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/CategoryTheory/CofilteredSystem.lean | CategoryTheory.Functor.IsMittagLeffler.toPreimages | [
{
"state_after": "case intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\n⊢ ∃ i_1 f,\n ∀ ⦃k : J⦄ (g : k ⟶ i_1), range ((Functor.toPreimages F s).map f) ⊆ range ((Functor.t... | [
253,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean | AntitoneOn.Iio | [] | [
101,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
11
] |
Mathlib/Data/Set/Finite.lean | Finset.bddAbove | [] | [
1641,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1640,
11
] |
Mathlib/GroupTheory/DoubleCoset.lean | Doset.mem_doset_of_not_disjoint | [
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b : G\nh : ∃ x, x ∈ doset a ↑H ↑K ∧ x ∈ doset b ↑H ↑K\n⊢ b ∈ doset a ↑H ↑K",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\n... | [
67,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
] |
Mathlib/Algebra/Group/Prod.lean | Prod.fst_inv | [] | [
138,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.nat_cast_lt | [
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ ↑m < ↑n ↔ m < n",
"tactic": "simp only [lt_iff_le_not_le, nat_cast_le]"
}
] | [
2311,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2310,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean | Polynomial.map_restriction | [
{
"state_after": "no goals",
"state_before": "R✝ : Type u\nS : Type ?u.71352\ninst✝¹ : Ring R✝\nR : Type u\ninst✝ : CommRing R\np : R[X]\nn : ℕ\n⊢ coeff (map (algebraMap { x // x ∈ Subring.closure ↑(frange p) } R) (restriction p)) n = coeff p n",
"tactic": "rw [coeff_map, Algebra.algebraMap_ofSubring_ap... | [
302,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
] |
Mathlib/Data/Real/CauSeq.lean | CauSeq.add_apply | [] | [
214,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/Algebra/Field/Basic.lean | div_add_div | [] | [
210,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | ENNReal.eventually_pow_one_div_le | [
{
"state_after": "case intro\ny : ℝ≥0∞\nhy : 1 < y\nx : ℝ≥0\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ y",
"state_before": "x : ℝ≥0∞\nhx : x ≠ ⊤\ny : ℝ≥0∞\nhy : 1 < y\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y",
"tactic": "lift x to ℝ≥0 using hx"
},
{
"state_after": "case pos\ny : ℝ≥0∞\nhy : 1 < y\nx... | [
500,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
492,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | Real.arcsin_nonneg | [
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ sin 0 ≤ x ↔ 0 ≤ x",
"tactic": "rw [sin_zero]"
}
] | [
204,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
202,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean | PiCountable.dist_le_dist_pi_of_dist_lt | [
{
"state_after": "no goals",
"state_before": "E : ℕ → Type ?u.306614\nι : Type u_1\ninst✝¹ : Encodable ι\nF : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (F i)\nx y : (i : ι) → F i\ni : ι\nh : dist x y < (1 / 2) ^ encode i\n⊢ dist (x i) (y i) ≤ dist x y",
"tactic": "simpa only [not_le.2 h, false_or_iff]... | [
850,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
848,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | norm_sum_le | [] | [
1478,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1476,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean | MeasureTheory.map_mul_right_eq_self | [] | [
93,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
] |
Mathlib/Analysis/Analytic/Basic.lean | FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum | [
{
"state_after": "𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.1266934\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinea... | [
1155,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1151,
1
] |
Mathlib/LinearAlgebra/CrossProduct.lean | triple_product_eq_det | [
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nu v w : Fin 3 → R\n⊢ u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 +\n u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 +\n u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2... | [
124,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
] |
Mathlib/Algebra/Regular/Basic.lean | isRegular_mul_iff | [
{
"state_after": "R : Type u_1\ninst✝ : CommSemigroup R\na b : R\n⊢ IsRegular (a * b) ↔ IsRegular (a * b) ∧ IsRegular (b * a)",
"state_before": "R : Type u_1\ninst✝ : CommSemigroup R\na b : R\n⊢ IsRegular (a * b) ↔ IsRegular a ∧ IsRegular b",
"tactic": "refine' Iff.trans _ isRegular_mul_and_mul_iff"
}... | [
296,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/Logic/Denumerable.lean | nonempty_denumerable | [] | [
382,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
381,
1
] |
Mathlib/Data/Real/Hyperreal.lean | Hyperreal.infinitesimal_inv_of_infinite | [] | [
756,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
754,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean | Nat.ascFactorial_of_sub | [
{
"state_after": "n k : ℕ\nh : k < n\nt : ℕ := n - succ k\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)",
"state_before": "n k : ℕ\nh : k < n\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)",
"tactic": "let t := n - k.succ"
},
{
"state_after": ... | [
282,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.spanFinset_ne_zero | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.1280273\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1280480\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRin... | [
1281,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1280,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.coeff_inv_aux | [
{
"state_after": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\n⊢ (if single () n = 0 then a\n else\n -a *\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSe... | [
1907,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1867,
1
] |
Mathlib/Data/Set/Image.lean | Set.image_diff | [] | [
444,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
441,
1
] |
Mathlib/Topology/Order/Basic.lean | pi_Ici_mem_nhds' | [] | [
1473,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1472,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean | PadicInt.coe_sub | [] | [
137,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | DifferentiableAt.rpow_const | [] | [
475,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
473,
1
] |
Mathlib/Data/Polynomial/Eval.lean | Polynomial.eval₂_mul_noncomm | [
{
"state_after": "case ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nq r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nhf : ∀ (k : ℕ), Commute (↑f (coeff q k)) x\np : AddMonoidAlgebra R ℕ\n⊢ eval₂ f x ({ toFinsupp := p } * q) = eval₂ f x... | [
184,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
] |
Mathlib/Algebra/Group/Conj.lean | ConjClasses.exists_rep | [] | [
204,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
] |
Mathlib/Data/List/Rdrop.lean | List.rdrop_eq_reverse_drop_reverse | [
{
"state_after": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ take (length l - n) l = reverse (drop n (reverse l))",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ rdrop l n = reverse (drop n (reverse l))",
"tactic": "rw [rdrop]"
},
{
"state_after": "case H0\nα : Type u_1... | [
62,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
56,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.mul_inv_cancel | [
{
"state_after": "no goals",
"state_before": "z : ℂ\nh : z ≠ 0\n⊢ z * z⁻¹ = 1",
"tactic": "rw [inv_def, ← mul_assoc, mul_conj, ← ofReal_mul, mul_inv_cancel (mt normSq_eq_zero.1 h),\n ofReal_one]"
}
] | [
757,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
755,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Real.pi_pos | [
{
"state_after": "no goals",
"state_before": "⊢ 0 < 2",
"tactic": "norm_num"
}
] | [
166,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
165,
1
] |
Mathlib/Data/Num/Lemmas.lean | ZNum.gcd_to_nat | [
{
"state_after": "a b : ZNum\n⊢ Nat.gcd (natAbs ↑a) (natAbs ↑b) = Int.gcd ↑a ↑b",
"state_before": "a b : ZNum\n⊢ Nat.gcd ↑(abs a) ↑(abs b) = Int.gcd ↑a ↑b",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "a b : ZNum\n⊢ Nat.gcd (natAbs ↑a) (natAbs ↑b) = Int.gcd ↑a ↑b",
"... | [
1774,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1773,
1
] |
Mathlib/Data/Finset/Image.lean | Finset.image_val | [] | [
316,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
1
] |
Mathlib/Topology/ContinuousOn.lean | continuousWithinAt_insert_self | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.325537\nδ : Type ?u.325540\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\nx : α\ns : Set α\n⊢ ContinuousWithinAt f (insert x s) x ↔ Continuous... | [
766,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
763,
1
] |
Mathlib/Data/TypeVec.lean | TypeVec.toSubtype_of_subtype | [
{
"state_after": "case a.h\nn : ℕ\nα : TypeVec n\np : α ⟹ repeat n Prop\ni : Fin2 n\nx : Subtype_ p i\n⊢ (toSubtype p ⊚ ofSubtype p) i x = id i x",
"state_before": "n : ℕ\nα : TypeVec n\np : α ⟹ repeat n Prop\n⊢ toSubtype p ⊚ ofSubtype p = id",
"tactic": "ext (i x)"
},
{
"state_after": "case a.h... | [
798,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
794,
1
] |
Mathlib/Init/CcLemmas.lean | if_eq_of_eq_true | [] | [
86,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Data/ZMod/Basic.lean | ZMod.ringHom_rightInverse | [] | [
1172,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1170,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean | derivWithin_sub | [] | [
317,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | pow_lt_pow_of_lt_one | [] | [
523,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
522,
1
] |
Mathlib/Data/Rat/NNRat.lean | NNRat.coe_mul | [] | [
135,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.coeFn_mul | [] | [
646,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
645,
1
] |
Mathlib/Topology/ContinuousOn.lean | continuousWithinAt_pi | [] | [
573,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
570,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean | OreLocalization.oreDiv_mul_oreDiv | [] | [
257,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
255,
1
] |
Mathlib/Dynamics/FixedPoints/Basic.lean | Function.fixedPoints_subset_range | [] | [
154,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
154,
1
] |
Mathlib/Algebra/GCDMonoid/Multiset.lean | Multiset.gcd_singleton | [] | [
148,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
147,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.lift_prod | [
{
"state_after": "case intro\nα β ι : Type u\nc : ι → Type v\n⊢ lift (prod fun i => #c i) = prod fun i => lift ((fun i => #c i) i)",
"state_before": "α β ι : Type u\nc : ι → Cardinal\n⊢ lift (prod c) = prod fun i => lift (c i)",
"tactic": "lift c to ι → Type v using fun _ => trivial"
},
{
"state... | [
1076,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1072,
1
] |
Mathlib/RingTheory/Localization/FractionRing.lean | IsFractionRing.mk'_eq_div | [] | [
165,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Std/Data/List/Lemmas.lean | List.isInfix.trans | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl l₁ r₁ l₂ r₂ : List α\n⊢ l₂ ++ l₁ ++ l ++ (r₁ ++ r₂) = l₂ ++ (l₁ ++ l ++ r₁) ++ r₂",
"tactic": "simp only [append_assoc]"
}
] | [
1598,
95
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1597,
1
] |
Mathlib/CategoryTheory/Sites/CoverPreserving.lean | CategoryTheory.compatiblePreservingOfDownwardsClosed | [
{
"state_after": "case Compatible\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : Fa... | [
171,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean | Real.tendsto_comp_exp_atBot | [
{
"state_after": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf : ℝ → α\n⊢ Tendsto (fun x => f (exp x)) atBot l ↔ Tendsto (f ∘ exp) atBot l",
"state_before": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf : ℝ → α\n⊢ Tendsto (fun x => f (exp x)) atBot l ↔ Tendsto f (𝓝[Set.Ioi 0] 0) l",
"tactic": "rw [← map_exp_at... | [
334,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | intervalIntegral.continuousOn_primitive_interval_left | [
{
"state_after": "ι : Type ?u.18752176\n𝕜 : Type ?u.18752179\nE : Type u_1\nF : Type ?u.18752185\nA : Type ?u.18752188\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b b₀ b₁ b₂ : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f [[b, a... | [
1236,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1232,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.blsub_le | [] | [
1812,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1811,
1
] |
Mathlib/Data/List/Sort.lean | List.length_split_lt | [
{
"state_after": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁ l₂ : List α\nh : split (a :: b :: l) = (l₁, l₂)\nl₁' l₂' : List α\ne : split l = (l₁', l₂')\n⊢ length l₁ < length (a :: b :: l) ∧ length l₂ < length (a :: b :: l)",
"state_before": "α : Type uu\nr : α → α → Prop\... | [
333,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
328,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean | measurable_const_smul_iff | [
{
"state_after": "no goals",
"state_before": "M : Type ?u.3449887\nβ : Type u_2\nα : Type u_1\ninst✝⁹ : MeasurableSpace M\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : Monoid M\ninst✝⁶ : MulAction M β\ninst✝⁵ : MeasurableSMul M β\ninst✝⁴ : MeasurableSpace α\nf : α → β\nμ : MeasureTheory.Measure α\nG : Type u_3\nins... | [
716,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
715,
1
] |
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.symm_toNonUnitalRingHom_apply_toNonUnitalRingHom_apply | [] | [
642,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
640,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | Polynomial.cyclotomic'.monic | [] | [
102,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean | DoubleQuot.quotQuotEquivComm_algebraMap | [] | [
668,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
666,
1
] |
Mathlib/Data/List/Prime.lean | Prime.not_dvd_prod | [] | [
46,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
44,
1
] |
Mathlib/Topology/Algebra/Constructions.lean | Units.continuous_val | [] | [
155,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
154,
1
] |
Mathlib/Order/BoundedOrder.lean | top_inf_eq | [] | [
492,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
491,
1
] |
Std/Data/List/Basic.lean | List.sections_eq_sectionsTR | [
{
"state_after": "case h.h\nα : Type u_1\nL : List (List α)\n⊢ sections L = sectionsTR L",
"state_before": "⊢ @sections = @sectionsTR",
"tactic": "funext α L"
},
{
"state_after": "case h.h\nα : Type u_1\nL : List (List α)\n⊢ sections L = bif any L isEmpty then [] else (foldr sectionsTR.go #[[]] ... | [
928,
40
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
922,
10
] |
Mathlib/Algebra/Group/Prod.lean | Units.embedProduct_injective | [] | [
780,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
779,
1
] |
Mathlib/Topology/Algebra/Constructions.lean | Units.continuous_coe_inv | [] | [
170,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
] |
Mathlib/Algebra/Invertible.lean | mul_invOf_eq_iff_eq_mul_right | [
{
"state_after": "no goals",
"state_before": "α : Type u\nc a b : α\ninst✝¹ : Monoid α\ninst✝ : Invertible c\n⊢ a * ⅟c = b ↔ a = b * c",
"tactic": "rw [← mul_right_inj_of_invertible (c := c), mul_invOf_mul_self_cancel]"
}
] | [
317,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean | Subalgebra.mul_self | [
{
"state_after": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ ↑toSubmodule S * ↑toSubmodule S ≤ ↑toSubmodule S\n\ncase a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra... | [
47,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
40,
1
] |
Mathlib/Topology/Order/Hom/Basic.lean | ContinuousOrderHom.cancel_right | [] | [
192,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
190,
1
] |
Mathlib/Analysis/Seminorm.lean | Seminorm.comp_zero | [] | [
334,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
333,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.tan_ofReal_im | [
{
"state_after": "no goals",
"state_before": "x✝ y : ℂ\nx : ℝ\n⊢ (tan ↑x).im = 0",
"tactic": "rw [← ofReal_tan_ofReal_re, ofReal_im]"
}
] | [
1002,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1002,
1
] |
Mathlib/Data/Multiset/Nodup.lean | Multiset.Nodup.mem_erase_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8207\nγ : Type ?u.8210\nr : α → α → Prop\ns t : Multiset α\na✝ : α\ninst✝ : DecidableEq α\na b : α\nl : Multiset α\nd : Nodup l\n⊢ a ∈ Multiset.erase l b ↔ a ≠ b ∧ a ∈ l",
"tactic": "rw [d.erase_eq_filter b, mem_filter, and_comm]"
... | [
183,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | HomogeneousLocalization.NumDenSameDeg.den_smul | [] | [
276,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
275,
1
] |
Mathlib/Data/Finset/Sups.lean | Finset.image_subset_sups_right | [] | [
105,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.fp_unbounded | [
{
"state_after": "case h.e'_3\nf : Ordinal → Ordinal\nH : IsNormal f\n⊢ fixedPoints f = ⋂ (i : Unit), fixedPoints f",
"state_before": "f : Ordinal → Ordinal\nH : IsNormal f\n⊢ Set.Unbounded (fun x x_1 => x < x_1) (fixedPoints f)",
"tactic": "convert fp_family_unbounded fun _ : Unit => H"
},
{
"s... | [
502,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
500,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.coe_map | [] | [
1167,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1166,
1
] |
Mathlib/Data/List/Perm.lean | List.Subperm.perm_of_length_le | [] | [
443,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
442,
1
] |
Mathlib/Data/Num/Bitwise.lean | PosNum.shiftl_succ_eq_bit0_shiftl | [] | [
111,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | fderivWithin_zero_of_not_differentiableWithinAt | [
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.37783\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.37878\ni... | [
222,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean | strictMonoOn_Iic_of_lt_succ | [
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ny : α\nhy : y ∈ Iic n\nhxy : x < y\n⊢ ψ x < ψ y",
"state_before": "α : Type u_1\nβ... | [
256,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
233,
1
] |
Mathlib/Logic/Basic.lean | funext₂ | [] | [
602,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
601,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean | differentiableOn_sub_const_iff | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.509886\ninst✝³ : NormedAddCommGroup G\ninst✝² : Normed... | [
576,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
574,
1
] |
Mathlib/Topology/Sequences.lean | IsCompact.tendsto_subseq | [] | [
302,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.Bounded.prod | [
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.492392\nι : Type ?u.492395\ninst✝¹ : PseudoMetricSpace α\nx : α\ns✝ t✝ : Set α\nr : ℝ\ninst✝ : PseudoMetricSpace β\ns : Set α\nt : Set β\nhs : IsBounded s\nht : IsBounded t\n⊢ IsBounded (s ×ˢ t)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.4923... | [
2389,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2386,
11
] |
Mathlib/Data/Fin/Tuple/Basic.lean | Fin.isSome_find_iff | [
{
"state_after": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝ : DecidablePred p\nh : ∃ a, find p = some a\n⊢ ∃ i, p i",
"state_before": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝ : DecidablePred p\nh : Option.isSome (find p) = true\n⊢ ∃ i, p i",
"tactic": "rw [Option.isSome_iff_exists] at h"
},
{
"state... | [
884,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
868,
1
] |
Mathlib/LinearAlgebra/Matrix/Circulant.lean | Matrix.circulant_mul_comm | [
{
"state_after": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j : n\n⊢ (circulant v ⬝ circulant w) i j = (circulant w ⬝ circulant v) i j",
"state_... | [
152,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/RingTheory/Localization/Away/Basic.lean | selfZpow_sub_cast_nat | [
{
"state_after": "case pos\nR : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℕ\nh : m ≤ n\n⊢ selfZpow x B (↑n - ↑m) = mk' B (x ^ n) (Submonoid.pow x m)\n\ncase neg\nR : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\nin... | [
254,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
247,
1
] |
Std/Logic.lean | or_congr | [] | [
255,
99
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
255,
1
] |
Mathlib/Order/WithBot.lean | WithTop.coe_lt_coe | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.87442\nγ : Type ?u.87445\nδ : Type ?u.87448\ninst✝ : LT α\na b : α\n⊢ ↑a < ↑b ↔ a < b",
"tactic": "simp only [← toDual_lt_toDual_iff, toDual_apply_coe, WithBot.coe_lt_coe, toDual_lt_toDual]"
}
] | [
1056,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1055,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean | PerfectClosure.mk_inv | [] | [
504,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
503,
1
] |
Mathlib/Data/Num/Lemmas.lean | ZNum.cast_neg | [] | [
1057,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1056,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.filterMap_le_filterMap | [] | [
2185,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2183,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.inclusion.injective | [
{
"state_after": "ι : Sort ?u.172133\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nx y : Subgraph G\nh✝ : x ≤ y\nv w : ↑x.verts\nh : ↑(inclusion h✝) v = ↑(inclusion h✝) w\n⊢ v = w",
"state_before": "ι : Sort ?u.172133\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\... | [
732,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
729,
1
] |
Mathlib/Data/Finset/Slice.lean | Set.Sized.subsingleton' | [] | [
86,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
] |
Mathlib/Data/Real/Basic.lean | Real.isCauSeq_iff_lift | [
{
"state_after": "no goals",
"state_before": "x y : ℝ\nf : ℕ → ℚ\nH : IsCauSeq abs f\nε : ℝ\nε0 : ε > 0\nδ : ℚ\nδ0 : 0 < δ\nδε : ↑δ < ε\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abs (f j - f i) < δ\nj : ℕ\nij : j ≥ i\n⊢ abs ((fun i => ↑(f i)) j - (fun i => ↑(f i)) i) < ↑δ",
"tactic": "simpa using (@Rat.cast_lt ℝ ... | [
662,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
656,
1
] |
Mathlib/FieldTheory/Separable.lean | Polynomial.separable_prod' | [
{
"state_after": "R : Type u\ninst✝ : CommRing R\nι : Type u_1\nf : ι → R[X]\ns✝ : Finset ι\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih :\n (∀ (x : ι), x ∈ s → ∀ (y : ι), y ∈ s → x ≠ y → IsCoprime (f x) (f y)) →\n (∀ (x : ι), x ∈ s → Separable (f x)) → Separable (∏ x in s, f x)\nh2 : Separable (f a) ∧ ∀ (x : ι),... | [
193,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
186,
1
] |
Mathlib/Data/FunLike/Equiv.lean | EquivLike.comp_surjective | [] | [
224,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
223,
1
] |
Mathlib/Order/SuccPred/Limit.lean | Order.isSuccLimit_iff_succ_lt | [] | [
161,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
] |
Mathlib/Algebra/GroupWithZero/Power.lean | zpow_add_one₀ | [
{
"state_after": "no goals",
"state_before": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (↑n + 1) = a ^ ↑n * a",
"tactic": "simp only [← Int.ofNat_succ, zpow_ofNat, pow_succ']"
},
{
"state_after": "no goals",
"state_before": "G₀ : Type u_1\ninst✝ : GroupWithZer... | [
83,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.uniformContinuousOn_iff | [] | [
827,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
824,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean | Monotone.Icc | [] | [
106,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
11
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | Filter.Tendsto.inner | [] | [
2262,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2260,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | strictConvexOn_pow | [
{
"state_after": "n : ℕ\nhn : 2 ≤ n\n⊢ StrictMonoOn (deriv fun x => x ^ n) (interior (Ici 0))",
"state_before": "n : ℕ\nhn : 2 ≤ n\n⊢ StrictConvexOn ℝ (Ici 0) fun x => x ^ n",
"tactic": "apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)"
},
{
"state_after": "n : ℕ\nh... | [
48,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
43,
1
] |
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