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/Order/OmegaCompletePartialOrder.lean | OmegaCompletePartialOrder.ωSup_le_ωSup_of_le | [
{
"state_after": "case intro\nα : Type u\nβ : Type v\nγ : Type ?u.8099\ninst✝ : OmegaCompletePartialOrder α\nc₀ c₁ : Chain α\nh✝ : c₀ ≤ c₁\ni w✝ : ℕ\nh : ↑c₀ i ≤ ↑c₁ w✝\n⊢ ↑c₀ i ≤ ωSup c₁",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.8099\ninst✝ : OmegaCompletePartialOrder α\nc₀ c₁ : Chain α\nh : c... | [
218,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
1
] |
Mathlib/Data/Set/Function.lean | Set.InjOn.mono | [] | [
629,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
628,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | Subring.mk'_toAddSubgroup | [] | [
324,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
] |
Mathlib/Data/Num/Lemmas.lean | PosNum.le_to_nat | [
{
"state_after": "α : Type ?u.168918\nm n : PosNum\n⊢ ¬↑n < ↑m ↔ m ≤ n",
"state_before": "α : Type ?u.168918\nm n : PosNum\n⊢ ↑m ≤ ↑n ↔ m ≤ n",
"tactic": "rw [← not_lt]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.168918\nm n : PosNum\n⊢ ¬↑n < ↑m ↔ m ≤ n",
"tactic": "exa... | [
199,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
] |
Mathlib/Data/Fin/VecNotation.lean | Matrix.tail_sub | [] | [
522,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean | AddSubmonoid.mul_bot | [
{
"state_after": "α : Type ?u.249186\nG : Type ?u.249189\nM : Type ?u.249192\nR : Type u_1\nA : Type ?u.249198\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS : AddSubmonoid R\nm : R\nx✝ : m ∈ S\nn : R\nhn : n = 0\n⊢ m * n = 0",
"state_before": "α : Type ?u.249186\nG : Type ... | [
576,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
574,
1
] |
Mathlib/Data/Fintype/BigOperators.lean | Finset.prod_univ_pi | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\n⊢ ∀ (a : (a : α) → a ∈ univ → δ a) (ha : a ∈ pi univ t), (fun x x_1 a => x a (_ : a ∈ univ)) a ha ∈... | [
184,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean | MulEquiv.piCongrRight_symm | [] | [
668,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
667,
1
] |
Mathlib/Data/Polynomial/Basic.lean | Polynomial.C_mul_monomial | [
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ ↑C a * ↑(monomial n) b = ↑(monomial n) (a * b)",
"tactic": "simp only [← monomial_zero_left, monomial_mul_monomial, zero_add]"
}
] | [
541,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
540,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean | smul_closedBall' | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\n⊢ c • closedBall x r = closedBall (c • x) (‖c‖ * r)",
"tactic": "simp only [← ball_union_sphere, Set.smul_set... | [
112,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Data/Bool/Basic.lean | Bool.false_le | [] | [
336,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
335,
1
] |
Mathlib/Data/Matrix/Basic.lean | Matrix.diagonal_injective | [
{
"state_after": "no goals",
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455,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean | Zsqrtd.smul_re | [
{
"state_after": "no goals",
"state_before": "d a : ℤ\nb : ℤ√d\n⊢ (↑a * b).re = a * b.re",
"tactic": "simp"
}
] | [
319,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
319,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | MeasureTheory.Lp.simpleFunc.memℒp | [] | [
630,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
629,
11
] |
Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.HasBasis.totallyBounded_iff | [] | [
488,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
484,
1
] |
Mathlib/Order/Bounds/Basic.lean | isGreatest_Iic | [] | [
515,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
514,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.eval₂_comp_right | [
{
"state_after": "case h_C\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np ... | [
1254,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1245,
1
] |
Mathlib/Algebra/Symmetrized.lean | SymAlg.sym_ne_one_iff | [] | [
258,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
] |
Mathlib/Topology/Separation.lean | RegularSpace.ofBasis | [] | [
1512,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1509,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | MonoidWithZero.inverse_apply | [] | [
270,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean | exp_add_of_commute_of_mem_ball | [
{
"state_after": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball... | [
281,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
265,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.sup_le_sup | [
{
"state_after": "case intro\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Or... | [
1427,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1415,
9
] |
Mathlib/LinearAlgebra/SModEq.lean | SModEq.comap | [] | [
107,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.concat_eq_append | [] | [
193,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/Analysis/Complex/Basic.lean | Complex.ofRealClm_coe | [] | [
404,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
403,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | map_mul_right_nhds | [] | [
832,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
831,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean | derivWithin_neg | [] | [
277,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
276,
1
] |
Mathlib/Data/PNat/Xgcd.lean | PNat.XgcdType.reduce_isSpecial | [
{
"state_after": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : r u = 0\nhs : IsSpecial x✝\n⊢ IsSpecial (finish u)",
"state_before": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : r u = 0\nhs : IsSpecial x✝\n⊢ IsSpecial (reduce x✝)",
"tactic": "rw [reduce_a h]"
},
{
"state_after": "no goals",
"state... | [
386,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
377,
1
] |
Mathlib/Data/Matrix/Basis.lean | Matrix.StdBasisMatrix.apply_of_row_ne | [
{
"state_after": "no goals",
"state_before": "l : Type ?u.22747\nm : Type u_1\nn : Type u_3\nR : Type ?u.22756\nα : Type u_2\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Semiring α\ni✝ : m\nj✝ : n\nc : α\ni'✝ : m\nj'✝ : n\ni i' : m\nhi : i ≠ i'\nj j' : n\na : α\n⊢ stdBasi... | [
137,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean | StrictMonoOn.continuousAt_of_exists_between | [] | [
220,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
1
] |
Mathlib/Data/PFun.lean | PFun.image_union | [] | [
421,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
420,
1
] |
src/lean/Init/SimpLemmas.lean | Bool.beq_to_eq | [
{
"state_after": "no goals",
"state_before": "a b : Bool\n⊢ ((a == b) = true) = (a = b)",
"tactic": "cases a <;> cases b <;> decide"
}
] | [
132,
58
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
131,
9
] |
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | MeasureTheory.SimpleFunc.edist_approxOn_y0_le | [] | [
189,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/Algebra/Algebra/Hom.lean | AlgHom.congr_arg | [] | [
219,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
11
] |
Std/Data/List/Lemmas.lean | List.disjoint_take_drop | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm n : Nat\nx✝¹ : Nodup []\nx✝ : m ≤ n\n⊢ Disjoint (take m []) (drop n [])",
"tactic": "simp"
},
{
"state_after": "case succ.zero\nα : Type u_1\nx : α\nxs : List α\nhl : Nodup (x :: xs)\nn✝ : Nat\nh : succ n✝ ≤ zero\n⊢ False\n\ncase succ... | [
1761,
61
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1753,
1
] |
Mathlib/Topology/Inseparable.lean | SeparationQuotient.continuous_lift | [
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.105005\nα : Type ?u.105008\nι : Type ?u.105011\nπ : ι → Type ?u.105016\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\... | [
575,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
573,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.toReal_add | [
{
"state_after": "case intro\ny : EReal\nhy : y ≠ ⊤\nh'y : y ≠ ⊥\nx : ℝ\nhx : ↑x ≠ ⊤\nh'x : ↑x ≠ ⊥\n⊢ toReal (↑x + y) = toReal ↑x + toReal y",
"state_before": "x y : EReal\nhx : x ≠ ⊤\nh'x : x ≠ ⊥\nhy : y ≠ ⊤\nh'y : y ≠ ⊥\n⊢ toReal (x + y) = toReal x + toReal y",
"tactic": "lift x to ℝ using ⟨hx, h'x⟩"
... | [
676,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
672,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | CategoryTheory.Limits.IsTerminal.isIso_from | [] | [
192,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
] |
Mathlib/Logic/Function/Basic.lean | Function.sometimes_spec | [
{
"state_after": "no goals",
"state_before": "p : Prop\nα : Sort u_1\ninst✝ : Nonempty α\nP : α → Prop\nf : p → α\na : p\nh : P (f a)\n⊢ P (sometimes f)",
"tactic": "rwa [sometimes_eq]"
}
] | [
979,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
977,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | isAdjointPair_toLinearMap₂ | [
{
"state_after": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁... | [
591,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
577,
1
] |
Mathlib/Order/Iterate.lean | Monotone.seq_le_seq | [
{
"state_after": "case zero\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)\nhx : ∀ (k : ℕ), k < Nat.zero → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.zero → f (y k) ≤ ... | [
50,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
44,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.toNat_le_of_le_of_lt_aleph0 | [] | [
1705,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1703,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean | EuclideanGeometry.dist_smul_vadd_sq | [
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nr : ℝ\nv : V\np₁ p₂ : P\n⊢ r * (r * inner v v) + 2 * (r * inner v (p₁ -ᵥ p₂)) + inner (p₁ -ᵥ p₂) (p₁ -ᵥ p₂) =\n inner v v * r * r + 2 * inner v (p₁... | [
139,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean | LinearMap.ker_eq_bot_iff_range_eq_top | [
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\n⊢ ker f = ⊥ ↔ range f = ⊤",
"tactic": "rw [range_eq_top... | [
926,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
924,
1
] |
Mathlib/Data/Set/Sups.lean | Set.iUnion_image_sup_left | [] | [
179,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
1
] |
Mathlib/LinearAlgebra/Determinant.lean | Basis.det_map' | [] | [
635,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
633,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.IsPullback.paste_horiz | [] | [
519,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
515,
1
] |
Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | Path.Homotopy.continuous_transAssocReparamAux | [
{
"state_after": "case refine'_9\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I\nhx : ↑x = 1 / 4\n⊢ 2 * ↑x = if ↑x ≤ 1 / 2 then ↑x + 1 / 4 else 1 / 2 * (↑x + 1)",
"state_before": "case refine'_9\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ ... | [
210,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean | EuclideanGeometry.inner_nonneg_of_dist_le_radius | [
{
"state_after": "case inl\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : dist p₂ s.center ≤ s.radius\nh : 0 < inner (p₁ -ᵥ p₂) (p₁ -ᵥ s.center)\n⊢ 0 ≤ inner (p₁ -ᵥ p₂) ... | [
366,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
362,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean | StieltjesFunction.length_Ioc | [
{
"state_after": "f : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')",
"state_before": "f : StieltjesFunction\na b : ℝ\n⊢ length f (Ioc a b) = ofReal (↑f b - ↑f a)",
"tactic": "refine'\n le_antisymm (iInf_le_of_le a <| iInf₂_le b... | [
330,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.diam_le_of_forall_dist_le_of_nonempty | [] | [
2634,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2629,
1
] |
Mathlib/Analysis/Calculus/Deriv/Pow.lean | differentiableOn_pow | [] | [
87,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/Data/Real/Hyperreal.lean | Hyperreal.epsilon_lt_pos | [] | [
218,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
] |
Mathlib/Data/Real/ENatENNReal.lean | ENat.toENNReal_sub | [] | [
121,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean | AlgebraicGeometry.StructureSheaf.const_mul_rev | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nf g : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu₁ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g ∈ Ideal.primeCompl x.asIdeal\nhu₂ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → f ∈ Ideal.primeCompl x.asIdeal\n⊢ const R f g U hu₁ * const R g... | [
397,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
396,
1
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Mathlib/Data/Ordmap/Ordset.lean | Ordset.empty_iff | [
{
"state_after": "case refl\nα : Type u_1\ninst✝ : Preorder α\n⊢ empty ↑∅ = true",
"state_before": "α : Type u_1\ninst✝ : Preorder α\ns : Ordset α\nh : s = ∅\n⊢ empty ↑s = true",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nα : Type u_1\ninst✝ : Preorder α\... | [
1728,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1726,
1
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Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsLittleO.trans_eventuallyEq | [] | [
403,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
401,
1
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Mathlib/Logic/Basic.lean | not_iff_comm | [] | [
433,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
433,
1
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Mathlib/Data/List/Indexes.lean | List.mapIdx_eq_ofFn | [
{
"state_after": "case nil\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ f : ℕ → α → β\n⊢ mapIdx f [] = ofFn fun i => f (↑i) (get [] i)\n\ncase cons\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ : ℕ → α → β\nhd : α\ntl : List α\nIH : ∀ (f : ℕ → α → β), mapIdx f tl = ofFn fun i => f (↑i) ... | [
209,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
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Mathlib/Algebra/Opposites.lean | MulOpposite.op_injective | [] | [
153,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
] |
Mathlib/Data/Matrix/Kronecker.lean | Matrix.det_kroneckerMapBilinear | [
{
"state_after": "no goals",
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256,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
239,
1
] |
Mathlib/Order/Filter/Bases.lean | FilterBasis.hasBasis | [] | [
343,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
341,
11
] |
Mathlib/RepresentationTheory/Basic.lean | Representation.ofMulAction_def | [] | [
276,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
275,
1
] |
Mathlib/Data/List/Perm.lean | List.Perm.pmap | [
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\np✝ : α → Prop\nf : (a : α) → p✝ a → β\nl₁ l₂ : List α\np : l₁ ~ l₂\nH₁ : ∀ (a : α), a ∈ l₁ → p✝ a\nH₂ : ∀ (a : α), a ∈ l₂ → p✝ a\n⊢ List.pmap f l₁ H₁ ~ List.pmap f l₂ H₂",
"tactic": "induction p with\n| nil => simp... | [
271,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
263,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean | Finset.prod_pow_eq_pow_sum | [
{
"state_after": "case empty\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\nx : β\nf : α → ℕ\n⊢ ∏ i in ∅, x ^ f i = x ^ ∑ x in ∅, f x\n\ncase insert\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid... | [
45,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
40,
1
] |
Mathlib/Data/Polynomial/EraseLead.lean | Polynomial.map_natDegree_eq_natDegree | [
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝² : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst✝¹ : Semiring S\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nφ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → natDegree (↑φ (↑(monomial n) c)) = n\n⊢ ∀ (n : ℕ) (c : R), c ≠ 0 → natDeg... | [
289,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
285,
1
] |
Mathlib/Data/Set/Finite.lean | Set.Finite.toFinset_insert | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝ : DecidableEq α\ns : Set α\na : α\nhs : Set.Finite (insert a s)\n⊢ ∀ (a_1 : α), a_1 ∈ Finite.toFinset hs ↔ a_1 ∈ insert a (Finite.toFinset (_ : Set.Finite s))",
"tactic": "simp"
}
] | [
1061,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1059,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.mul_lt_mul_right | [] | [
1044,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1043,
1
] |
Mathlib/Order/Filter/Pointwise.lean | Filter.vsub_mem_vsub | [] | [
1093,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1092,
1
] |
Mathlib/Data/Polynomial/Laurent.lean | LaurentPolynomial.degree_zero | [] | [
485,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
484,
1
] |
Mathlib/Algebra/Ring/Prod.lean | RingHom.fst_comp_prod | [] | [
232,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
] |
Mathlib/Algebra/Group/TypeTags.lean | toAdd_ofAdd | [] | [
103,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Order/Filter/Germ.lean | Filter.Germ.liftRel_const | [] | [
325,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/Logic/Equiv/Defs.lean | Equiv.Perm.congr_fun | [] | [
142,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
11
] |
Mathlib/Order/CompleteLattice.lean | le_iSup' | [] | [
761,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
760,
1
] |
Mathlib/FieldTheory/Separable.lean | Polynomial.exists_finset_of_splits | [
{
"state_after": "case intro\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ ∃ s, map i f = ↑C (↑i (leadingCoeff f)) * ∏ a in s... | [
460,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
452,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean | Measurable.pow_const | [] | [
232,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean | QuotientGroup.rangeKerLift_injective | [
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝¹ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝ : Group H\nφ : G →* H\na✝ b✝ : G ⧸ ker φ\na b : G\nh : ↑(rangeRestrict φ) a = ↑(rangeRestrict φ) b\n⊢ Setoid.r a b",
"tactic": "rw [leftRel_apply, ← ker_rangeRestrict, me... | [
385,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
381,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearMap.map_smul_of_tower | [] | [
512,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
509,
1
] |
Mathlib/NumberTheory/Bernoulli.lean | bernoulli'_one | [
{
"state_after": "A : Type ?u.182698\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ 1 - ∑ k in range 1, ↑(Nat.choose 1 k) / (↑1 - ↑k + 1) * bernoulli' k = 1 / 2",
"state_before": "A : Type ?u.182698\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ bernoulli' 1 = 1 / 2",
"tactic": "rw [bernoulli'_def]"
},
... | [
114,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | Asymptotics.isLittleO_sum_range_of_tendsto_zero | [
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\nh : Tendsto f atTop (𝓝 0)\nthis :\n Tendsto (fun n => ∑ i in range n, 1) atTop atTop → (fun n => ∑ i in range n, f i) =o[atTop] fun n => ∑ i in range n, 1\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ↑n",
"state_before": "α :... | [
147,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
] |
Mathlib/Control/Applicative.lean | Functor.Comp.map_pure | [
{
"state_after": "no goals",
"state_before": "F : Type u → Type w\nG : Type v → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα β γ : Type v\nf : α → β\nx : α\n⊢ run (f <$> pure x) = run (pure (f x))",
"tactic": "simp"
}
] | [
97,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean | AntitoneOn.const_mul' | [] | [
1307,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1306,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.two_le_iff | [
{
"state_after": "no goals",
"state_before": "α β : Type u\n⊢ 2 ≤ (#α) ↔ ∃ x y, x ≠ y",
"tactic": "rw [← Nat.cast_two, nat_succ, succ_le_iff, Nat.cast_one, one_lt_iff_nontrivial, nontrivial_iff]"
}
] | [
2230,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2229,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | intervalIntegral.inv_smul_integral_comp_div | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.14846073\n𝕜 : Type ?u.14846076\nE : Type u_1\nF : Type ?u.14846082\nA : Type ?u.14846085\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b c✝ d : ℝ\nf : ℝ → E\nc : ℝ\n⊢ (c⁻¹ • ∫ (x : ℝ) in a..b, f (x / c)) = ∫ (x... | [
733,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
731,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean | pow_bit0' | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.37531\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\nn : ℕ\n⊢ a ^ bit0 n = (a * a) ^ n",
"tactic": "rw [pow_bit0, (Commute.refl a).mul_pow]"
}
] | [
211,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.subtypeDomain_sum | [] | [
1109,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1107,
1
] |
Mathlib/Order/Hom/Set.lean | StrictMono.orderIsoOfSurjective_symm_apply_self | [] | [
129,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.integrable_of_integrable_trim | [
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Me... | [
1175,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1169,
1
] |
Mathlib/Order/Lattice.lean | sup_lt_iff | [] | [
865,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
863,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.lift_of | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : CommRing S\ni : R →+* S\na : S\nh : eval₂ i a f = 0\nx : R\n⊢ ↑(lift i a h) (↑(of f) x) = ↑i x",
"tactic": "rw [← mk_C x, lift_mk, eval₂_C]"
}
] | [
294,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/Algebra/Order/Floor.lean | Int.ceil_intCast | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.210900\nα : Type u_1\nβ : Type ?u.210906\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz✝ : ℤ\na✝ : α\nz a : ℤ\n⊢ ⌈↑z⌉ ≤ a ↔ z ≤ a",
"tactic": "rw [ceil_le, Int.cast_le]"
}
] | [
1131,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1130,
1
] |
Mathlib/Topology/UniformSpace/AbstractCompletion.lean | AbstractCompletion.map_comp | [] | [
238,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
236,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean | Nat.ArithmeticFunction.intCoe_mul | [
{
"state_after": "case h\nR : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\nn : ℕ\n⊢ ↑↑(f * g) n = ↑(↑f * ↑g) n",
"state_before": "R : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\n⊢ ↑(f * g) = ↑f * ↑g",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case ... | [
295,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
292,
1
] |
Mathlib/Algebra/Hom/Ring.lean | NonUnitalRingHom.coe_mulHom_injective | [] | [
199,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
] |
Mathlib/Data/List/Card.lean | List.card_cons_of_mem | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Sort ?u.20954\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\na : α\nas : List α\nh : a ∈ as\n⊢ card (a :: as) = card as",
"tactic": "simp [card, h]"
}
] | [
82,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
9
] |
Mathlib/Algebra/Associated.lean | Associates.mk_eq_mk_iff_associated | [] | [
750,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
749,
1
] |
Std/Data/Int/Lemmas.lean | Int.mul_lt_mul | [] | [
1190,
95
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1188,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.two_nsmul_toReal_eq_two_mul_sub_two_pi | [
{
"state_after": "θ : Angle\n⊢ toReal (2 • ↑(toReal θ)) = 2 * toReal θ - 2 * π ↔ π / 2 < toReal θ",
"state_before": "θ : Angle\n⊢ toReal (2 • θ) = 2 * toReal θ - 2 * π ↔ π / 2 < toReal θ",
"tactic": "nth_rw 1 [← coe_toReal θ]"
},
{
"state_after": "θ : Angle\n⊢ π < 2 * toReal θ ∧ 2 * toReal θ ≤ 3... | [
704,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
698,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.bsup_eq_blsub_iff_lt_bsup | [
{
"state_after": "α : Type ?u.356399\nβ : Type ?u.356402\nγ : Type ?u.356405\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nh : bsup o f = blsub o f\ni : Ordinal\n⊢ ∀ (hi : i < o), f i hi < blsub o f",
"state_before": "α : Type ?u.356399\nβ : Type ?u... | [
1867,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1863,
1
] |
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