file_path stringlengths 11 79 | full_name stringlengths 2 100 | traced_tactics list | end list | commit stringclasses 4
values | url stringclasses 4
values | start list |
|---|---|---|---|---|---|---|
Mathlib/Data/Num/Lemmas.lean | PosNum.one_le_cast | [
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nn : PosNum\n⊢ 1 ≤ ↑n",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nn : PosNum\n⊢ 1 ≤ ↑n",
"tactic": "rw [← cast_to_nat, ← Nat.cast_one, Nat.cast_le (α := α)]"
},
{
"state_after": "no goals",
"state_before": ... | [
679,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
678,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | bddAbove_range_of_tendsto_atTop_atBot | [] | [
79,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
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Mathlib/Data/Polynomial/Degree/Lemmas.lean | Polynomial.leadingCoeff_comp | [
{
"state_after": "no goals",
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"tactic": "rw [← coeff_comp_degree_mul_degree h... | [
366,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
364,
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Mathlib/Analysis/NormedSpace/LinearIsometry.lean | SemilinearIsometryClass.diam_image | [] | [
121,
52
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119,
1
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Mathlib/Algebra/Module/Submodule/Basic.lean | Submodule.ext | [] | [
103,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Analysis/Convex/Star.lean | StarConvex.is_linear_preimage | [] | [
227,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
225,
1
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Mathlib/GroupTheory/Perm/Support.lean | Equiv.Perm.disjoint_inv_right_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf g h : Perm α\n⊢ Disjoint f g⁻¹ ↔ Disjoint f g",
"tactic": "rw [disjoint_comm, disjoint_inv_left_iff, disjoint_comm]"
}
] | [
108,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.normalCore_idempotent | [] | [
2572,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2571,
1
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Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | affineSpan_insert_affineSpan | [
{
"state_after": "no goals",
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"tactic": "rw... | [
1385,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1383,
1
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Mathlib/Topology/MetricSpace/EMetricSpace.lean | EMetric.diam_union' | [
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.302983\ninst✝ : PseudoEMetricSpace α\nx✝ y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns t✝ t : Set α\nh : Set.Nonempty (s ∩ t)\nx : α\nxs : x ∈ s\nxt : x ∈ t\n⊢ diam (s ∪ t) ≤ diam s + diam t",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.302983\ninst✝ : PseudoEMetr... | [
968,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
966,
1
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Mathlib/LinearAlgebra/ProjectiveSpace/Basic.lean | Projectivization.exists_smul_eq_mk_rep | [] | [
120,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
119,
1
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Mathlib/Analysis/Seminorm.lean | Seminorm.continuous_of_le | [
{
"state_after": "R : Type ?u.1437726\nR' : Type ?u.1437729\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1437735\n𝕜₃ : Type ?u.1437738\n𝕝 : Type ?u.1437741\nE : Type u_1\nE₂ : Type ?u.1437747\nE₃ : Type ?u.1437750\nF : Type ?u.1437753\nG : Type ?u.1437756\nι : Type ?u.1437759\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : S... | [
1180,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1173,
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Mathlib/Topology/ContinuousFunction/Algebra.lean | ContinuousMap.zpow_comp | [] | [
251,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
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Mathlib/CategoryTheory/CofilteredSystem.lean | CategoryTheory.Functor.eventualRange_eq_range_precomp | [
{
"state_after": "case a\nJ : Type u\ninst✝ : Category J\nF : J ⥤ Type v\ni j k : J\ns : Set (F.obj i)\nf : i ⟶ j\ng : j ⟶ k\nh : eventualRange F k = range (F.map g)\n⊢ eventualRange F k ⊆ range (F.map (f ≫ g))\n\ncase a\nJ : Type u\ninst✝ : Category J\nF : J ⥤ Type v\ni j k : J\ns : Set (F.obj i)\nf : i ⟶ j\ng... | [
172,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
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] |
Mathlib/Logic/Equiv/Defs.lean | Equiv.ext_iff | [] | [
131,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
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Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.le_trim_iff | [
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm m₁ m₂ : OuterMeasure α\nme : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s\n⊢ m₁ ≤ trim m₂ ↔ ∀ (s : Set α), MeasurableSet s → ↑m₁ s ≤ ↑m₂ s",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm m₁ m₂ : OuterMeasure α\n⊢ m₁ ≤ trim m₂ ↔ ∀ (s : Se... | [
1642,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1633,
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Mathlib/Data/List/Perm.lean | List.nil_perm | [] | [
174,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
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Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | CategoryTheory.Abelian.pseudoEqual_trans | [
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP : C\nf g h : Over P\nR : C\np : R ⟶ f.left\nq : R ⟶ g.left\nep : Epi p\nEq : Epi q\ncomm : p ≫ f.hom = q ≫ g.hom\nR' : C\np' : R' ⟶ g.left\nq' : R' ⟶ h.left\nep' : Epi p'\neq' : Epi q'\ncomm' : p' ≫ g.hom = q' ≫ h.hom\n⊢ PseudoEqual P f h",... | [
130,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | toIocMod_add_toIocDiv_zsmul | [
{
"state_after": "no goals",
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"tactic": "rw [toIocMod, sub_add_cancel]"
}
] | [
161,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
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] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.limsup_ae_eq_of_forall_ae_eq | [
{
"state_after": "α : Type u_1\nβ : Type ?u.75562\nγ : Type ?u.75565\nδ : Type ?u.75568\nι : Type ?u.75571\nR : Type ?u.75574\nR' : Type ?u.75577\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ns : ℕ → Set α\nt : Set α\nh : ∀ (n : ℕ), ↑↑μ (s n \\ t) = 0 ∧ ↑↑μ (t \\ s n) = 0\n⊢ ↑↑μ (limsup s at... | [
622,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
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Mathlib/Topology/Category/TopCat/Basic.lean | TopCat.of_homeoOfIso | [
{
"state_after": "X Y : TopCat\nf : X ≅ Y\n⊢ Iso.mk\n (Homeomorph.toContinuousMap\n (Homeomorph.mk\n { toFun := (forget TopCat).map f.hom, invFun := (forget TopCat).map f.inv,\n left_inv := (_ : ∀ (x : ↑X), (forget TopCat).map f.inv ((forget TopCat).map f.hom x) = x),\n ... | [
158,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
154,
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Mathlib/MeasureTheory/Measure/Complex.lean | MeasureTheory.ComplexMeasure.toComplexMeasure_to_signedMeasure | [] | [
78,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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1
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Mathlib/Order/JordanHolder.lean | CompositionSeries.append_castAdd | [
{
"state_after": "no goals",
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"tactic": "rw [coe_appe... | [
523,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/Data/List/Lattice.lean | List.nil_bagInter | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl✝ l₁ l₂ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nl : List α\n⊢ List.bagInter [] l = []",
"tactic": "cases l <;> rfl"
}
] | [
209,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | Matrix.nonsing_inv_apply | [
{
"state_after": "no goals",
"state_before": "l : Type ?u.101841\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit (det A)\n⊢ A⁻¹ = ↑(IsUnit.unit h)⁻¹ • adjugate A",
"tactic": "rw [inv_def, ← Ring.inverse_unit h.unit, IsU... | [
265,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
264,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean | Matrix.det_updateColumn_add | [
{
"state_after": "m : Type ?u.1639172\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\nj : n\nu v : n → R\n⊢ det (updateRow Mᵀ j u) + det (updateRow Mᵀ j v) = det (updateColumn M j u) + det (updateColumn M j ... | [
399,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
396,
1
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Mathlib/Data/Polynomial/Expand.lean | Polynomial.Monic.expand | [
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nf : R[X]\nhp : 0 < p\nh : Monic f\n⊢ (if p ∣ natDegree f * p then coeff f (natDegree f * p / p) else 0) = 1",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q ... | [
178,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
] |
Mathlib/CategoryTheory/Limits/EssentiallySmall.lean | CategoryTheory.Limits.hasProductsOfShape_of_small | [] | [
44,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
] |
Mathlib/Topology/MetricSpace/IsometricSMul.lean | nndist_smul | [] | [
346,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
344,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.lt_iff_exists_real_btwn | [] | [
621,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
617,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean | Set.smul_set_symm_diff₀ | [] | [
1055,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1054,
1
] |
Mathlib/CategoryTheory/IsConnected.lean | CategoryTheory.isPreconnected_induction | [
{
"state_after": "case intro\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsPreconnected J\nZ : J → Sort u_1\nh₁ : {j₁ j₂ : J} → (j₁ ⟶ j₂) → Z j₁ → Z j₂\nh₂ : {j₁ j₂ : J} → (j₁ ⟶ j₂) → Z j₂ → Z j₁\nj₀ : J\nx : Z j₀\nj j₁✝ j₂✝ : J\nf : j₁✝ ⟶ j₂✝\ny : Z j₁✝\n⊢ j₂✝ ∈ {j | Nonempty (... | [
206,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
195,
1
] |
Mathlib/Data/List/Pairwise.lean | List.pairwise_of_forall | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3869\nR S T : α → α → Prop\na : α\nl✝ l : List α\nH : ∀ (x y : α), R x y\n⊢ Pairwise R l",
"tactic": "induction l <;> [exact Pairwise.nil; simp only [*, pairwise_cons, forall₂_true_iff, and_true_iff]]"
}
] | [
107,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean | ContDiff.dist | [] | [
220,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
] |
Mathlib/Combinatorics/Colex.lean | Colex.sum_two_pow_lt_iff_lt | [
{
"state_after": "α : Type ?u.84084\nA B : Finset ℕ\nz : ∀ (A B : Finset ℕ), toColex A < toColex B → ∑ i in A, 2 ^ i < ∑ i in B, 2 ^ i\nh : ∑ i in A, 2 ^ i < ∑ i in B, 2 ^ i\nh₁ : A = B ∨ B < A\n⊢ A = B → False",
"state_before": "α : Type ?u.84084\nA B : Finset ℕ\nz : ∀ (A B : Finset ℕ), toColex A < toColex... | [
420,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
397,
1
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Mathlib/LinearAlgebra/Ray.lean | sameRay_smul_right_iff | [] | [
535,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
533,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.coe_bot | [] | [
642,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
641,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineIsometryEquiv.diam_image | [] | [
674,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
673,
1
] |
Mathlib/Topology/Order/Basic.lean | comap_coe_Ioi_nhdsWithin_Ioi | [] | [
2502,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2501,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean | MvPolynomial.hom_congr_vars | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.682541\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ f₂ : MvPolynomial σ R →+* S\np₁ p₂ : MvPolynomial σ R\nhC : RingHom.comp f₁ C = RingHom.comp ... | [
869,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
863,
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Mathlib/Analysis/Calculus/ContDiffDef.lean | contDiffAt_succ_iff_hasFDerivAt | [
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAdd... | [
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45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1363,
1
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Mathlib/Data/Set/Function.lean | Set.injective_piecewise_iff | [
{
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1515,
70
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1509,
1
] |
Mathlib/Logic/Equiv/Basic.lean | Equiv.swap_comp_apply | [
{
"state_after": "case mk\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\ntoFun✝ invFun✝ : α → α\nleft_inv✝ : LeftInverse invFun✝ toFun✝\nright_inv✝ : Function.RightInverse invFun✝ toFun✝\n⊢ ↑({ toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }.trans (swap a b)) x =\n if ↑... | [
1595,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1592,
1
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Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | isAddCyclic_of_card_pow_eq_one_le | [
{
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410,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
407,
1
] |
Mathlib/Data/Rat/Order.lean | Rat.mul_nonneg | [
{
"state_after": "a✝ b✝ c a b : ℚ\nha : Rat.Nonneg a\nhb : Rat.Nonneg b\n⊢ Rat.Nonneg (a * b)",
"state_before": "a✝ b✝ c a b : ℚ\nha : 0 ≤ a\nhb : 0 ≤ b\n⊢ 0 ≤ a * b",
"tactic": "rw [← nonneg_iff_zero_le] at ha hb⊢"
},
{
"state_after": "no goals",
"state_before": "a✝ b✝ c a b : ℚ\nha : Rat.N... | [
253,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
252,
11
] |
Mathlib/Logic/Function/Iterate.lean | Function.Commute.iterate_left | [] | [
132,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean | padicNormE.norm_int_le_one | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ\nthis : ‖↑↑z‖ ≤ 1\n⊢ ‖↑z‖ ≤ 1",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ\n⊢ ¬p ∣ (↑z).den",
"tactic": "simp [hp.1.ne_one]"
}
] | [
917,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
915,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean | dvd_antisymm_of_normalize_eq | [] | [
194,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Std/Data/List/Lemmas.lean | List.pairwise_map | [
{
"state_after": "case nil\nα : Type u_1\nα✝ : Type u_2\nf : α → α✝\nR : α✝ → α✝ → Prop\n⊢ Pairwise R (map f []) ↔ Pairwise (fun a b => R (f a) (f b)) []\n\ncase cons\nα : Type u_1\nα✝ : Type u_2\nf : α → α✝\nR : α✝ → α✝ → Prop\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : Pairwise R (map f tail✝) ↔ Pairwise (fun a b ... | [
1285,
58
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1281,
1
] |
Mathlib/Order/UpperLower/Basic.lean | UpperSet.prod_self_le_prod_self | [] | [
1616,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1615,
1
] |
Mathlib/Logic/Equiv/Fin.lean | finSuccEquiv'_below | [
{
"state_after": "no goals",
"state_before": "m✝ n : ℕ\ni : Fin (n + 1)\nm : Fin n\nh : ↑Fin.castSucc m < i\n⊢ ↑(finSuccEquiv' i) (↑Fin.castSucc m) = some m",
"tactic": "rw [← Fin.succAbove_below _ _ h, finSuccEquiv'_succAbove]"
}
] | [
153,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Icc_subset_Ioc_iff | [] | [
262,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.cycleFactorsFinset_eq_singleton_self_iff | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2751377\nα : Type u_1\nβ : Type ?u.2751383\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f : Perm α\n⊢ cycleFactorsFinset f = {f} ↔ IsCycle f",
"tactic": "simp [cycleFactorsFinset_eq_finset]"
}
] | [
1453,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1452,
1
] |
Mathlib/Order/BooleanAlgebra.lean | sdiff_sdiff_sup_sdiff' | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.33026\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ z \\ (x \\ y) ⊓ z \\ (y \\ x) = (z \\ x ⊔ z ⊓ x ⊓ y) ⊓ (z \\ y ⊔ z ⊓ y ⊓ x)",
"tactic": "rw [sdiff_sdiff_right, sdiff_sdiff_right]"
},
{
"state_after": "no goals",
... | [
420,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
414,
1
] |
Std/Logic.lean | apply_ite | [] | [
699,
43
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
697,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | Complex.tan_eq | [
{
"state_after": "case h.e'_2.h.e'_1\nz : ℂ\nh :\n ((∀ (k : ℤ), ↑z.re ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), ↑z.im * I ≠ (2 * ↑l + 1) * ↑π / 2) ∨\n (∃ k, ↑z.re = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, ↑z.im * I = (2 * ↑l + 1) * ↑π / 2\n⊢ z = ↑z.re + ↑z.im * I",
"state_before": "z : ℂ\nh :\n ((∀ (k : ℤ), ↑z.re ≠ ... | [
153,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Data/Real/CauSeq.lean | CauSeq.ext | [] | [
153,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
] |
Mathlib/Data/Finset/Interval.lean | Finset.card_Ioc_finset | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\n⊢ card (Ioc s t) = 2 ^ (card t - card s) - 1",
"tactic": "rw [card_Ioc_eq_card_Icc_sub_one, card_Icc_finset h]"
}
] | [
114,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
113,
1
] |
Mathlib/Tactic/NormNum/Basic.lean | Mathlib.Meta.NormNum.isRat_sub | [
{
"state_after": "α : Type u_1\ninst✝ : Ring α\nf : α → α → α\na b : α\nna nb nc : ℤ\nda db dc k : ℕ\nhf : f = HSub.hSub\nra : IsRat a na da\nrb : IsRat b nb db\nh₁ : Int.sub (Int.mul na ↑db) (Int.mul nb ↑da) = Int.mul (↑k) nc\nh₂ : Nat.mul da db = Nat.mul k dc\n⊢ IsRat (a + -b) nc dc",
"state_before": "α :... | [
307,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.eval_C | [] | [
1143,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1142,
1
] |
Mathlib/RingTheory/IntegralClosure.lean | isIntegral_trans_aux | [
{
"state_after": "R : Type u_3\nA : Type u_1\nB : Type u_2\nS✝ : Type ?u.1370556\nT : Type ?u.1370559\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : CommRing S✝\ninst✝² : CommRing T\ninst✝¹ : Algebra A B\ninst✝ : Algebra R B\nf : R →+* S✝\ng : S✝ →+* T\nx : B\np : A[X]\npmonic : Monic ... | [
991,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
968,
1
] |
Mathlib/MeasureTheory/Integral/IntervalAverage.lean | interval_average_eq_div | [
{
"state_after": "no goals",
"state_before": "E : Type ?u.12725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → ℝ\na b : ℝ\n⊢ (⨍ (x : ℝ) in Ι a b, f x) = (∫ (x : ℝ) in a..b, f x) / (b - a)",
"tactic": "rw [interval_average_eq, smul_eq_mul, div_eq_inv_mul]"
}
... | [
57,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1801194\nγ : Type ?u.1801197\nδ : Type ?u.1801200\nm m0 : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (x : α) in s, g x ∂Measure.withDensity μ f) = ∫⁻ (x... | [
1781,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1778,
1
] |
Mathlib/RepresentationTheory/Basic.lean | Representation.asAlgebraHom_single | [
{
"state_after": "no goals",
"state_before": "k : Type u_2\nG : Type u_3\nV : Type u_1\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nr : k\n⊢ ↑(asAlgebraHom ρ) (Finsupp.single g r) = r • ↑ρ g",
"tactic": "simp only [asAlgebraH... | [
94,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.comap_const_of_mem | [] | [
2078,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2077,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean | le_geom | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.38546\nβ : Type ?u.38549\nι : Type ?u.38552\nu : ℕ → ℝ\nc : ℝ\nhc : 0 ≤ c\nn : ℕ\nh : ∀ (k : ℕ), k < n → u (k + 1) ≤ c * u k\n⊢ u n ≤ c ^ n * u 0",
"tactic": "apply (monotone_mul_left_of_nonneg hc).seq_le_seq n _ h _ <;>\n simp [_root_.pow_succ,... | [
151,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearEquiv.map_smulₛₗ | [] | [
1867,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1866,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean | CategoryTheory.Comonad.algebra_mono_of_mono | [] | [
515,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
514,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean | Submodule.coe_smul_of_tower | [] | [
325,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean | MulChar.IsQuadratic.pow_even | [
{
"state_after": "case intro\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nn : ℕ\nhn : Even (2 * n)\n⊢ χ ^ (2 * n) = 1",
"state_before": "R : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' :... | [
524,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/GroupTheory/Sylow.lean | Sylow.card_normalizer_modEq_card | [
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\nthis : { x // x ∈ subgroupOf H (normalizer H) } ≃ { x // x ∈ H }\n⊢ Fintype.card { x // x ∈ normalizer H } ≡ Fintype.card G [MOD ... | [
563,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
557,
1
] |
Mathlib/Data/Set/Intervals/OrderIso.lean | OrderIso.preimage_Ioc | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ↑e ⁻¹' Ioc a b = Ioc (↑(symm e) a) (↑(symm e) b)",
"tactic": "simp [← Ioi_inter_Iic]"
}
] | [
61,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean | StrictMono.const_mul' | [] | [
1377,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1376,
1
] |
Mathlib/Algebra/Algebra/Operations.lean | Submodule.pow_eq_span_pow_set | [
{
"state_after": "no goals",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n✝ : A\nn : ℕ\n⊢ M ^ n = span R (↑M ^ n)",
"tactic": "rw [← span_pow, span_eq]"
}
] | [
408,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
407,
1
] |
Mathlib/Topology/Semicontinuous.lean | lowerSemicontinuousAt_iSup | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nβ : Type ?u.119406\ninst✝² : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nι : Sort u_3\nδ : Type u_2\nδ' : Type ?u.119438\ninst✝¹ : CompleteLinearOrder δ\ninst✝ : ConditionallyCompleteLinearOrder δ'\nf : ι → α... | [
586,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
584,
1
] |
Mathlib/CategoryTheory/Conj.lean | CategoryTheory.Iso.refl_conj | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nα : X ≅ Y\nf : End X\n⊢ ↑(conj (refl X)) f = f",
"tactic": "rw [conj_apply, Iso.refl_inv, Iso.refl_hom, Category.id_comp, Category.comp_id]"
}
] | [
101,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
] |
Mathlib/Data/Polynomial/Monic.lean | Polynomial.monic_zero_iff_subsingleton | [] | [
39,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
38,
1
] |
Mathlib/NumberTheory/PythagoreanTriples.lean | coprime_sq_sub_sq_sum_of_odd_odd | [
{
"state_after": "case intro\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 1\nhn : n % 2 = 1\nm0 : ℤ\nhm2 : m - 1 = m0 * 2\n⊢ 2 ∣ m ^ 2 + n ^ 2 ∧\n 2 ∣ m ^ 2 - n ^ 2 ∧ (m ^ 2 - n ^ 2) / 2 % 2 = 0 ∧ Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1",
"state_before": "m n : ℤ\nh : Int.gcd m n = 1\nhm :... | [
435,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
405,
9
] |
Mathlib/Topology/Algebra/ValuedField.lean | Valued.valuedCompletion_apply | [] | [
371,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
370,
1
] |
Mathlib/Topology/ContinuousOn.lean | tendsto_const_nhdsWithin | [] | [
183,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
] |
Mathlib/Algebra/DirectSum/Internal.lean | DirectSum.coeRingHom_of | [] | [
155,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
153,
1
] |
Mathlib/Data/Matrix/Basic.lean | Matrix.conjTranspose_smul_non_comm | [
{
"state_after": "no goals",
"state_before": "l : Type ?u.972645\nm : Type u_2\nn : Type u_3\no : Type ?u.972654\nm' : o → Type ?u.972659\nn' : o → Type ?u.972664\nR : Type u_1\nS : Type ?u.972670\nα : Type v\nβ : Type w\nγ : Type ?u.972677\ninst✝³ : Star R\ninst✝² : Star α\ninst✝¹ : SMul R α\ninst✝ : SMul ... | [
2182,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2179,
1
] |
Mathlib/Data/Matrix/Basic.lean | Matrix.star_mul | [] | [
2369,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2367,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometry.one_def | [] | [
438,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
437,
1
] |
Mathlib/Algebra/Order/Sub/Basic.lean | OrderIso.map_tsub | [
{
"state_after": "α : Type ?u.7326\nβ : Type ?u.7329\nM : Type u_1\nN : Type u_2\ninst✝⁷ : Preorder M\ninst✝⁶ : Add M\ninst✝⁵ : Sub M\ninst✝⁴ : OrderedSub M\ninst✝³ : PartialOrder N\ninst✝² : Add N\ninst✝¹ : Sub N\ninst✝ : OrderedSub N\ne : M ≃o N\nh_add : ∀ (a b : M), ↑e (a + b) = ↑e a + ↑e b\na b : M\ne_add :... | [
54,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
48,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.mem_spanSingleton | [
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1298308\ninst✝³ : CommRing R₁\nK : Type ?u.1298314\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx y : P\n⊢ x ∈ { val := ... | [
1308,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1306,
1
] |
Mathlib/Order/Filter/Pointwise.lean | Filter.zero_smul_filter | [
{
"state_after": "F : Type ?u.443848\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.443857\nδ : Type ?u.443860\nε : Type ?u.443863\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : SMulWithZero α β\nf : Filter α\ng : Filter β\nhg : NeBot g\ns : Set β\nhs : s ∈ g\n⊢ {0} ∈ 0",
"state_before": "F : Type ?u.443848\nα : Type... | [
1375,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1371,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.conj_ofReal | [
{
"state_after": "no goals",
"state_before": "r : ℝ\n⊢ (↑(starRingEnd ℂ) ↑r).re = (↑r).re ∧ (↑(starRingEnd ℂ) ↑r).im = (↑r).im",
"tactic": "simp [star]"
}
] | [
517,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
516,
1
] |
Mathlib/Topology/Algebra/Field.lean | Filter.tendsto_cocompact_mul_right₀ | [] | [
37,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
35,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean | LinearPMap.sSup_le | [] | [
618,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
11
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean | CategoryTheory.Preadditive.IsIso.comp_left_eq_zero | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Preadditive C\nP Q R : C\nf f' : P ⟶ Q\ng g' : Q ⟶ R\ninst✝ : IsIso f\n⊢ f ≫ g = 0 ↔ g = 0",
"tactic": "rw [← IsIso.eq_inv_comp, Limits.comp_zero]"
}
] | [
269,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
1
] |
Mathlib/Algebra/Order/SMul.lean | monotone_smul_left | [] | [
138,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/Data/Nat/Pairing.lean | Set.iUnion_unpair | [] | [
215,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/CategoryTheory/Limits/Lattice.lean | CategoryTheory.Limits.CompleteLattice.finite_limit_eq_finset_univ_inf | [] | [
75,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean | AddLECancellable.tsub_le_tsub_iff_left | [
{
"state_after": "α : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nha : AddLECancellable a\nhc : AddLECancellable c\nh : c ≤ a\n⊢ a - b ≤ a - c → c ≤ b",
"state_before": "α : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : Ord... | [
371,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
366,
11
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.lt_aleph0_of_finite | [] | [
1477,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1476,
1
] |
Mathlib/Data/Setoid/Partition.lean | IndexedPartition.disjoint | [] | [
379,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
378,
1
] |
Mathlib/Data/PFunctor/Multivariate/M.lean | MvPFunctor.M.map_dest | [
{
"state_after": "n : ℕ\nP : MvPFunctor (n + 1)\nα β : TypeVec n\ng : α ::: M P α ⟹ β ::: M P β\nx : M P α\nh : ∀ (x : M P α), lastFun g x = dropFun g <$$> x\n⊢ g <$$> dest P x = (dropFun g ::: fun x => dropFun g <$$> x) <$$> dest P x",
"state_before": "n : ℕ\nP : MvPFunctor (n + 1)\nα β : TypeVec n\ng : α ... | [
386,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
381,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean | Polynomial.coeff_sub_eq_left_of_lt | [
{
"state_after": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Ring R\np q : R[X]\ndg : natDegree (-q) < n\n⊢ coeff (p - q) n = coeff p n",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Ring R\np q : R[X]\ndg : natDegree q < n\n⊢ coeff (p - q) n = coeff p n"... | [
312,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
310,
1
] |
Mathlib/Data/Part.lean | Part.some_get | [] | [
227,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
226,
1
] |
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