b : ℤ\nw : a ∣ b\nh : natAbs b < natAbs a\n⊢ b = 0",
"tactic": "rw [← natAbs_dvd, ← dvd_natAbs, coe_nat_dvd] at w"
},
{
"state_after": "a b : ℤ\nw : natAbs a ∣ natAbs b\nh : natAbs b < natAb... | [
39,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
35,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.IsBoundedUnder.bddAbove_range_of_cofinite | [
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.11552\nι : Type ?u.11555\nr : α → α → Prop\nf✝ g : Filter α\ninst✝ : SemilatticeSup β\nf : α → β\nb : β\nhb : ∀ᶠ (x : β) in map f cofinite, (fun x x_1 => x ≤ x_1) x b\n⊢ BddAbove (range f)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ ... | [
147,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
] |
Mathlib/Data/Nat/Choose/Multinomial.lean | Nat.multinomial_nil | [
{
"state_after": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\nn : ℕ\n⊢ 1 / 1 = 1",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\nn : ℕ\n⊢ multinomial ∅ f = 1",
"tactic": "dsimp [multinomial]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Finset α\n... | [
63,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.mem_Icc_of_Ioo | [] | [
655,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
654,
1
] |
Mathlib/Data/Finset/Lattice.lean | map_finset_inf | [
{
"state_after": "no goals",
"state_before": "F : Type u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.109238\nι : Type u_4\nκ : Type ?u.109244\ninst✝⁴ : SemilatticeInf α\ninst✝³ : OrderTop α\ns✝¹ s₁ s₂ : Finset β\nf✝ g✝ : β → α\na : α\ninst✝² : SemilatticeInf β\ninst✝¹ : OrderTop β\ninst✝ : InfTopHomClass F α... | [
375,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
372,
1
] |
Std/Data/String/Lemmas.lean | String.Iterator.ValidFor.of_eq | [] | [
525,
31
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
524,
1
] |
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint | [
{
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... | [
225,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
220,
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Mathlib/Algebra/Order/ToIntervalMod.lean | toIocDiv_wcovby_toIcoDiv | [
{
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"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α... | [
701,
14
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696,
1
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Mathlib/Data/Matrix/Basic.lean | Matrix.of_symm_apply | [] | [
129,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
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Mathlib/Algebra/Order/Rearrangement.lean | MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff | [
{
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188,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
184,
1
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Mathlib/Data/Sum/Basic.lean | Sum.not_liftRel_inr_inl | [] | [
410,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
409,
1
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Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | CategoryTheory.isIso_of_mono_of_strongEpi | [
{
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"tactic": "simp"
},
{
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"state_before": "C : Type u\ninst✝² : Category C\nP Q : C\nf : P ⟶ Q\ninst✝¹ : Mon... | [
192,
70
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191,
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Mathlib/Order/Monotone/Basic.lean | StrictAnti.comp_strictAntiOn | [] | [
706,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
704,
11
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Mathlib/Order/LiminfLimsup.lean | OrderIso.limsup_apply | [
{
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1232,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1217,
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Mathlib/MeasureTheory/Function/Jacobian.lean | MeasureTheory.aemeasurable_toNNReal_abs_det_fderivWithin | [
{
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776,
43
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770,
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Mathlib/RingTheory/IntegralClosure.lean | IsIntegralClosure.algebraMap_equiv | [] | [
949,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
948,
1
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Mathlib/LinearAlgebra/Prod.lean | Submodule.prod_comap_inl | [
{
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571,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
571,
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