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/Computability/Primrec.lean | Primrec.list_join | [
{
"state_after": "α : Type u_1\nβ : Type ?u.314731\nγ : Type ?u.314734\nσ : Type ?u.314737\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nl : List (List α)\n⊢ List.foldr (fun b s => b ++ s) [] l = List.join l",
"state_before": "α : Type u_1\nβ : Type ?u.31473... | [
1123,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1121,
1
] |
Mathlib/Data/Finset/Card.lean | Finset.strongDownwardInductionOn_eq | [
{
"state_after": "α : Type u_1\nβ : Type ?u.89625\ns✝ t : Finset α\nf : α → β\nn : ℕ\np : Finset α → Sort u_2\ns : Finset α\nH : (t₁ : Finset α) → ({t₂ : Finset α} → card t₂ ≤ n → t₁ ⊂ t₂ → p t₂) → card t₁ ≤ n → p t₁\n⊢ (fun a => strongDownwardInduction H s a) = H s fun {t} ht x => strongDownwardInduction H t h... | [
706,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
702,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.erase_eq_empty_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.272209\nγ : Type ?u.272212\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝ b : α\ns : Finset α\na : α\n⊢ erase s a = ∅ ↔ s = ∅ ∨ s = {a}",
"tactic": "rw [← sdiff_singleton_eq_erase, sdiff_eq_empty_iff_subset, subset_singleton_iff]"
... | [
2343,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2342,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.bot_mul_of_neg | [
{
"state_after": "x : EReal\nh : x < 0\n⊢ x * ⊥ = ⊤",
"state_before": "x : EReal\nh : x < 0\n⊢ ⊥ * x = ⊤",
"tactic": "rw [EReal.mul_comm]"
},
{
"state_after": "no goals",
"state_before": "x : EReal\nh : x < 0\n⊢ x * ⊥ = ⊤",
"tactic": "exact mul_bot_of_neg h"
}
] | [
979,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
977,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_finset_coe | [] | [
915,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
914,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.EventuallyLE.congr | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.204158\nι : Sort x\ninst✝ : LE β\nl : Filter α\nf f' g g' : α → β\nH✝ : f ≤ᶠ[l] g\nhf✝ : f =ᶠ[l] f'\nhg✝ : g =ᶠ[l] g'\nx : α\nhf : f x = f' x\nhg : g x = g' x\nH : f x ≤ g x\n⊢ f' x ≤ g' x",
"tactic": "rwa [hf,... | [
1633,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1631,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean | Dfinsupp.neLocus_eq_support_sub | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → AddGroup (N a)\nf f₁ f₂ g g₁ g₂ : Π₀ (a : α), N a\n⊢ neLocus f g = support (f - g)",
"tactic": "rw [← @neLocus_add_right α N _ _ _ _ _ (-g), add_r... | [
149,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigO.eventually_mul_div_cancel | [] | [
1911,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1909,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean | Submodule.toSubMulAction_eq | [] | [
162,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/Algebra/Order/Field/Power.lean | zpow_bit1_nonneg_iff | [] | [
166,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
165,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean | Memℓp.sub | [
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf g : (i : α) → E i\nhf : Memℓp f p\nhg : Memℓp g p\n⊢ Memℓp (f + -g) p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf g : (i : α) → E i\nhf... | [
250,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
] |
Mathlib/Algebra/BigOperators/Order.lean | Finset.prod_le_one | [
{
"state_after": "case h.e'_4\nι : Type u_1\nα : Type ?u.152557\nβ : Type ?u.152560\nM : Type ?u.152563\nN : Type ?u.152566\nG : Type ?u.152569\nk : Type ?u.152572\nR : Type u_2\ninst✝ : OrderedCommSemiring R\nf g : ι → R\ns t : Finset ι\nh0 : ∀ (i : ι), i ∈ s → 0 ≤ f i\nh1 : ∀ (i : ι), i ∈ s → f i ≤ 1\n⊢ ∏ i i... | [
621,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
619,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_or | [] | [
1346,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1339,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | aemeasurable_of_tendsto_metrizable_ae' | [] | [
123,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/Data/Set/Sups.lean | Set.infs_left_comm | [] | [
358,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
357,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.mk_powerset | [] | [
645,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
644,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean | HasDerivWithinAt.const_smul | [] | [
137,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
8
] |
Mathlib/Order/Concept.lean | Concept.inf_snd | [] | [
347,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
346,
1
] |
Mathlib/Data/Set/Basic.lean | Set.ite_empty_left | [
{
"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 : Set α\n⊢ Set.ite t ∅ s = s \\ t",
"tactic": "simp [Set.ite]"
}
] | [
2287,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2287,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | Filter.Tendsto.bddBelow_range_of_cofinite | [] | [
99,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
] |
Mathlib/Data/Nat/Factors.lean | Nat.mem_factors_mul_right | [
{
"state_after": "p a b : ℕ\nhpb : p ∈ factors b\nha : a ≠ 0\n⊢ p ∈ factors (b * a)",
"state_before": "p a b : ℕ\nhpb : p ∈ factors b\nha : a ≠ 0\n⊢ p ∈ factors (a * b)",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "p a b : ℕ\nhpb : p ∈ factors b\nha : a ≠ 0\n⊢ ... | [
296,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
293,
1
] |
Mathlib/Data/Int/Log.lean | Int.log_inv | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\n⊢ log b r⁻¹ = -clog b r",
"tactic": "rw [← inv_inv r, clog_inv, neg_neg, inv_inv]"
}
] | [
223,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Mathlib/Data/List/Indexes.lean | List.mapIdx_cons | [
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nl : List α\nf : ℕ → α → β\na : α\n⊢ mapIdx f (a :: l) = f 0 a :: mapIdx (fun i => f (i + 1)) l",
"tactic": "simp [mapIdx_eq_enum_map, enum_eq_zip_range, map_uncurry_zip_eq_zipWith,\n range_succ_eq_map, z... | [
180,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
Std/Data/Int/Lemmas.lean | Int.add_right_neg | [
{
"state_after": "no goals",
"state_before": "a : Int\n⊢ a + -a = 0",
"tactic": "rw [Int.add_comm, Int.add_left_neg]"
}
] | [
303,
38
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
302,
25
] |
Mathlib/Order/Antichain.lean | IsAntichain.top_mem_iff | [] | [
257,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Data/Finset/Prod.lean | Finset.coe_offDiag | [] | [
312,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
1
] |
Mathlib/Tactic/Abel.lean | Mathlib.Tactic.Abel.unfold_sub | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : SubtractionMonoid α\na b c : α\nh : a + -b = c\n⊢ a - b = c",
"tactic": "rw [sub_eq_add_neg, h]"
}
] | [
252,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
251,
1
] |
Mathlib/Topology/Order/Basic.lean | Ioi_mem_nhds | [] | [
333,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
332,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | fderiv_arctan | [] | [
190,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Std/Data/Int/Lemmas.lean | Int.lt_iff_le_and_ne | [
{
"state_after": "a b : Int\nx✝ : a ≤ b ∧ a ≠ b\naleb : a ≤ b\naneb : a ≠ b\n⊢ a < b",
"state_before": "a b : Int\n⊢ a < b ↔ a ≤ b ∧ a ≠ b",
"tactic": "refine ⟨fun h => ⟨Int.le_of_lt h, Int.ne_of_lt h⟩, fun ⟨aleb, aneb⟩ => ?_⟩"
},
{
"state_after": "a b : Int\nx✝ : a ≤ b ∧ a ≠ b\naleb : a ≤ b\nan... | [
640,
82
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
636,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | CategoryTheory.Limits.IsZero.of_mono_eq_zero | [
{
"state_after": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Mono 0\n⊢ IsZero X",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\nh : f =... | [
207,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
] |
Std/Data/List/Lemmas.lean | List.empty_eq | [] | [
209,
52
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
209,
9
] |
Mathlib/Data/MvPolynomial/Derivation.lean | MvPolynomial.mkDerivationₗ_X | [
{
"state_after": "no goals",
"state_before": "σ : Type u_2\nR : Type u_3\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Module (MvPolynomial σ R) A\nf : σ → A\ni : σ\n⊢ (1 •\n Finsupp.sum (Finsupp.single i 1) fun i_1 k =>\n ↑(monomial (Finsupp.sing... | [
59,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean | LocalRing.eq_maximalIdeal | [] | [
136,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean | CategoryTheory.Limits.hasLimit_of_hasColimit_op | [] | [
251,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Data/PNat/Xgcd.lean | PNat.XgcdType.step_isSpecial | [
{
"state_after": "u : XgcdType\nhs : u.wp + u.zp + u.wp * u.zp = u.x * u.y\n⊢ u.y * q u + u.zp + u.wp + (u.y * q u + u.zp) * u.wp = u.y * ((u.wp + 1) * q u + u.x)",
"state_before": "u : XgcdType\nhs : IsSpecial u\n⊢ IsSpecial (step u)",
"tactic": "dsimp [IsSpecial, step] at hs⊢"
},
{
"state_afte... | [
321,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.closure_induction | [] | [
1142,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1140,
1
] |
Mathlib/LinearAlgebra/Matrix/Circulant.lean | Matrix.Fin.circulant_isSymm_apply | [] | [
210,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
] |
Mathlib/Data/Real/Hyperreal.lean | Hyperreal.exists_st_iff_not_infinite | [] | [
338,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
337,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean | min_abs_abs_le_abs_max | [] | [
333,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.zero_rightMoves | [] | [
341,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | CategoryTheory.Limits.biprod.isIso_inl_iff_id_eq_fst_comp_inl | [
{
"state_after": "case mp\nJ : Type w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nP Q X Y : C\ninst✝ : HasBinaryBiproduct X Y\n⊢ IsIso inl → 𝟙 (X ⊞ Y) = fst ≫ inl\n\ncase mpr\nJ : Type w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nP Q X Y : C\ninst✝ : HasBinaryBiproduct X ... | [
1592,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1584,
1
] |
Std/Logic.lean | true_imp_iff | [] | [
120,
87
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
120,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.Ico_subset_Ico_iff | [] | [
250,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean | LieSubmodule.lie_bot | [
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\n... | [
138,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
1
] |
Mathlib/Topology/Algebra/Algebra.lean | Subalgebra.le_topologicalClosure | [] | [
117,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean | Finsupp.multiset_sum_sum_index | [
{
"state_after": "no goals",
"state_before": "α : Type u_3\nι : Type ?u.559428\nγ : Type ?u.559431\nA : Type ?u.559434\nB : Type ?u.559437\nC : Type ?u.559440\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t ... | [
544,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
539,
1
] |
Mathlib/Topology/Instances/Matrix.lean | summable_matrix_blockDiagonal | [] | [
396,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
392,
1
] |
Mathlib/Data/Dfinsupp/Multiset.lean | Multiset.toDfinsupp_replicate | [
{
"state_after": "case h\nα : Type u_1\nβ : α → Type ?u.32641\ninst✝ : DecidableEq α\na : α\nn : ℕ\ni : α\n⊢ ↑(↑toDfinsupp (replicate n a)) i = ↑(Dfinsupp.single a n) i",
"state_before": "α : Type u_1\nβ : α → Type ?u.32641\ninst✝ : DecidableEq α\na : α\nn : ℕ\n⊢ ↑toDfinsupp (replicate n a) = Dfinsupp.singl... | [
75,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Topology/Order/Basic.lean | comap_coe_nhdsWithin_Iio_of_Ioo_subset | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nhb : s ⊆ Iio b\nhs : Set.Nonempty s → ∃ a, a < b ∧ Ioo a b ⊆ s\n✝ : Nontrivial (Filter { x // x ∈ s })\n⊢ comap Subtype.val (𝓝[Iio b... | [
2460,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2445,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean | MulSemiringActionHom.coe_fn_coe | [] | [
495,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
494,
1
] |
Std/Data/Array/Init/Lemmas.lean | Array.getElem_map | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β\narr : Array α\ni : Nat\nh : i < size (map f arr)\nthis :\n (∀ (i : Fin (size arr)), SatisfiesM (fun b => b = f arr[i]) (f arr[i])) →\n SatisfiesM (fun arr_1 => ∃ eq, ∀ (i : Nat) (h : i < size arr), arr_1[i] = f arr[{ val := i, isLt := h }])\n (ma... | [
186,
35
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
182,
9
] |
Mathlib/Data/List/Rotate.lean | List.cyclicPermutations_eq_singleton_iff | [
{
"state_after": "α : Type u\nl✝ l' l : List α\nx : α\nh : cyclicPermutations l = [[x]]\n⊢ l = [x]",
"state_before": "α : Type u\nl✝ l' l : List α\nx : α\n⊢ cyclicPermutations l = [[x]] ↔ l = [x]",
"tactic": "refine' ⟨fun h => _, fun h => by simp [cyclicPermutations, h, dropLast_eq_take]⟩"
},
{
... | [
653,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
650,
1
] |
Mathlib/RingTheory/Localization/LocalizationLocalization.lean | IsLocalization.localization_localization_surj | [
{
"state_after": "case intro.mk\nR : Type u_3\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.84572\ninst✝⁶ : CommRing P\nN : Submonoid S\nT : Type u_2\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst... | [
92,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | Matrix.separatingLeft_toLinearMap₂'_iff | [] | [
700,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
697,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean | IsPrimitiveRoot.isUnit | [
{
"state_after": "M : Type u_1\nN : Type ?u.1712947\nG : Type ?u.1712950\nR : Type ?u.1712953\nS : Type ?u.1712956\nF : Type ?u.1712959\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\n⊢ ζ * ζ ^ (k - 1) = 1",
"state_b... | [
369,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
367,
1
] |
Mathlib/Order/Interval.lean | Interval.pure_ne_bot | [] | [
405,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
404,
1
] |
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | ModularGroup.coe_T | [] | [
361,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
360,
1
] |
Mathlib/LinearAlgebra/Ray.lean | SameRay.sameRay_pos_smul_right | [] | [
131,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Order/JordanHolder.lean | CompositionSeries.mem_def | [] | [
192,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
] |
Mathlib/Algebra/Order/Archimedean.lean | pow_unbounded_of_one_lt | [] | [
146,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
1
] |
Mathlib/Algebra/DualNumber.lean | DualNumber.inr_eq_smul_eps | [] | [
90,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
] |
Mathlib/Data/Set/Finite.lean | Set.Finite.ofFinset | [] | [
91,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
90,
11
] |
Mathlib/Algebra/GroupPower/Order.lean | pow_le_one_iff_of_nonneg | [
{
"state_after": "β : Type ?u.237030\nA : Type ?u.237033\nG : Type ?u.237036\nM : Type ?u.237039\nR : Type u_1\ninst✝ : LinearOrderedSemiring R\na✝ b a : R\nha : 0 ≤ a\nn : ℕ\nhn : n ≠ 0\n⊢ a ^ n ≤ 1 → a ≤ 1",
"state_before": "β : Type ?u.237030\nA : Type ?u.237033\nG : Type ?u.237036\nM : Type ?u.237039\nR... | [
567,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
564,
1
] |
Mathlib/Combinatorics/Additive/Behrend.lean | Behrend.card_sphere_le_rothNumberNat | [
{
"state_after": "case zero\nα : Type ?u.508512\nβ : Type ?u.508515\nn d✝ k✝ N : ℕ\nx : Fin n → ℕ\nd k : ℕ\n⊢ card (sphere zero d k) ≤ ↑rothNumberNat ((2 * d - 1) ^ zero)\n\ncase succ\nα : Type ?u.508512\nβ : Type ?u.508515\nn d✝ k✝ N : ℕ\nx : Fin n → ℕ\nd k n✝ : ℕ\n⊢ card (sphere (succ n✝) d k) ≤ ↑rothNumberNa... | [
231,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
216,
1
] |
Mathlib/Order/Lattice.lean | inf_lt_left | [] | [
463,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
462,
1
] |
Mathlib/RingTheory/Artinian.lean | isArtinian_of_surjective | [] | [
94,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean | Ordinal.type_ne_zero_iff_nonempty | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.25081\nγ : Type ?u.25084\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\ninst✝ : IsWellOrder α r\n⊢ type r ≠ 0 ↔ Nonempty α",
"tactic": "simp"
}
] | [
235,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/Data/List/ToFinsupp.lean | List.toFinsupp_cons_apply_zero | [] | [
103,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.map_congr | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.111985\nf g : α → β\ns : Multiset α\nh : ∀ (x : α), x ∈ s → f x = g x\n⊢ map f s = map g s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.111985\nf g : α → β\ns t : Multiset α\n⊢ s = t → (∀ (x : α), x ∈ t → f x = g x) → map f s = map g t"... | [
1151,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1147,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | lt_mul_of_one_lt_of_le_of_pos | [] | [
877,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
875,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.length_takeUntil_le | [
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\np : Walk G v w\nh : u ∈ support p\nthis : length (append (takeUntil p u h) (dropUntil p u h)) = length p\n⊢ length (takeUntil p u h) ≤ length p",
"state_b... | [
1171,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1167,
1
] |
Mathlib/Algebra/Quaternion.lean | Quaternion.im_imK | [] | [
814,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
814,
9
] |
Mathlib/CategoryTheory/Extensive.lean | CategoryTheory.finitaryExtensive_of_preserves_and_reflects_isomorphism | [
{
"state_after": "J : Type v'\ninst✝⁸ : Category J\nC : Type u\ninst✝⁷ : Category C\nX Y : C\nD : Type u''\ninst✝⁶ : Category D\nF : C ⥤ D\ninst✝⁵ : FinitaryExtensive D\ninst✝⁴ : HasFiniteCoproducts C\ninst✝³ : HasPullbacks C\ninst✝² : PreservesLimitsOfShape WalkingCospan F\ninst✝¹ : PreservesColimitsOfShape (D... | [
586,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
579,
1
] |
Std/Data/Int/Lemmas.lean | Int.ofNat_succ_pos | [] | [
606,
86
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
606,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | mul_le_mul_of_le_of_le | [] | [
410,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
] |
Mathlib/Order/Lattice.lean | inf_sup_self | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Lattice α\na b c d : α\n⊢ a ⊓ (a ⊔ b) = a",
"tactic": "simp"
}
] | [
711,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
711,
1
] |
Mathlib/Data/Set/Finite.lean | Finite.Set.subset | [
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\ninst✝ : Finite ↑s\nh : t ⊆ s\n⊢ Finite ↑{x | x ∈ s ∧ x ∈ t}",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\ninst✝ : Finite ↑s\nh : t ⊆ s\n⊢ Finite ↑t",
"tactic": "rw [← sep_eq_of_subset h]"
}... | [
604,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
602,
11
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.lf_zero | [
{
"state_after": "x : PGame\n⊢ ((∃ i,\n (∀ (i' : LeftMoves x), moveLeft x i' ⧏ moveLeft 0 i) ∧\n ∀ (j : RightMoves (moveLeft 0 i)), x ⧏ moveRight (moveLeft 0 i) j) ∨\n ∃ j,\n (∀ (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i ⧏ 0) ∧\n ∀ (j' : RightMoves 0), moveR... | [
687,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
685,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean | Dfinsupp.neLocus_self_sub_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → AddGroup (N a)\nf f₁ f₂ g g₁ g₂ : Π₀ (a : α), N a\n⊢ neLocus f (f - g) = support g",
"tactic": "rw [sub_eq_add_neg, neLocus_self_add_right, suppor... | [
174,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
] |
Std/Data/String/Lemmas.lean | Substring.Valid.data_takeWhile | [
{
"state_after": "no goals",
"state_before": "p : Char → Bool\nx✝ : Substring\nh✝ : Valid x✝\nw✝² w✝¹ w✝ : List Char\nh : ValidFor w✝² w✝¹ w✝ x✝\n⊢ (toString (Substring.takeWhile x✝ p)).data = List.takeWhile p (toString x✝).data",
"tactic": "simp [(h.takeWhile _).toString, h.toString]"
}
] | [
1073,
91
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1071,
1
] |
Mathlib/Deprecated/Subgroup.lean | Group.closure.isSubgroup | [] | [
539,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
536,
1
] |
Mathlib/Order/BooleanAlgebra.lean | sdiff_eq_self_iff_disjoint' | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.20985\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x \\ y = x ↔ Disjoint x y",
"tactic": "rw [sdiff_eq_self_iff_disjoint, disjoint_comm]"
}
] | [
318,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
1
] |
Mathlib/Algebra/Hom/Aut.lean | AddAut.coe_one | [] | [
204,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | ENNReal.zero_rpow_of_neg | [
{
"state_after": "y : ℝ\nh : y < 0\n⊢ Option.some 0 ^ y = ⊤",
"state_before": "y : ℝ\nh : y < 0\n⊢ 0 ^ y = ⊤",
"tactic": "rw [← ENNReal.coe_zero, ← ENNReal.some_eq_coe]"
},
{
"state_after": "no goals",
"state_before": "y : ℝ\nh : y < 0\n⊢ (if 0 = 0 ∧ y < 0 then ⊤ else ↑(NNReal.rpow 0 y)) = ⊤... | [
334,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.normSq_to_real | [
{
"state_after": "no goals",
"state_before": "K : Type ?u.7355735\nE : Type ?u.7355738\ninst✝ : IsROrC K\nx : ℝ\n⊢ ↑normSq x = x * x",
"tactic": "simp [IsROrC.normSq]"
}
] | [
859,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
859,
1
] |
Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | SimpleGraph.singletonFinsubgraph_le_adj_right | [
{
"state_after": "no goals",
"state_before": "V : Type u\nW : Type v\nG : SimpleGraph V\nF : SimpleGraph W\nu v : V\ne : Adj G u v\n⊢ singletonFinsubgraph v ≤ finsubgraphOfAdj e",
"tactic": "simp [singletonFinsubgraph, finsubgraphOfAdj]"
}
] | [
103,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_ite_irrel | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.417321\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\np : Prop\ninst✝ : Decidable p\ns : Finset α\nf g : α → β\n⊢ (∏ x in s, if p then f x else g x) = if p then ∏ x in s, f x else ∏ x in s, g x"... | [
1113,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1111,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean | Prod.snd_exp | [] | [
546,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
545,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | toIcoDiv_sub_eq_toIcoDiv_add | [
{
"state_after": "case h\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c✝ : α\nn : ℤ\na b c : α\n⊢ b - c - toIcoDiv hp (a + c) b • p ∈ Set.Ico a (a + p)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 ... | [
362,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
358,
1
] |
Mathlib/Order/Partition/Finpartition.lean | Finpartition.exists_mem | [
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\na : α\nha : a ∈ sup P.parts id\n⊢ ∃ t, t ∈ P.parts ∧ a ∈ t",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\na : α\nha : a ∈ s\n⊢ ∃ t, t ∈ P.parts ∧ a ∈ t",
"tactic": "sim... | [
470,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
468,
1
] |
Mathlib/Order/Lattice.lean | sup_idem | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : SemilatticeSup α\na b c d : α\n⊢ a ⊔ a = a",
"tactic": "simp"
}
] | [
242,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
242,
1
] |
Mathlib/MeasureTheory/Measure/Content.lean | MeasureTheory.Content.measure_eq_content_of_regular | [
{
"state_after": "case refine'_1\nG : Type w\ninst✝³ : TopologicalSpace G\nμ : Content G\ninst✝² : MeasurableSpace G\ninst✝¹ : T2Space G\ninst✝ : BorelSpace G\nH : ContentRegular μ\nK : Compacts G\n⊢ ↑↑(Content.measure μ) ↑K ≤ (fun s => ↑(toFun μ s)) K\n\ncase refine'_2\nG : Type w\ninst✝³ : TopologicalSpace G\... | [
455,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
439,
1
] |
Mathlib/Topology/Homotopy/Basic.lean | ContinuousMap.HomotopicRel.trans | [] | [
697,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
695,
1
] |
Mathlib/Topology/Connected.lean | isPreconnected_closed_iff | [
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.23251\nπ : ι → Type ?u.23256\ninst✝ : TopologicalSpace α\ns✝ t✝ u v s : Set α\nh : IsPreconnected s\nt t' : Set α\nht : IsClosed t\nht' : IsClosed t'\nhtt' : s ⊆ t ∪ t'\nx : α\nxs : x ∈ s\nxt : x ∈ t\ny : α\nys : y ∈ s\nyt' : y ... | [
363,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean | Bool.or_eq_false_eq_eq_false_and_eq_false | [
{
"state_after": "no goals",
"state_before": "a b : Bool\n⊢ ((a || b) = false) = (a = false ∧ b = false)",
"tactic": "cases a <;> cases b <;> simp"
}
] | [
106,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
] |
Mathlib/Data/List/Sigma.lean | List.nodupKeys_cons | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ns : Sigma β\nl : List (Sigma β)\n⊢ NodupKeys (s :: l) ↔ ¬s.fst ∈ keys l ∧ NodupKeys l",
"tactic": "simp [keys, NodupKeys]"
}
] | [
107,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
] |
Mathlib/Order/Hom/Lattice.lean | InfHom.cancel_left | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.80850\nι : Type ?u.80853\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.80865\ninst✝³ : Inf α\ninst✝² : Inf β\ninst✝¹ : Inf γ\ninst✝ : Inf δ\ng : InfHom β γ\nf₁ f₂ : InfHom α β\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a... | [
621,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
618,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean | CategoryTheory.GrothendieckTopology.arrow_trans | [
{
"state_after": "C : Type u\ninst✝ : Category C\nX Y : C\nS✝ R✝ : Sieve X\nJ : GrothendieckTopology C\nf : Y ⟶ X\nS R : Sieve X\nh : Covers J S f\nk : ∀ {Z : C} (g : Z ⟶ X), S.arrows g → Covers J R g\n⊢ Covers J R f",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nS✝ R✝ : Sieve X\nJ : Grothendie... | [
214,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
] |
Mathlib/Data/Nat/PartENat.lean | PartENat.le_of_lt_add_one | [
{
"state_after": "case a\nx y : PartENat\nh✝ : x < y + 1\nh : x < ⊤ + 1\n⊢ x ≤ ⊤\n\ncase a\nx y : PartENat\nh✝ : x < y + 1\nn : ℕ\nh : x < ↑n + 1\n⊢ x ≤ ↑n",
"state_before": "x y : PartENat\nh : x < y + 1\n⊢ x ≤ y",
"tactic": "induction' y using PartENat.casesOn with n"
},
{
"state_after": "case... | [
495,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
490,
1
] |
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