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/Topology/Algebra/Order/LiminfLimsup.lean | iUnion_Ici_eq_Ioi_of_lt_of_tendsto | [
{
"state_after": "α : Type u\nβ : Type v\nι✝ : Type ?u.62603\nR : Type u_2\ninst✝³ : CompleteLinearOrder R\ninst✝² : TopologicalSpace R\ninst✝¹ : OrderTopology R\nι : Type u_1\nx : R\nas : ι → R\nx_lt : ∀ (i : ι), x < as i\nF : Filter ι\ninst✝ : NeBot F\nas_lim : Tendsto as F (𝓝 x)\nobs : ¬x ∈ range as\n⊢ (⋃ (... | [
444,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
433,
1
] |
Mathlib/LinearAlgebra/Basic.lean | Submodule.map_iInf_comap_of_surjective | [] | [
926,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
924,
1
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Mathlib/Topology/UniformSpace/Separation.lean | isClosed_of_spaced_out | [
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : SeparatedSpace α\nV₀ : Set (α × α)\nV₀_in : V₀ ∈ 𝓤 α\ns : Set α\nhs : Set.Pairwise s fun x y => ¬(x, y) ∈ V₀\nV₁ : Set (α × α)\nV₁_in : V₁ ∈ 𝓤 α\nV₁_... | [
239,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
225,
1
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Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearEquiv.norm_pos | [] | [
1957,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1955,
1
] |
Std/Data/Int/Lemmas.lean | Int.negSucc_mul_subNatNat | [
{
"state_after": "no goals",
"state_before": "m n k : Nat\n⊢ -[m+1] * subNatNat n k = subNatNat (succ m * k) (succ m * n)",
"tactic": "cases n.lt_or_ge k with\n| inl h =>\n have h' : succ m * n < succ m * k := Nat.mul_lt_mul_of_pos_left h (Nat.succ_pos m)\n rw [subNatNat_of_lt h, subNatNat_of_le (Nat.... | [
441,
88
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
429,
1
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Mathlib/Data/List/ProdSigma.lean | List.mem_sigma | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8468\nσ : α → Type u_2\nl₁ : List α\nl₂ : (a : α) → List (σ a)\na : α\nb : σ a\n⊢ { fst := a, snd := b } ∈ List.sigma l₁ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ a",
"tactic": "simp [List.sigma, mem_bind, mem_map, exists_prop, exists_and_left, and_left... | [
87,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Dynamics/PeriodicPts.lean | Function.minimalPeriod_iterate_eq_div_gcd' | [] | [
477,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
474,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.disjoint_iUnion_right | [] | [
2082,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2080,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.not_le_iff | [
{
"state_after": "no goals",
"state_before": "z w : ℂ\n⊢ ¬z ≤ w ↔ w.re < z.re ∨ z.im ≠ w.im",
"tactic": "rw [le_def, not_and_or, not_le]"
}
] | [
1176,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1175,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | Subring.map_sup | [] | [
1063,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1062,
1
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Mathlib/Algebra/Homology/Opposite.lean | imageToKernel_op | [
{
"state_after": "no goals",
"state_before": "V : Type ?u.24\ninst✝¹ : Category V\ninst✝ : Abelian V\nX Y Z : V\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ g.op ≫ f.op = 0",
"tactic": "rw [← op_comp, w, op_zero]"
},
{
"state_after": "no goals",
"state_before": "V : Type ?u.24\ninst✝¹ : Category ... | [
52,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
] |
Mathlib/Order/Lattice.lean | left_or_right_lt_sup | [] | [
218,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
] |
Std/Data/List/Lemmas.lean | List.mem_reverseAux | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nx : α\nx✝ : List α\n⊢ x ∈ reverseAux [] x✝ ↔ x ∈ [] ∨ x ∈ x✝",
"tactic": "simp"
},
{
"state_after": "α : Type u_1\nx a : α\ntail✝ x✝ : List α\n⊢ x ∈ tail✝ ∨ x = a ∨ x ∈ x✝ ↔ (x = a ∨ x ∈ tail✝) ∨ x ∈ x✝",
"state_before": "α : Type u... | [
883,
77
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
881,
9
] |
Mathlib/Data/Fin/Tuple/Basic.lean | Fin.insertNth_le_iff | [
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\ninst✝ : (i : Fin (n + 1)) → Preorder (α i)\ni : Fin (n + 1)\nx : α i\np : (j : Fin n) → α (↑(succAbove i) j)\nq : (j : Fin (n + 1)) → α j\n⊢ insertNth i x p ≤ q ↔ x ≤ q i ∧ p ≤ fun j => q (↑(succAbove i) j)",
"t... | [
804,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
802,
1
] |
Mathlib/Order/Disjoint.lean | disjoint_right_comm | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\na b c d : α\n⊢ Disjoint (a ⊓ b) c ↔ Disjoint (a ⊓ c) b",
"tactic": "simp_rw [disjoint_iff_inf_le, inf_right_comm]"
}
] | [
149,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean | CategoryTheory.Bicategory.comp_whiskerLeft_symm | [
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh h' : c ⟶ d\nη : h ⟶ h'\n⊢ f ◁ g ◁ η = (α_ f g h).inv ≫ (f ≫ g) ◁ η ≫ (α_ f g h').hom",
"tactic": "simp"
}
] | [
385,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
384,
1
] |
Mathlib/Algebra/Module/Equiv.lean | AddEquiv.coe_toLinearEquiv | [] | [
787,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
786,
1
] |
Mathlib/Analysis/Convex/Gauge.lean | Absorbent.gauge_set_nonempty | [] | [
86,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/ModelTheory/Satisfiability.lean | FirstOrder.Language.Theory.models_iff_not_satisfiable | [
{
"state_after": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nφ : Sentence L\n⊢ (∀ (M : ModelType T), ↑M ⊨ φ) ↔ ¬Nonempty (ModelType (T ∪ {Formula.not φ}))",
"state_before": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nφ : Sentence L\n⊢ T ⊨ᵇ φ ↔ ¬IsSatisfiable (T ∪ {Formula.not φ})",
"tactic": "rw... | [
344,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
329,
1
] |
Mathlib/Data/Nat/Parity.lean | Nat.odd_add' | [
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ Odd (m + n) ↔ (Odd n ↔ Even m)",
"tactic": "rw [add_comm, odd_add]"
}
] | [
190,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
190,
1
] |
Mathlib/LinearAlgebra/Alternating.lean | MultilinearMap.alternization_map_eq_zero_of_eq_aux | [
{
"state_after": "R : Type u_1\ninst✝¹² : Semiring R\nM : Type u_2\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nN : Type ?u.568657\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R N\nP : Type ?u.568687\ninst✝⁷ : AddCommMonoid P\ninst✝⁶ : Module R P\nM' : Type ?u.568717\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Modul... | [
859,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
850,
9
] |
Mathlib/Topology/Semicontinuous.lean | lowerSemicontinuous_tsum | [] | [
657,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
656,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | TopCat.Presheaf.SheafConditionEqualizerProducts.res_π | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\ni : ι\n⊢ res F U ≫ limit.π (Discrete.functor fun i => F.obj (U i).op) { as := i } = F.map (leSupr U i).op",
"tactic": "rw [res, limit.lift_π,... | [
85,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean | BoxIntegral.integralSum_fiberwise | [] | [
114,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Data/Finsupp/Antidiagonal.lean | Finsupp.swap_mem_antidiagonal | [
{
"state_after": "no goals",
"state_before": "α : Type u\nn : α →₀ ℕ\nf : (α →₀ ℕ) × (α →₀ ℕ)\n⊢ Prod.swap f ∈ antidiagonal n ↔ f ∈ antidiagonal n",
"tactic": "simp only [mem_antidiagonal, add_comm, Prod.swap]"
}
] | [
57,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/Data/Nat/Pow.lean | Nat.not_pos_pow_dvd | [
{
"state_after": "no goals",
"state_before": "p k : ℕ\nhp : 1 < succ p\nhk : 1 < succ k\nh : succ p ^ succ k ∣ succ p\n⊢ succ p * succ p ^ k ∣ succ p * 1",
"tactic": "simpa [pow_succ'] using h"
},
{
"state_after": "no goals",
"state_before": "p k : ℕ\nhp : 1 < succ p\nhk : 1 < succ k\nh : su... | [
228,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | ENNReal.measurable_ofReal | [] | [
1856,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1855,
1
] |
Mathlib/Algebra/CharP/ExpChar.lean | expChar_one_iff_char_zero | [
{
"state_after": "case mp\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\np q : ℕ\ninst✝¹ : CharP R p\ninst✝ : ExpChar R q\n⊢ q = 1 → p = 0\n\ncase mpr\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\np q : ℕ\ninst✝¹ : CharP R p\ninst✝ : ExpChar R q\n⊢ p = 0 → q = 1",
"state_before": "R : T... | [
91,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean | tendsto_atTop_ciSup | [
{
"state_after": "case inl\nα : Type u_2\nβ : Type ?u.16816\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLattice α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_mono : Monotone f\nhbdd : BddAbove (range f)\nh✝ : IsEmpty ι\n⊢ Tendsto f atTop (𝓝 (⨆ (i : ι),... | [
132,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Data/Polynomial/Eval.lean | Polynomial.eval_sum | [] | [
471,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
469,
1
] |
Mathlib/RingTheory/MvPolynomial/Symmetric.lean | MvPolynomial.esymm_zero | [
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\nτ : Type ?u.96806\nS : Type ?u.96809\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\n⊢ esymm σ R 0 = 1",
"tactic": "simp only [esymm, powersetLen_zero, sum_singleton, prod_empty]"
}
] | [
195,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean | MeasureTheory.set_integral_mono_set | [] | [
710,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
708,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean | MultilinearMap.compLinearMap_injective | [
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁴ : Semiring R\ninst✝¹³ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝¹² : (i : ι) → AddCommMonoid (M₁ i)\ninst✝¹¹ : AddCommMo... | [
432,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
429,
1
] |
Mathlib/CategoryTheory/Monoidal/Opposite.lean | CategoryTheory.unmop_mop | [] | [
115,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
114,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Reachable.refl | [] | [
1869,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1869,
11
] |
Mathlib/Data/W/Constructions.lean | WType.rightInverse_nat | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ toNat (ofNat (Nat.succ n)) = Nat.succ n",
"tactic": "rw [ofNat, toNat, rightInverse_nat n]"
}
] | [
82,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean | lt_div_iff_mul_lt' | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c d : α\n⊢ b < c / a ↔ a * b < c",
"tactic": "rw [lt_div_iff_mul_lt, mul_comm]"
}
] | [
977,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
977,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean | Set.mem_inv | [] | [
182,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
] |
Mathlib/Topology/Connected.lean | IsPreconnected.subset_or_subset | [
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.28526\nπ : ι → Type ?u.28531\ninst✝ : TopologicalSpace α\ns t u v : Set α\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s ⊆ u ∪ v\nhs : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\n⊢ s ⊆ u ∨ s ⊆ v",
"state_before"... | [
425,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
417,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean | integral_one | [
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, 1) = b - a",
"tactic": "simp only [mul_one, smul_eq_mul, integral_const]"
}
] | [
419,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
418,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasSum_le | [] | [
47,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
46,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.setToFun_measure_zero' | [] | [
1441,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1439,
1
] |
Mathlib/Order/SuccPred/Basic.lean | Order.pred_lt_top | [] | [
901,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
900,
1
] |
Mathlib/Algebra/Free.lean | FreeSemigroup.traverse_eq | [] | [
678,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
678,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.Valid'.balance | [
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH : ∃ l' r', BalancedSz l' r' ∧ (Nat.dist (size l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l')\n⊢ Valid' o₁ (Ordnode.balance' l ... | [
1347,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1343,
1
] |
Mathlib/Data/Int/Order/Units.lean | Int.units_coe_mul_self | [
{
"state_after": "no goals",
"state_before": "u : ℤˣ\n⊢ ↑u * ↑u = 1",
"tactic": "rw [← Units.val_mul, units_mul_self, Units.val_one]"
}
] | [
48,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | MeasureTheory.SimpleFunc.nearestPtInd_zero | [] | [
81,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Mathlib/RingTheory/Int/Basic.lean | Int.natAbs_lcm | [] | [
171,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
170,
1
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Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_isDescendingCentralSeries | [
{
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331,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
327,
1
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Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | NNReal.tendsto_rpow_atTop | [
{
"state_after": "y : ℝ\nhy : 0 < y\n⊢ ∀ (b : ℝ), ∃ i, ∀ (a : ℝ≥0), i ≤ a → b ≤ ↑a ^ y",
"state_before": "y : ℝ\nhy : 0 < y\n⊢ Tendsto (fun x => ↑x ^ y) atTop atTop",
"tactic": "rw [Filter.tendsto_atTop_atTop]"
},
{
"state_after": "y : ℝ\nhy : 0 < y\nb : ℝ\n⊢ ∃ i, ∀ (a : ℝ≥0), i ≤ a → b ≤ ↑a ^ y... | [
124,
58
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117,
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Mathlib/Algebra/Module/Torsion.lean | Submodule.torsionBySet_singleton_eq | [
{
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272,
34
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269,
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Mathlib/Algebra/Algebra/Operations.lean | Submodule.le_one_toAddSubmonoid | [
{
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"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA... | [
94,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
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Mathlib/Data/Int/Order/Basic.lean | Int.emod_lt | [
{
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"state_before": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\nH : b ≠ 0\n⊢ a % b < abs b",
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},
{
"state_after": "no goals",
"state_before": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\nH : b ≠ 0\n⊢ a % abs b < abs b",
... | [
239,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
238,
1
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Mathlib/Topology/Algebra/Module/LocallyConvex.lean | locallyConvexSpace_iff_zero | [] | [
98,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
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Mathlib/Data/Nat/Factors.lean | Nat.mem_factors_mul_of_coprime | [
{
"state_after": "case inl\nb p : ℕ\nhab : coprime 0 b\n⊢ p ∈ factors (0 * b) ↔ p ∈ factors 0 ∪ factors b\n\ncase inr\na b : ℕ\nhab : coprime a b\np : ℕ\nha : a > 0\n⊢ p ∈ factors (a * b) ↔ p ∈ factors a ∪ factors b",
"state_before": "a b : ℕ\nhab : coprime a b\np : ℕ\n⊢ p ∈ factors (a * b) ↔ p ∈ factors a ... | [
279,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
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Mathlib/MeasureTheory/Integral/IntegrableOn.lean | MeasureTheory.Integrable.integrableOn | [] | [
167,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
166,
1
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Mathlib/CategoryTheory/Monoidal/Subcategory.lean | CategoryTheory.MonoidalCategory.fullMonoidalClosedSubcategory_ihom_obj | [] | [
238,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
236,
1
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Mathlib/Topology/Instances/ENNReal.lean | ENNReal.nhdsWithin_Ioi_coe_neBot | [] | [
211,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
] |
Mathlib/Algebra/Order/Pointwise.lean | sSup_mul | [] | [
80,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
] |
Mathlib/Topology/Perfect.lean | Perfect.small_diam_splitting | [
{
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261,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
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Mathlib/ModelTheory/Syntax.lean | FirstOrder.Language.BoundedFormula.IsQF.not | [] | [
724,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
723,
1
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Mathlib/Data/Finset/MulAntidiagonal.lean | Finset.mulAntidiagonal_mono_right | [] | [
90,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean | UniformInducing.uniformContinuous_iff | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ng : β → γ\nhg : UniformInducing g\n⊢ map (fun x => (f x.fst, f x.snd)) (𝓤 α) ≤ 𝓤 β ↔ map (fun x => ((g ∘ f) x.fst, (g ∘ f) x.snd)) (𝓤 α) ≤ 𝓤 γ",
"state_before": "α :... | [
101,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
1
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Mathlib/RingTheory/AlgebraicIndependent.lean | AlgebraicIndependent.map | [
{
"state_after": "ι : Type u_4\nι' : Type ?u.209986\nR : Type u_1\nK : Type ?u.209992\nA : Type u_2\nA' : Type u_3\nA'' : Type ?u.210001\nV : Type u\nV' : Type ?u.210006\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'... | [
156,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
1
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Mathlib/Order/Filter/Bases.lean | Filter.Tendsto.basis_left | [] | [
881,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
879,
1
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Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | NNReal.rpow_add' | [] | [
85,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Deprecated/Subring.lean | IsSubring.inter | [] | [
82,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
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Mathlib/GroupTheory/GroupAction/Sigma.lean | Sigma.smul_def | [] | [
40,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | ZFSet.ext_iff | [
{
"state_after": "no goals",
"state_before": "x y : ZFSet\nh : x = y\n⊢ ∀ (z : ZFSet), z ∈ x ↔ z ∈ y",
"tactic": "simp [h]"
}
] | [
801,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
800,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | div_neg_of_neg_of_pos | [] | [
705,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
704,
1
] |
Mathlib/Data/List/Basic.lean | List.pmap_map | [
{
"state_after": "no goals",
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3107,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3105,
1
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Mathlib/CategoryTheory/Limits/ConcreteCategory.lean | CategoryTheory.Limits.Concrete.colimit_rep_eq_iff_exists | [] | [
302,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
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Mathlib/Data/List/Basic.lean | List.bind_assoc | [
{
"state_after": "no goals",
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"tactic": "induction l <;> ... | [
513,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
512,
1
] |
Mathlib/Data/Set/Finite.lean | Set.forall_finite_image_eval_iff | [] | [
1105,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1103,
1
] |
Mathlib/Algebra/Hom/Iterate.lean | Commute.function_commute_mul_left | [] | [
247,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
245,
1
] |
Mathlib/CategoryTheory/Monoidal/Category.lean | CategoryTheory.MonoidalCategory.hom_inv_id_tensor' | [
{
"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⊢ (f ⊗ g) ≫ (inv f ⊗ h) = (𝟙 V ⊗ g) ≫ (𝟙 V ⊗ h)",
"... | [
380,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
378,
1
] |
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | CategoryTheory.Abelian.Pseudoelement.pseudoZero_iff | [
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP : C\na : Over P\n⊢ Quot.mk (PseudoEqual P) a = 0 ↔ a ≈ Over.mk 0",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP : C\na : Over P\n⊢ Quot.mk (PseudoEqual P) a = 0 ↔ a.hom = 0",
"tactic": "rw [← pseudoZero_aux... | [
261,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
259,
1
] |
Mathlib/Data/TypeVec.lean | TypeVec.repeat_eq_append1 | [
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nβ : Type u_1\nn : ℕ\nα : TypeVec n\n⊢ repeatEq (α ::: β) = splitFun (repeatEq α) (uncurry Eq)",
"tactic": "induction n <;> rfl"
}
] | [
462,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
459,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Finset.summable_compl_iff | [] | [
884,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
882,
11
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.deleteVerts_empty | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.270663\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : Subgraph G\ns : Set V\n⊢ deleteVerts G' ∅ = G'",
"tactic": "simp [deleteVerts]"
}
] | [
1229,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1228,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.le_inf'_iff | [] | [
950,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
949,
1
] |
Mathlib/Control/Fold.lean | Traversable.toList_spec | [
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nxs : t α\n⊢ ↑FreeMonoid.toList (foldMap FreeMonoid.of xs) =\n ↑FreeMonoid.toList (List.reverse (List.reverse (foldMap FreeMonoid.of xs)))",
"tactic": "simp only [L... | [
349,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
336,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean | mul_inv_le_mul_inv_iff' | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b",
"tactic": "rw [mul_comm c, mul_inv_le_inv_mul_iff, mul_comm]"
}
] | [
542,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
] |
Mathlib/Data/Pi/Algebra.lean | Pi.bit0_apply | [] | [
157,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/Order/UpperLower/Basic.lean | LowerSet.mem_iInf_iff | [
{
"state_after": "α : Type u_1\nβ : Type ?u.76981\nγ : Type ?u.76984\nι : Sort u_2\nκ : ι → Sort ?u.76992\ninst✝ : LE α\nS : Set (LowerSet α)\ns t : LowerSet α\na : α\nf : ι → LowerSet α\n⊢ (a ∈ ⋂ (i : ι), ↑(f i)) ↔ ∀ (i : ι), a ∈ f i",
"state_before": "α : Type u_1\nβ : Type ?u.76981\nγ : Type ?u.76984\nι ... | [
760,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
758,
1
] |
Mathlib/Topology/Filter.lean | Filter.HasBasis.nhds' | [
{
"state_after": "no goals",
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"tactic": "simpa only [Iic_principal] using h.nhds"
}
] | [
104,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/Data/List/Intervals.lean | List.Ico.zero_bot | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ Ico 0 n = range n",
"tactic": "rw [Ico, tsub_zero, range_eq_range']"
}
] | [
45,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
] |
Mathlib/RingTheory/Localization/Basic.lean | IsLocalization.map_eq | [] | [
615,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
614,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.div_apply | [] | [
524,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
523,
1
] |
Mathlib/Topology/ContinuousOn.lean | ContinuousOn.isOpen_preimage | [
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.343466\nδ : Type ?u.343469\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nt : Set β\nh : ContinuousOn f s\nhs : IsOpen s\nhp : f ⁻¹' t ⊆ s\nht : IsOpe... | [
1015,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1012,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.restrict_congr_set | [] | [
1553,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1552,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean | HolderWith.dist_le_of_le | [
{
"state_after": "case intro\nX : Type u_1\nY : Type u_2\nZ : Type ?u.969277\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : dist x y ≤ ↑d\n⊢ dist (f x) (f y) ≤ ↑C * ↑d ^ ↑r",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Typ... | [
259,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.iUnion₂_mono | [] | [
380,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
378,
1
] |
Mathlib/Algebra/Order/Archimedean.lean | exists_nat_one_div_lt | [
{
"state_after": "case intro\nα : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : Archimedean α\nx y ε✝ ε : α\nhε : 0 < ε\nn : ℕ\nhn : 1 / ε < ↑n\n⊢ ∃ n, 1 / (↑n + 1) < ε",
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312,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
306,
1
] |
Mathlib/CategoryTheory/Simple.lean | CategoryTheory.simple_iff_subobject_isSimpleOrder | [
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX : C\nh : Simple X\n⊢ IsSimpleOrder (Subobject X)",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX : C\n⊢ Simple X → IsSimpleOrder (Subobject X... | [
261,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean | LocalEquiv.source_inter_preimage_target_inter | [] | [
521,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
519,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiffWithinAt.comp_of_mem | [] | [
705,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
702,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean | Antitone.rightLim_le | [] | [
315,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
1
] |
Mathlib/Order/SuccPred/Basic.lean | Order.Iic_succ | [] | [
449,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
448,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | ContinuousLinearEquiv.uniqueDiffOn_preimage_iff | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nf' : E →L[𝕜] F\ne : F ≃L[𝕜] E\n⊢ UniqueDiff... | [
535,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
533,
1
] |
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