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/Nat/Choose/Multinomial.lean | Nat.multinomial_insert_one | [
{
"state_after": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\nn : ℕ\ninst✝ : DecidableEq α\nh : ¬a ∈ s\nh₁ : f a = 1\n⊢ (∑ i in insert a s, f i)! / ∏ i in insert a s, (f i)! = succ (Finset.sum s f) * ((∑ i in s, f i)! / ∏ i in s, (f i)!)",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\... | [
77,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
72,
1
] |
Mathlib/Topology/Algebra/Monoid.lean | Submonoid.mem_nhds_one | [] | [
723,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
721,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean | Pell.d_pos | [
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\n⊢ 0 < 1",
"tactic": "decide"
}
] | [
101,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.apply_apply | [] | [
914,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
913,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean | multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity | [
{
"state_after": "case intro\nR : Type u_1\nA : Type ?u.1422103\nK : Type ?u.1422106\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing A\ninst✝⁸ : Field K\ninst✝⁷ : IsDomain A\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsPrincipalIdealRing R\ninst✝⁴ : DecidableEq R\ninst✝³ : DecidableEq (Ideal R)\ninst✝² : NormalizationMonoid R\nin... | [
1508,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1501,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean | Valuation.map_one_add_of_lt | [
{
"state_after": "K : Type ?u.2382416\nF : Type ?u.2382419\nR : Type u_2\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type ?u.2382431\nΓ''₀ : Type ?u.2382434\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : ↑v x < ↑v... | [
342,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineSubspace.isometryEquivMap.coe_apply | [] | [
913,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
911,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean | finprod_induction | [
{
"state_after": "G : Type ?u.40810\nM : Type u_1\nN : Type ?u.40816\nα : Sort u_2\nβ : Sort ?u.40822\nι : Sort ?u.40825\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : M → Prop\nhp₀ : p 1\nhp₁ : ∀ (x y : M), p x → p y → p (x * y)\nhp₂ : ∀ (i : α), p (f i)\n⊢ p (if h : Set.Finite (mulSupport ((fun ... | [
280,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
276,
1
] |
Mathlib/Data/Finset/Image.lean | Finset.image_mono | [] | [
450,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
450,
1
] |
Mathlib/Topology/Homeomorph.lean | Homeomorph.self_trans_symm | [
{
"state_after": "case H\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.35926\nδ : Type ?u.35929\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nx✝ : α\n⊢ ↑(Homeomorph.trans h (Homeomorph.symm h)) x✝ = ↑(Homeomorph.refl α) x✝",
"state... | [
162,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean | MeasurableSpace.generateFrom_le_iff | [] | [
396,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
394,
1
] |
Std/Logic.lean | and_congr_left_iff | [
{
"state_after": "no goals",
"state_before": "a c b : Prop\n⊢ (a ∧ c ↔ b ∧ c) ↔ c → (a ↔ b)",
"tactic": "simp only [and_comm, ← and_congr_right_iff]"
}
] | [
223,
46
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
222,
9
] |
Mathlib/Data/PNat/Factors.lean | PNat.count_factorMultiset | [
{
"state_after": "m : ℕ+\np : Nat.Primes\nk : ℕ\n⊢ Pow.pow (↑p) k ∣ m ↔ k ≤ Multiset.count p (factorMultiset m)",
"state_before": "m : ℕ+\np : Nat.Primes\nk : ℕ\n⊢ Pow.pow (↑p) k ∣ m ↔ k ≤ Multiset.count p (factorMultiset m)",
"tactic": "intros"
},
{
"state_after": "m : ℕ+\np : Nat.Primes\nk : ℕ... | [
406,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
395,
1
] |
Mathlib/Order/UpperLower/Basic.lean | LowerSet.map_map | [
{
"state_after": "case a.h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.107147\nκ : ι → Sort ?u.107152\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf✝ : α ≃o β\ns t : LowerSet α\na : α\nb : β\ng : β ≃o γ\nf : α ≃o β\nx✝ : γ\n⊢ x✝ ∈ ↑(↑(map g) (↑(map f) s)) ↔ x✝ ∈ ↑(↑(map (OrderIso.tr... | [
1034,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1032,
1
] |
Mathlib/Logic/Equiv/Defs.lean | Equiv.Perm.sigmaCongrRight_trans | [] | [
784,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
781,
9
] |
Mathlib/LinearAlgebra/Basis.lean | LinearMap.exists_rightInverse_of_surjective | [
{
"state_after": "ι : Type ?u.1199606\nι' : Type ?u.1199609\nR : Type ?u.1199612\nR₂ : Type ?u.1199615\nK : Type u_1\nM : Type ?u.1199621\nM' : Type ?u.1199624\nM'' : Type ?u.1199627\nV : Type u\nV' : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst... | [
1603,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1596,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.weightedVSubOfPoint_apply | [
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_1\nP : Type u_4\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type ?u.8363\ns₂ : Finset ι₂\nw : ι → k\np : ι → P\nb : P\n⊢ ↑(weightedVSubOfPoint s p b) w = ∑ i in s, ... | [
83,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/Analysis/Complex/PhragmenLindelof.lean | PhragmenLindelof.eqOn_horizontal_strip | [] | [
271,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
262,
1
] |
Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.algebraMap_mem | [] | [
157,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/Topology/Order/Basic.lean | nhds_basis_zero_abs_sub_lt | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrderedAddCommGroup α\ninst✝¹ : OrderTopology α\nl : Filter β\nf g : β → α\ninst✝ : NoMaxOrder α\n⊢ HasBasis (𝓝 0) (fun ε => 0 < ε) fun ε => {b | abs b < ε}",
"tactic": "simpa u... | [
1928,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1926,
1
] |
Mathlib/Data/Sigma/Basic.lean | PSigma.mk.inj_iff | [] | [
231,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
227,
1
] |
Mathlib/Topology/Instances/Sign.lean | continuousAt_sign_of_pos | [
{
"state_after": "α : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α\ninst✝² : PartialOrder α\ninst✝¹ : DecidableRel fun x x_1 => x < x_1\ninst✝ : OrderTopology α\na : α\nh : 0 < a\n⊢ (fun x => 1) =ᶠ[nhds a] ↑SignType.sign",
"state_before": "α : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α... | [
38,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
35,
1
] |
Mathlib/Data/Finsupp/ToDfinsupp.lean | sigmaFinsuppEquivDfinsupp_symm_apply | [] | [
313,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
1
] |
Mathlib/RingTheory/Subring/Pointwise.lean | Subring.pointwise_smul_subset_iff | [] | [
132,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/Algebra/Lie/Abelian.lean | LieModule.coe_maxTrivHom_apply | [] | [
184,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | CategoryTheory.Limits.biprod.inr_fst | [] | [
1341,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1339,
1
] |
Mathlib/Algebra/Divisibility/Basic.lean | dvd_mul_right | [] | [
81,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Mathlib/Data/MvPolynomial/Monad.lean | MvPolynomial.hom_bind₁ | [
{
"state_after": "no goals",
"state_before": "σ : Type u_4\nτ : Type u_2\nR : Type u_1\nS : Type u_3\nT : Type ?u.758101\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf✝ : σ → MvPolynomial τ R\nf : MvPolynomial τ R →+* S\ng : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ ↑f (↑(b... | [
290,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
288,
1
] |
Mathlib/Init/Data/Subtype/Basic.lean | Subtype.ne_of_val_ne | [] | [
37,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
36,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean | OpenSubgroup.coe_toOpens | [] | [
99,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean | isBigO_pow_pow_of_le_left | [] | [
105,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/Data/ULift.lean | ULift.exists | [] | [
132,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean | cauchySeq_of_le_geometric_two | [] | [
410,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
409,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | mem_spanPoints | [] | [
112,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Computability/Primrec.lean | list_cons' | [] | [
922,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
920,
9
] |
Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.le_nhds_lim | [] | [
438,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
436,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.radical_mul | [
{
"state_after": "R : Type u\nι : Type ?u.325271\ninst✝ : CommSemiring R\nI J K L : Ideal R\n⊢ radical (I * J) ≤ radical I ⊓ radical J",
"state_before": "R : Type u\nι : Type ?u.325271\ninst✝ : CommSemiring R\nI J K L : Ideal R\n⊢ radical (I * J) = radical I ⊓ radical J",
"tactic": "refine le_antisymm ?... | [
938,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
933,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean | Nat.ord_proj_of_not_prime | [
{
"state_after": "no goals",
"state_before": "n p : ℕ\nhp : ¬Prime p\n⊢ p ^ ↑(factorization n) p = 1",
"tactic": "simp [factorization_eq_zero_of_non_prime n hp]"
}
] | [
360,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
359,
1
] |
Mathlib/Order/Disjoint.lean | Codisjoint.mono | [] | [
253,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
252,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean | DifferentiableOn.const_add | [] | [
295,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean | DirectSum.decompose_mul_add_left | [] | [
157,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
154,
1
] |
Mathlib/Algebra/ModEq.lean | AddCommGroup.modEq_sub_iff_add_modEq | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz : ℤ\n⊢ c + a ≡ b [PMOD p] ↔ a + c ≡ b [PMOD p]",
"tactic": "rw [add_comm]"
}
] | [
269,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
1
] |
Mathlib/Data/Set/Basic.lean | Set.ite_compl | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t✝ t₁ t₂ u t s s' : Set α\n⊢ Set.ite (tᶜ) s s' = Set.ite t s' s",
"tactic": "rw [Set.ite, Set.ite, diff_compl, union_comm, diff_eq]"
}
] | [
2252,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2251,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.cons_erase | [] | [
1035,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1034,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.lt_sub_one_iff | [
{
"state_after": "case mk.zero\nn✝ m n : ℕ\nhk : zero < n + 2\n⊢ { val := zero, isLt := hk } < { val := zero, isLt := hk } - 1 ↔ { val := zero, isLt := hk } = 0\n\ncase mk.succ\nn✝ m n k : ℕ\nhk : Nat.succ k < n + 2\n⊢ { val := Nat.succ k, isLt := hk } < { val := Nat.succ k, isLt := hk } - 1 ↔ { val := Nat.succ... | [
1991,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1985,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean | sub_orthogonalProjection_mem_orthogonal | [
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.498264\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhw : w ∈ K\n⊢ inner w (v - ↑(↑(... | [
491,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
488,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.disjoint_iUnion₂_left | [] | [
2089,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2087,
1
] |
Std/Logic.lean | iff_of_false | [] | [
61,
71
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
61,
1
] |
Mathlib/Data/Real/GoldenRatio.lean | inv_gold | [
{
"state_after": "this : 1 + sqrt 5 ≠ 0\n⊢ φ⁻¹ = -ψ",
"state_before": "⊢ φ⁻¹ = -ψ",
"tactic": "have : 1 + Real.sqrt 5 ≠ 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))"
},
{
"state_after": "this : 1 + sqrt 5 ≠ 0\n⊢ 2 * 2 = sqrt 5 * sqrt 5 - 1",
"state_before": "this ... | [
53,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
] |
Std/Data/List/Lemmas.lean | List.ne_of_not_mem_cons | [] | [
96,
89
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
96,
1
] |
Mathlib/RingTheory/WittVector/WittPolynomial.lean | constantCoeff_xInTermsOfW | [
{
"state_after": "p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst✝ : Invertible ↑p\nn : ℕ\n⊢ ∀ (n : ℕ), (∀ (m : ℕ), m < n → ↑constantCoeff (xInTermsOfW p R m) = 0) → ↑constantCoeff (xInTermsOfW p R n) = 0",
"state_before": "p : ℕ\nR : Type u_1\ninst✝² : CommRi... | [
243,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
227,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean | Nat.ArithmeticFunction.IsMultiplicative.int_cast | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nf : ArithmeticFunction ℤ\ninst✝ : Ring R\nh : IsMultiplicative f\n⊢ ↑↑f 1 = 1",
"tactic": "simp [h]"
},
{
"state_after": "case intro\nR : Type u_1\nf : ArithmeticFunction ℤ\ninst✝ : Ring R\nm n : ℕ\ncop : coprime m n\nleft✝ : ↑f 1 = 1\n... | [
622,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
619,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.strictMono_subseq_of_id_le | [] | [
598,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
596,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | GroupTopology.toTopologicalSpace_inf | [] | [
1942,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1940,
1
] |
Mathlib/RingTheory/FiniteType.lean | AlgHom.FiniteType.comp_surjective | [] | [
309,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
] |
Mathlib/Topology/Constructions.lean | isClosed_sum_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.119189\nδ : Type ?u.119192\nε : Type ?u.119195\nζ : Type ?u.119198\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns : Set (α ⊕ β)\n⊢ IsClosed s ↔ IsClosed (i... | [
899,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
897,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean | CategoryTheory.Subobject.bot_arrow | [] | [
343,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
342,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.coe_sub_iff_lt | [
{
"state_after": "case zero\nn m : ℕ\na b : Fin zero\n⊢ ↑(a - b) = zero + ↑a - ↑b ↔ a < b\n\ncase succ\nn✝ m n : ℕ\na b : Fin (Nat.succ n)\n⊢ ↑(a - b) = Nat.succ n + ↑a - ↑b ↔ a < b",
"state_before": "n✝ m n : ℕ\na b : Fin n\n⊢ ↑(a - b) = n + ↑a - ↑b ↔ a < b",
"tactic": "cases' n with n"
},
{
"s... | [
1981,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1971,
1
] |
Mathlib/Data/Vector3.lean | Vector3.append_add | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm✝ n✝ m : ℕ\nv : Vector3 α (succ m)\nn : ℕ\nw : Vector3 α n\ni : Fin2 n\n_a : α\nt : Vector3 α m\n⊢ ((_a :: t) +-+ w) (add i (succ m)) = w i",
"tactic": "simp [append_add, add]"
}
] | [
183,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
180,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean | UniformCauchySeqOn.prod | [] | [
536,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
533,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean | OrderMonoidHom.coe_mk | [] | [
324,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/Analysis/Convex/Basic.lean | Convex.translate | [] | [
223,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Mathlib/Algebra/Order/Field/Power.lean | Odd.zpow_nonneg_iff | [
{
"state_after": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nn k : ℤ\nhk : n = 2 * k + 1\n⊢ 0 ≤ a ^ n ↔ 0 ≤ a",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhn : Odd n\n⊢ 0 ≤ a ^ n ↔ 0 ≤ a",
"tactic": "cases' hn with k hk"
},
{
"state... | [
194,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
11
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | Real.abs_rpow_of_nonneg | [
{
"state_after": "x y : ℝ\nhx_nonneg : 0 ≤ x\nh_rpow_nonneg : 0 ≤ x ^ y\n⊢ abs (x ^ y) = abs x ^ y",
"state_before": "x y : ℝ\nhx_nonneg : 0 ≤ x\n⊢ abs (x ^ y) = abs x ^ y",
"tactic": "have h_rpow_nonneg : 0 ≤ x ^ y := Real.rpow_nonneg_of_nonneg hx_nonneg _"
},
{
"state_after": "no goals",
"... | [
151,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
149,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean | Trivialization.sourceHomeomorphBaseSetProd_symm_apply | [] | [
492,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
489,
1
] |
Mathlib/Data/Set/Finite.lean | Set.infinite_univ_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\n⊢ Set.Infinite univ ↔ Infinite α",
"tactic": "rw [Set.Infinite, finite_univ_iff, not_finite_iff_infinite]"
}
] | [
1274,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1273,
1
] |
Mathlib/Computability/TMToPartrec.lean | Turing.ToPartrec.cont_eval_fix | [
{
"state_after": "f : Code\nk : Cont\nv : List ℕ\nfok : Code.Ok f\nx : Cfg\n⊢ x ∈ eval step (stepNormal f (Cont.fix f k) v) ↔\n x ∈ do\n let v ← Code.eval (Code.fix f) v\n eval step (Cfg.ret k v)",
"state_before": "f : Code\nk : Cont\nv : List ℕ\nfok : Code.Ok f\n⊢ eval step (stepNormal f (Cont... | [
698,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
628,
1
] |
Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | SimpleGraph.adjMatrix_vecMul_apply | [
{
"state_after": "V : Type u_2\nα : Type u_1\nβ : Type ?u.43997\nG : SimpleGraph V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype V\ninst✝ : NonAssocSemiring α\nv : V\nvec : V → α\n⊢ (vec ⬝ᵥ fun i => adjMatrix α G i v) = (fun u => vec u) ⬝ᵥ adjMatrix α G v",
"state_before": "V : Type u_2\nα : Type u_1\nβ : ... | [
224,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
220,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.LipschitzWith.integrable_comp_iff_of_antilipschitz | [
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.992421\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nK K' : ℝ≥0\nf : α → β\ng : β → γ\nhg : LipschitzWith K g\nhg' : Antilip... | [
817,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
814,
1
] |
Mathlib/Combinatorics/Hindman.lean | Ultrafilter.eventually_mul | [] | [
65,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean | Submodule.IsPrincipal.span_singleton_generator | [] | [
106,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean | LinearMap.IsRefl.ortho_comm | [] | [
182,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean | Set.Ioo.lt_one | [] | [
350,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
349,
1
] |
Mathlib/Algebra/CharZero/Quotient.lean | AddSubgroup.nsmul_mem_zmultiples_iff_exists_sub_div | [
{
"state_after": "R : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\np r : R\nn : ℕ\nhn : n ≠ 0\n⊢ (∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p) ↔ ∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p",
"state_before": "R : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\np r : R\nn : ℕ\nhn : n ≠ 0\n⊢ n • r ∈ zmult... | [
51,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
46,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean | padicNormE.nonarchimedean | [
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ_[p]\n⊢ ↑(↑padicNormE (q + r)) ≤ max ↑(↑padicNormE q) ↑(↑padicNormE r)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ_[p]\n⊢ ‖q + r‖ ≤ max ‖q‖ ‖r‖",
"tactic": "dsimp [norm]"
},
{
"state_after": "no goals",
"state_before": "... | [
824,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
822,
1
] |
Mathlib/Order/Heyting/Regular.lean | Heyting.Regular.coe_himp | [] | [
165,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Path.count_edges_eq_one | [] | [
1316,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1314,
1
] |
Mathlib/Data/Polynomial/Monic.lean | Polynomial.Monic.natDegree_mul' | [
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\nhq : q ≠ 0\n⊢ Polynomial.leadingCoeff p * Polynomial.leadingCoeff q ≠ 0",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Moni... | [
190,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
8
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | le_mul_of_one_le_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c d : α\ninst✝³ : MulOneClass α\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : PosMulMono α\nha : 0 ≤ a\nh : 1 ≤ b\n⊢ a ≤ a * b",
"tactic": "simpa only [mul_one] using mul_le_mul_of_nonneg_left h ha"
}
] | [
682,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
681,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_const_sub_Ici | [] | [
222,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
221,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_eq_of_degree_eq_some | [
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : degree 0 = ↑n\nhp0 : p = 0\n⊢ False",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : degree p = ↑n\nhp0 : p = 0\n⊢ False",
"ta... | [
157,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean | Equiv.swap_apply_apply | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\n⊢ swap (↑f x) (↑f y) = f * swap x y * f⁻¹",
"tactic": "rw [mul_swap_eq_swap_mul, mul_inv_cancel_right]"
}
] | [
534,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
533,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | AffineEquiv.coe_constVSub | [] | [
468,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
467,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.normSq_eq_def' | [] | [
470,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
469,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.map_injective | [] | [
2857,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2856,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean | Cardinal.ord_lt_ord | [] | [
1418,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1417,
1
] |
Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean | Matrix.GLPos.det_ne_zero | [] | [
202,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/Logic/Function/Iterate.lean | Function.iterate_succ_apply' | [
{
"state_after": "α : Type u\nβ : Type v\nf : α → α\nn : ℕ\nx : α\n⊢ (f ∘ f^[n]) x = f ((f^[n]) x)",
"state_before": "α : Type u\nβ : Type v\nf : α → α\nn : ℕ\nx : α\n⊢ (f^[Nat.succ n]) x = f ((f^[n]) x)",
"tactic": "rw [iterate_succ']"
},
{
"state_after": "no goals",
"state_before": "α : Ty... | [
181,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean | Polynomial.dvd_C_mul | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nha : a ≠ 0\nx✝ : p ∣ ↑C a * q\nr : R[X]\nhr : ↑C a * q = p * r\n⊢ q = p * (↑C a⁻¹ * r)",
"tactic": "rw [mul_left_comm p, ← hr, ← mul_assoc, ← C.map_mul, _root_.inv_m... | [
452,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
447,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | limsup_eq_tendsto_sum_indicator_nat_atTop | [
{
"state_after": "case h\nα : Type u\nβ : Type v\ns : ℕ → Set α\nω : α\n⊢ ω ∈ limsup s atTop ↔ ω ∈ {ω | Tendsto (fun n => ∑ k in Finset.range n, Set.indicator (s (k + 1)) 1 ω) atTop atTop}",
"state_before": "α : Type u\nβ : Type v\ns : ℕ → Set α\n⊢ limsup s atTop = {ω | Tendsto (fun n => ∑ k in Finset.range... | [
514,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
459,
1
] |
Mathlib/Analysis/Convex/Strict.lean | StrictConvex.affine_preimage | [
{
"state_after": "𝕜 : Type u_2\n𝕝 : Type ?u.200813\nE : Type u_3\nF : Type u_1\nβ : Type ?u.200822\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ t : Set E\nx✝ y✝ : E\ns : Set F... | [
372,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
367,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.castPred_one | [
{
"state_after": "case zero\nm : ℕ\n⊢ castPred 1 = 1\n\ncase succ\nm n✝ : ℕ\n⊢ castPred 1 = 1",
"state_before": "n m : ℕ\n⊢ castPred 1 = 1",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\nm : ℕ\n⊢ castPred 1 = 1",
"tactic": "rfl"
},
{
"state_after... | [
2306,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2303,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | NonUnitalSubsemiring.coe_top | [] | [
266,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
265,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.ofReal_injective | [] | [
168,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean | MeasureTheory.Measure.lintegral_join | [
{
"state_after": "α : Type u_1\nβ : Type ?u.8987\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nm : Measure (Measure α)\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ (⨆ (n : ℕ),\n ∑ x in SimpleFunc.range (SimpleFunc.eapprox f n),\n x * ∫⁻ (μ : Measure α), ↑↑μ (↑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m) =... | [
150,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.frequently_lt_of_liminf_lt | [] | [
1167,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1163,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean | ContinuousMap.id_apply | [] | [
226,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
225,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsLittleO.of_const_mul_right | [] | [
1548,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1546,
1
] |
Mathlib/Data/List/NodupEquivFin.lean | List.Sorted.coe_getIso_apply | [] | [
104,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/Data/Nat/Dist.lean | Nat.dist_self | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ dist n n = 0",
"tactic": "simp [dist.def, tsub_self]"
}
] | [
35,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
35,
1
] |
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