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/Analysis/SpecialFunctions/Pow/Real.lean | Complex.ofReal_cpow | [
{
"state_after": "x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ ↑(if x = 0 then if y = 0 then 1 else 0 else Real.exp (Real.log x * y)) =\n if x = 0 then if y = 0 then 1 else 0 else exp (log ↑x * ↑y)",
"state_before": "x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ ↑(x ^ y) = ↑x ^ ↑y",
"tactic": "simp only [Real.rpow_def_of_nonneg hx, Co... | [
248,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
246,
1
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Mathlib/Data/Finset/Pointwise.lean | Finset.mem_smul | [] | [
1281,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1280,
1
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Mathlib/GroupTheory/Subgroup/Pointwise.lean | Subgroup.mul_inf_assoc | [
{
"state_after": "case h\nα : Type ?u.33826\nG : Type u_1\nA✝ : Type ?u.33832\nS : Type ?u.33835\ninst✝¹ : Group G\ninst✝ : AddGroup A✝\ns : Set G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ x✝ ∈ ↑A * ↑(B ⊓ C) ↔ x✝ ∈ ↑A * ↑B ⊓ ↑C",
"state_before": "α : Type ?u.33826\nG : Type u_1\nA✝ : Type ?u.33832\nS : Type... | [
208,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
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Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.insert_ae_eq_self | [] | [
3324,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3323,
1
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Mathlib/Data/List/Zip.lean | List.revzip_map_fst | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.93517\nγ : Type ?u.93520\nδ : Type ?u.93523\nε : Type ?u.93526\nl : List α\n⊢ map Prod.fst (revzip l) = l",
"tactic": "rw [← unzip_left, unzip_revzip]"
}
] | [
354,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
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Mathlib/Logic/Equiv/Set.lean | Equiv.ofInjective_symm_apply | [
{
"state_after": "case a\nα✝ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα : Sort u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\na : α\n⊢ ↑(ofInjective f hf) (↑(ofInjective f hf).symm { val := f a, property := (_ : ∃ y, f y = f a) }) =\n ↑(ofInjective f hf) a",
"state_before": "α✝ : Sort u\nβ✝ : Sort v\nγ : Sort... | [
621,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
618,
1
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Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.map_add_left_Ioo | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map ((fun x x_1 => x + x_1) c) (Ioo a b) = Ioo (c + a) (c + b)",
"tactic": "classical rw [Ioo, Ioo, ← Finset.image_add_left_Ioo, Finset.... | [
306,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
1
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Mathlib/RingTheory/HahnSeries.lean | HahnSeries.support_neg | [
{
"state_after": "case h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddGroup R\nx : HahnSeries Γ R\nx✝ : Γ\n⊢ x✝ ∈ support (-x) ↔ x✝ ∈ support x",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddGroup R\nx : HahnSeries Γ R\n⊢ support (-x) = support x",
... | [
464,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
462,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean | Rel.mk_mem_interedges_comm | [] | [
278,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/Data/Nat/Sqrt.lean | Nat.sqrt_add_eq | [
{
"state_after": "n a : ℕ\nh : a ≤ n + n\n⊢ n * n + a < succ (n * n + (n + n))",
"state_before": "n a : ℕ\nh : a ≤ n + n\n⊢ n * n + a < succ n * succ n",
"tactic": "rw [succ_mul, mul_succ, add_succ, add_assoc]"
},
{
"state_after": "no goals",
"state_before": "n a : ℕ\nh : a ≤ n + n\n⊢ n * n ... | [
148,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
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Mathlib/Data/Finset/Basic.lean | Finset.inter_biUnion | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.513265\ninst✝ : DecidableEq β\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : α → Finset β\nt : Finset β\ns : Finset α\nf : α → Finset β\n⊢ (Finset.biUnion s fun x => f x ∩ t) = Finset.biUnion s fun x => t ∩ f x",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Ty... | [
3590,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3587,
1
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Mathlib/Data/Nat/Basic.lean | Nat.find_eq_zero | [
{
"state_after": "no goals",
"state_before": "m n k : ℕ\np q : ℕ → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\nh : ∃ n, p n\n⊢ Nat.find h = 0 ↔ p 0",
"tactic": "simp [find_eq_iff]"
}
] | [
849,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
849,
1
] |
Mathlib/GroupTheory/Transfer.lean | MonoidHom.transfer_eq_prod_quotient_orbitRel_zpowers_quot | [
{
"state_after": "G : Type u_1\ninst✝³ : Group G\nH : Subgroup G\nA : Type u_2\ninst✝² : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝¹ : FiniteIndex H\ng : G\ninst✝ : Fintype (Quotient (orbitRel { x // x ∈ zpowers g } (G ⧸ H)))\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H... | [
146,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Mathlib/Analysis/Seminorm.lean | Seminorm.ball_sup | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.860119\nR' : Type ?u.860122\n𝕜 : Type u_1\n𝕜₂ : Type ?u.860128\n𝕜₃ : Type ?u.860131\n𝕝 : Type ?u.860134\nE : Type u_2\nE₂ : Type ?u.860140\nE₃ : Type ?u.860143\nF : Type ?u.860146\nG : Type ?u.860149\nι : Type ?u.860152\ninst✝² : SeminormedRing �... | [
711,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
709,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean | padicValInt.self | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : 1 < p\n⊢ padicValInt p ↑p = 1",
"tactic": "simp [padicValNat.self hp]"
}
] | [
138,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
1
] |
Mathlib/Data/Polynomial/Laurent.lean | LaurentPolynomial.degree_T | [
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nn : ℤ\n⊢ degree (↑C 1 * T n) = ↑n",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nn : ℤ\n⊢ degree (T n) = ↑n",
"tactic": "rw [← one_mul (T n), ← map_one C]"
},
{
"state_after": "no goals",
... | [
525,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
523,
1
] |
Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | Nat.doubleFactorial_add_one | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (n + 1)‼ = (n + 1) * (n - 1)‼",
"tactic": "cases n <;> rfl"
}
] | [
46,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
46,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean | lt_of_mul_lt_mul_of_nonpos_left | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.194298\ninst✝ : LinearOrderedRing α\na b c : α\nh : c * a < c * b\nhc : c ≤ 0\n⊢ -c * b < -c * a",
"tactic": "rwa [neg_mul, neg_mul, neg_lt_neg_iff]"
}
] | [
1151,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1150,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.intCast_re | [
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.6456011\ninst✝ : IsROrC K\nn : ℤ\n⊢ ↑re ↑n = ↑n",
"tactic": "rw [← ofReal_intCast, ofReal_re]"
}
] | [
669,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
669,
1
] |
Mathlib/Topology/PathConnected.lean | Path.prod_coe | [] | [
559,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
557,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.sum_smul_const_vsub_eq_vsub_affineCombination | [
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_3\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.348746\ns₂ : Finset ι₂\nw : ι → k\np₂ : ι → P\np₁ : P\nh : ∑ i in s, w i = 1\n⊢ ∑ i in s, w i • ... | [
532,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
530,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | div_le_div_of_le_of_nonneg | [] | [
357,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
356,
1
] |
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.piCongrRight_trans | [] | [
489,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
485,
1
] |
Mathlib/Data/Nat/Interval.lean | Nat.decreasing_induction_of_infinite | [] | [
354,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.direction_le | [
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns1 s2 : AffineSubspace k P\nh : s1 ≤ s2\n⊢ Submodule.span k (↑s1 -ᵥ ↑s1) ≤ Submodule.span k (↑s2 -ᵥ ↑s2)",
"state_before": "k : Type u_1\nV : Type u_2\nP ... | [
927,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
925,
1
] |
Mathlib/Computability/TMToPartrec.lean | Turing.PartrecToTM2.contSupp_cons₁ | [
{
"state_after": "no goals",
"state_before": "fs : Code\nk : Cont'\n⊢ contSupp (Cont'.cons₁ fs k) =\n trStmts₁\n (move₂ (fun x => false) main aux\n (move₂ (fun s => decide (s = Γ'.consₗ)) stack main\n (move₂ (fun x => false) aux stack (trNormal fs (Cont'.cons₂ k))))) ∪\n c... | [
1870,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1863,
1
] |
Mathlib/Algebra/Algebra/Basic.lean | NoZeroSMulDivisors.algebraMap_injective | [
{
"state_after": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Nontrivial A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\nthis : Function.Injective fun c => c • 1\n⊢ Function.Injective ↑(algebraMap R A)",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R... | [
817,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
812,
1
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Mathlib/Data/Multiset/Bind.lean | Multiset.product_zero | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.56883\nδ : Type ?u.56886\na : α\nb : β\ns : Multiset α\nt : Multiset β\n⊢ s ×ˢ 0 = 0",
"tactic": "simp [SProd.sprod, product]"
}
] | [
265,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
265,
1
] |
Mathlib/Combinatorics/Additive/Behrend.lean | Behrend.card_box | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.797\nβ : Type ?u.800\nn d k N : ℕ\nx : Fin n → ℕ\n⊢ card (box n d) = d ^ n",
"tactic": "simp [box]"
}
] | [
81,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/CategoryTheory/Monoidal/Limits.lean | CategoryTheory.Limits.limLax_μ | [
{
"state_after": "J : Type v\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasLimits C\ninst✝ : MonoidalCategory C\nF G : J ⥤ C\nj j' : J\nf : j ⟶ j'\n⊢ 𝟙 (limit F ⊗ limit G) ≫ (limit.π F j' ⊗ limit.π G j') = (limit.π F j ⊗ limit.π G j) ≫ (F.map f ⊗ G.map f)",
"state_before": "J : Ty... | [
145,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
] |
Mathlib/Algebra/GeomSum.lean | geom_sum_mul | [
{
"state_after": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\nthis : (∑ i in range n, x ^ i * 1 ^ (n - 1 - i)) * (x - 1) = x ^ n - 1 ^ n\n⊢ (∑ i in range n, x ^ i) * (x - 1) = x ^ n - 1",
"state_before": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\n⊢ (∑ i in range n, x ^ i) * (x - 1) = x ^ n - 1",
"tactic":... | [
222,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
] |
Mathlib/Order/Basic.lean | eq_iff_not_lt_of_le | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.4759\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.4770\nr : α✝ → α✝ → Prop\nα : Type u_1\ninst✝ : PartialOrder α\nx y : α\n⊢ x ≤ y → y = x ↔ ¬x < y",
"tactic": "rw [lt_iff_le_and_ne, not_and, Classical.not_not, eq_comm]"
}
] | [
382,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
381,
1
] |
Mathlib/Logic/Equiv/Set.lean | Equiv.apply_ofInjective_symm | [] | [
614,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
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] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean | isPreconnected_Ico | [] | [
462,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
461,
1
] |
Mathlib/Topology/Order/Basic.lean | interior_Iio | [] | [
316,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
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] |
Mathlib/Algebra/Order/Floor.lean | Int.ceil_eq_iff | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.222532\nα : Type u_1\nβ : Type ?u.222538\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ ⌈a⌉ = z ↔ ↑z - 1 < a ∧ a ≤ ↑z",
"tactic": "rw [← ceil_le, ← Int.cast_one, ← Int.cast_sub, ← lt_ceil, Int.sub_one_lt_iff, le_antisymm_iff,... | [
1208,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1206,
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] |
Mathlib/Data/List/Basic.lean | List.append_eq_has_append | [] | [
356,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
355,
1
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Mathlib/Data/Set/Intervals/Group.lean | Set.nonempty_Ico_sdiff | [
{
"state_after": "case inl\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nx dx y dy : α\nh : dy < dx\nhx : 0 < dx\nh' : x < y\n⊢ Nonempty ↑(Ico x (x + dx) \\ Ico y (y + dy))\n\ncase inr\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nx dx y dy : α\nh : dy < dx\nhx : 0 < dx\nh' : y ≤ x\n⊢ Nonempty ↑(Ico... | [
158,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasSum_iff_hasSum | [] | [
130,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
1
] |
Mathlib/Computability/Partrec.lean | Computable.list_length | [] | [
366,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
365,
1
] |
Mathlib/Order/Disjoint.lean | IsCompl.inf_eq_bot | [] | [
498,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
497,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.surjOn | [] | [
197,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
196,
11
] |
Std/Data/Int/Lemmas.lean | Int.lt_add_succ | [
{
"state_after": "a : Int\nn : Nat\n⊢ a + (↑n + 1) = a + ↑(succ n)",
"state_before": "a : Int\nn : Nat\n⊢ a + 1 + ↑n = a + ↑(succ n)",
"tactic": "rw [Int.add_comm, Int.add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "a : Int\nn : Nat\n⊢ a + (↑n + 1) = a + ↑(succ n)",
"tac... | [
591,
61
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
590,
1
] |
Mathlib/Topology/Algebra/Star.lean | ContinuousWithinAt.star | [] | [
75,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean | AddLECancellable.lt_tsub_of_add_lt_left | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.41593\ninst✝³ : PartialOrder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nha : AddLECancellable a\nh : a + c < b\n⊢ c + a < b",
"tactic": "rwa [add_comm]"
}
] | [
333,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
332,
11
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Mathlib/Analysis/Normed/Group/Hom.lean | NormedAddGroupHom.comp_assoc | [
{
"state_after": "case H\nV : Type ?u.478988\nV₁ : Type u_4\nV₂ : Type u_3\nV₃ : Type u_2\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup V₁\ninst✝² : SeminormedAddCommGroup V₂\ninst✝¹ : SeminormedAddCommGroup V₃\nf✝ g✝ : NormedAddGroupHom V₁ V₂\nV₄ : Type u_1\ninst✝ : SeminormedAddCommGroup... | [
705,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
701,
1
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Mathlib/Order/Filter/Lift.lean | Filter.lift'_inf_le | [] | [
410,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
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src/lean/Init/Data/Array/BasicAux.lean | List.toArray_eq_toArray_eq | [
{
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"state_before": "α : Type u_1\nas bs : List α\n⊢ (toArray as = toArray bs) = (as = bs)",
"tactic": "apply propext"
},
{
"state_after": "case a.mp\nα : Type u_1\nas bs : List α\n⊢ toArray as = toArray bs... | [
38,
20
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
35,
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Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometry.preimage_sphere | [] | [
315,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
1
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Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_right | [] | [
663,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
661,
1
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Mathlib/Algebra/Order/Hom/Basic.lean | abs_sub_map_le_div | [
{
"state_after": "ι : Type ?u.44336\nF : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44348\nδ : Type ?u.44351\ninst✝² : Group α\ninst✝¹ : LinearOrderedAddCommGroup β\ninst✝ : GroupSeminormClass F α β\nf : F\nx y : α\n⊢ ↑f x ≤ ↑f y + ↑f (x / y) ∧ ↑f y ≤ ↑f x + ↑f (x / y)",
"state_before": "ι : Type ?u.... | [
272,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
269,
1
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Mathlib/CategoryTheory/Monoidal/Category.lean | CategoryTheory.MonoidalCategory.tensor_inv_hom_id' | [
{
"state_after": "no goals",
"state_before": "C✝ : Type u\n𝒞 : Category C✝\ninst✝³ : MonoidalCategory C✝\nC : Type u\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nU V✝ W✝ X✝ Y✝ Z✝ V W X Y Z : C\nf : V ⟶ W\ninst✝ : IsIso f\ng : X ⟶ Y\nh : Y ⟶ Z\n⊢ (g ⊗ inv f) ≫ (h ⊗ f) = (g ⊗ 𝟙 W) ≫ (h ⊗ 𝟙 W)",
"... | [
398,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
396,
1
] |
Mathlib/Data/Polynomial/Eval.lean | Polynomial.eval_finset_sum | [] | [
476,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
474,
1
] |
Mathlib/ModelTheory/Satisfiability.lean | FirstOrder.Language.Theory.SemanticallyEquivalent.realize_bd_iff | [] | [
449,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
446,
1
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Mathlib/Data/Finset/Pointwise.lean | Finset.smul_eq_empty | [] | [
1317,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1316,
1
] |
Mathlib/Data/Rat/Floor.lean | Int.mod_nat_eq_sub_mul_floor_rat_div | [
{
"state_after": "no goals",
"state_before": "n : ℤ\nd : ℕ\n⊢ n % ↑d = n - ↑d * ⌊↑n / ↑d⌋",
"tactic": "rw [eq_sub_of_add_eq <| Int.emod_add_ediv n d, Rat.floor_int_div_nat_eq_div]"
}
] | [
98,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean | Lagrange.natDegree_basisDivisor_self | [
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝ : Field F\nx y : F\n⊢ natDegree (basisDivisor x x) = 0",
"tactic": "rw [basisDivisor_self, natDegree_zero]"
}
] | [
162,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Std/Data/String/Lemmas.lean | String.Iterator.ValidFor.curr | [
{
"state_after": "case refl\nl r : List Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ Iterator.curr { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } =\n List.headD r default",
"state_before": "l r : L... | [
546,
49
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
545,
1
] |
Mathlib/CategoryTheory/Preadditive/Generator.lean | CategoryTheory.isCoseparator_iff_faithful_preadditiveYoneda | [
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\n⊢ Faithful (preadditiveYoneda.obj G ⋙ forget AddCommGroupCat) ↔ Faithful (preadditiveYoneda.obj G)",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\n⊢ IsCoseparator G ↔ Faithful (preadditiveYoned... | [
74,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean | pow_three' | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.12288\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\n⊢ a ^ 3 = a * a * a",
"tactic": "rw [pow_succ', pow_two]"
}
] | [
112,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean | LieSubmodule.lie_le_left | [
{
"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... | [
131,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean | Balanced.neg | [] | [
226,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
225,
1
] |
Mathlib/Order/UpperLower/Basic.lean | IsLowerSet.compl | [] | [
88,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
] |
Mathlib/Data/Fintype/Card.lean | Fintype.one_lt_card_iff | [] | [
597,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
596,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | MeasureTheory.intervalIntegral_tendsto_integral_Ioi | [
{
"state_after": "ι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na✝ b : ι → ℝ\nf : ℝ → E\na : ℝ\nhfi : IntegrableOn f (Ioi a)\nhb : Tendsto b l atTop\nφ : ι → Set ℝ := fun i => Iic (b i)\... | [
654,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
645,
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] |
Mathlib/Data/Polynomial/Splits.lean | Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split | [
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ nextCoeff (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots P))) = -Multiset.sum (roots P)",
"state_before": "F : Type u\nK :... | [
482,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
477,
1
] |
Mathlib/Data/List/Rotate.lean | List.IsRotated.eqv | [] | [
469,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
468,
1
] |
Mathlib/Data/Polynomial/Splits.lean | Polynomial.natDegree_eq_card_roots' | [
{
"state_after": "case pos\nF : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\nhp : map i p = 0\n⊢ natDegree (map i p) = ↑Multiset.card (roots (map i p))\n\ncase neg\nF : Type u\nK : Type v\nL : Type w\ninst✝² : C... | [
222,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
207,
1
] |
Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.pow_left_iff | [
{
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"state_before": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt : Finset I\nm n : ℕ\nhm : 0 < m\n⊢ IsCoprime (x ^ m) ... | [
209,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
206,
1
] |
Mathlib/Analysis/NormedSpace/MStructure.lean | IsLprojection.mul_compl_self | [
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝² : NormedAddCommGroup X\nM : Type u_1\ninst✝¹ : Ring M\ninst✝ : Module M X\nP : { P // IsLprojection X P }\n⊢ ↑P * ↑(Pᶜ) = 0",
"tactic": "rw [coe_compl, mul_sub, mul_one, P.prop.proj.eq, sub_self]"
}
] | [
268,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.prod_mem | [] | [
181,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
11
] |
Mathlib/Deprecated/Subgroup.lean | IsGroupHom.one_ker_inv' | [
{
"state_after": "G : Type u_1\nH : Type u_2\nA : Type ?u.71750\na✝ a₁ a₂ b✝ c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na b : G\nh : (f a)⁻¹ * f b = 1\n⊢ f a = f b",
"state_before": "G : Type u_1\nH : Type u_2\nA : Type ?u.71750\na✝ a₁ a₂ b✝ c : G\ninst✝¹ : Group G\ninst✝ : Grou... | [
358,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
354,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean | DoubleQuot.quotQuotEquivQuotSup_symm_quotQuotMk | [] | [
577,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
575,
1
] |
Mathlib/Order/CompactlyGenerated.lean | CompleteLattice.independent_sUnion_of_directed | [
{
"state_after": "ι : Sort ?u.107868\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\ns : Set (Set α)\nhs : DirectedOn (fun x x_1 => x ⊆ x_1) s\nh : ∀ (a : Set α), a ∈ s → SetIndependent a\n⊢ SetIndependent (⋃ (i : ↑s), ↑i)",
... | [
461,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
458,
1
] |
Mathlib/Analysis/Seminorm.lean | Seminorm.coe_sup | [] | [
244,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
243,
1
] |
Mathlib/Order/CompleteLattice.lean | Antitone.map_iSup₂_le | [] | [
1016,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1014,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | DifferentiableAt.rpow | [] | [
440,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
438,
1
] |
Mathlib/Algebra/Parity.lean | Odd.pow_nonneg_iff | [] | [
487,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
486,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.EventuallyLE.rfl | [] | [
1657,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1656,
1
] |
Mathlib/Data/Real/ENatENNReal.lean | ENat.toENNReal_top | [] | [
54,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
] |
Std/Data/Int/Lemmas.lean | Int.sign_eq_neg_one_of_neg | [] | [
1274,
23
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1272,
1
] |
Mathlib/CategoryTheory/Idempotents/FunctorCategories.lean | CategoryTheory.Idempotents.app_comp_p | [] | [
54,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.toLinearIsometry_injective | [] | [
613,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean | Filter.lt_pure_iff | [] | [
422,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
421,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | MonoidHom.comap_ker | [] | [
2800,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2799,
1
] |
Mathlib/Topology/Basic.lean | mem_of_mem_nhds | [] | [
907,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
906,
1
] |
Mathlib/Data/Finsupp/Fin.lean | Finsupp.cons_ne_zero_of_left | [
{
"state_after": "n : ℕ\ni : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\nc : cons y s = 0\n⊢ y = 0",
"state_before": "n : ℕ\ni : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\nh : y ≠ 0\n⊢ cons y s ≠ 0",
"tactic": "contrapose! h wi... | [
83,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/Algebra/Free.lean | Magma.AssocQuotient.lift_comp_of | [] | [
419,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
419,
1
] |
Mathlib/RingTheory/Localization/Basic.lean | IsLocalization.eq_iff_exists | [] | [
134,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
] |
Mathlib/Data/List/AList.lean | AList.entries_toAList | [] | [
340,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
] |
Mathlib/Computability/Halting.lean | ComputablePred.not | [
{
"state_after": "case intro.intro\nα : Type u_1\nσ : Type ?u.283450\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf : α → Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n⊢ ComputablePred fun a => ¬(fun a => f a = true) a",
"state_before": "α : Type u_1\nσ : Type ?u.283450\ninst✝¹ : Prim... | [
187,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
11
] |
Mathlib/Order/Bounds/Basic.lean | BddAbove.bddAbove_image2_of_bddBelow | [
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α → β → γ\ns : Set α\nt : Set β\na✝ : α\nb✝ : β\nh₀ : ∀ (b : β), Monotone (swap f b)\nh₁ : ∀ (a : α), Antitone (f a)\na : α\nha : a ∈ upperBounds s\nb : β\nhb : b... | [
1430,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1427,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | CategoryTheory.Limits.initial.to_comp | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : HasInitial C\nP Q : C\nf : P ⟶ Q\n⊢ to P ≫ f = to Q",
"tactic": "aesop"
}
] | [
379,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
378,
1
] |
Mathlib/RingTheory/Polynomial/Dickson.lean | Polynomial.dickson_one_one_comp_comm | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.255623\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nm n : ℕ\n⊢ comp (dickson 1 1 m) (dickson 1 1 n) = comp (dickson 1 1 n) (dickson 1 1 m)",
"tactic": "rw [← dickson_one_one_mul, mul_comm, dickson_one_one_mul]"
}
] | [
199,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.enumOrd_strictMono | [] | [
2187,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2186,
1
] |
Mathlib/Topology/Order/Basic.lean | nhds_basis_Ioo' | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\ns : Set α\n⊢ (∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s) ↔ ∃ i, (i.fst < a ∧ a < i.snd) ∧ Ioo i.fst i.snd ⊆ s",
... | [
1330,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1328,
1
] |
Mathlib/ModelTheory/Graph.lean | FirstOrder.Language.Theory.simpleGraph_isSatisfiable | [] | [
138,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean | AEMeasurable.comp_quasiMeasurePreserving | [] | [
171,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
] |
Mathlib/Data/List/Chain.lean | List.Chain.rel | [
{
"state_after": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\ninst✝ : IsTrans α R\nhl : Pairwise R (a :: l)\nhb : b ∈ l\n⊢ R a b",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\ninst✝ : IsTrans α R\nhl : Chain R a l\nhb : b ∈ l\n⊢ R a b",
... | [
140,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
11
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean | MulChar.IsQuadratic.eq_of_eq_coe | [] | [
463,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
460,
1
] |
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