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/Compactification/OnePoint.lean | OnePoint.isClosed_image_coe | [
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt s : Set X\n⊢ IsClosed (some '' s) ↔ IsClosed s ∧ IsCompact s",
"tactic": "rw [← isOpen_compl_iff, isOpen_compl_image_coe]"
}
] | [
252,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Real.Icc_eq_closedBall | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.201326\nι : Type ?u.201329\ninst✝ : PseudoMetricSpace α\nx y : ℝ\n⊢ Icc x y = closedBall ((x + y) / 2) ((y - x) / 2)",
"tactic": "rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel', add_self_div_tw... | [
1412,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1410,
1
] |
Mathlib/Data/List/Pairwise.lean | List.pairwise_replicate | [
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.47232\nβ : Type ?u.47235\nR S T : α✝ → α✝ → Prop\na : α✝\nl : List α✝\nα : Type u_1\nr : α → α → Prop\nx : α\nhx : r x x\n⊢ Pairwise r (replicate 0 x)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.4... | [
347,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.ofMulEquivOfDom_eq | [] | [
1521,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1520,
1
] |
Mathlib/Computability/PartrecCode.lean | Nat.Partrec.Code.encode_lt_comp | [
{
"state_after": "cf cg : Code\nthis : ?m.357933\n⊢ encode cf < encode (comp cf cg) ∧ encode cg < encode (comp cf cg)\n\ncase this\ncf cg : Code\n⊢ ?m.357933",
"state_before": "cf cg : Code\n⊢ encode cf < encode (comp cf cg) ∧ encode cg < encode (comp cf cg)",
"tactic": "suffices"
},
{
"state_af... | [
217,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.im_eq_conj_sub | [
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.3572488\ninst✝ : IsROrC K\nz : K\n⊢ ↑(↑im z) = I * (↑(starRingEnd K) z - z) / 2",
"tactic": "rw [← neg_inj, ← ofReal_neg, ← I_mul_re, re_eq_add_conj, map_mul, conj_I, ← neg_div, ← mul_neg,\n neg_sub, mul_sub, neg_mul, sub_eq_add_ne... | [
406,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
404,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Function.Embedding.cardinal_le | [] | [
277,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
276,
1
] |
Mathlib/Data/List/Perm.lean | List.Perm.append_right | [] | [
115,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean | Set.einfsep_empty | [] | [
132,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean | AlgebraicGeometry.StructureSheaf.toBasicOpen_to_map | [] | [
646,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
643,
1
] |
Mathlib/Topology/Sets/Compacts.lean | TopologicalSpace.Compacts.coe_sup | [] | [
108,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
] |
Mathlib/Topology/ContinuousFunction/Polynomial.lean | polynomialFunctions.comap_compRightAlgHom_iccHomeoI | [
{
"state_after": "case h\nR : Type ?u.106978\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalSemiring R\na b : ℝ\nh : a < b\nf : C(↑(Set.Icc a b), ℝ)\n⊢ f ∈\n Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))))\n (polynomi... | [
207,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
] |
Mathlib/GroupTheory/Perm/Support.lean | Equiv.Perm.disjoint_one_right | [] | [
76,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.inter_iUnion | [] | [
522,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | expMapCircle_add_two_pi | [] | [
110,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean | Real.cosh_arsinh | [
{
"state_after": "no goals",
"state_before": "x✝ y x : ℝ\n⊢ cosh (arsinh x) = sqrt (1 + x ^ 2)",
"tactic": "rw [← sqrt_sq (cosh_pos _).le, cosh_sq', sinh_arsinh]"
}
] | [
87,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.one_lt_omega | [
{
"state_after": "no goals",
"state_before": "⊢ 1 < ω",
"tactic": "simpa only [Nat.cast_one] using nat_lt_omega 1"
}
] | [
2420,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2420,
1
] |
Mathlib/GroupTheory/Congruence.lean | Con.eq | [] | [
362,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
11
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | extChartAt_source_mem_nhds | [] | [
1073,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1072,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean | Set.Ioc.coe_pow | [] | [
304,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
303,
1
] |
Mathlib/CategoryTheory/Arrow.lean | CategoryTheory.Arrow.isIso_of_isIso_left_of_isIso_right | [
{
"state_after": "T : Type u\ninst✝² : Category T\nf g : Arrow T\nff : f ⟶ g\ninst✝¹ : IsIso ff.left\ninst✝ : IsIso ff.right\ninverse : g ⟶ f := CommaMorphism.mk (inv ff.left) (inv ff.right)\n⊢ ∃ inv, ff ≫ inv = 𝟙 f ∧ inv ≫ ff = 𝟙 g",
"state_before": "T : Type u\ninst✝² : Category T\nf g : Arrow T\nff : f... | [
146,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean | SimpleGraph.interedges_empty_left | [] | [
340,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
] |
Mathlib/Probability/Independence/Basic.lean | ProbabilityTheory.IndepSets.indepSet_of_mem | [] | [
618,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
614,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.cofinite.bliminf_set_eq | [
{
"state_after": "α : Type u_1\nβ : Type ?u.181072\nγ : Type ?u.181075\nι : Type u_2\np : ι → Prop\ns : ι → Set α\n⊢ bliminf s cofinite pᶜ = {x | Set.Finite {n | p n ∧ ¬x ∈ s n}}ᶜ",
"state_before": "α : Type u_1\nβ : Type ?u.181072\nγ : Type ?u.181075\nι : Type u_2\np : ι → Prop\ns : ι → Set α\n⊢ bliminf s ... | [
1067,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1063,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | IsPiSystem.prod | [
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.23\nβ : Type u_2\nβ' : Type ?u.29\nγ : Type ?u.32\nE : Type ?u.35\nC : Set (Set α)\nD : Set (Set β)\nhC : IsPiSystem C\nhD : IsPiSystem D\ns₁ : Set α\nt₁ : Set β\nhs₁ : s₁ ∈ C\nht₁ : t₁ ∈ D\ns₂ : Set α\nt₂ : Set β... | [
79,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Data/List/Basic.lean | List.getD_eq_get | [
{
"state_after": "case nil\nι : Type ?u.501723\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nx : α\nxs : List α\nd : α\nn✝¹ n✝ : ℕ\nhn✝ : n✝ < length l\nn : ℕ\nhn : n < length []\n⊢ getD [] n d = get [] { val := n, isLt := hn }\n\ncase cons\nι : Type ?u.501723\nα : Type u\nβ : Type v\nγ : T... | [
4394,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
4389,
1
] |
Mathlib/Data/Set/Prod.lean | Set.forall_prod_set | [] | [
103,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Analysis/Normed/Order/Lattice.lean | lipschitzWith_pos | [] | [
205,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean | Submodule.mem_carrier | [] | [
225,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
224,
1
] |
Std/Data/Int/DivMod.lean | Int.neg_div_neg | [
{
"state_after": "no goals",
"state_before": "a b : Int\n⊢ div (-a) (-b) = div a b",
"tactic": "simp [Int.div_neg, Int.neg_div, Int.neg_neg]"
}
] | [
90,
47
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
89,
11
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Iic_subset_Iic | [] | [
424,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
] |
Mathlib/Algebra/Hom/Aut.lean | MulAut.mul_apply | [] | [
93,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean | UniformSpace.Completion.extension₂_coe_coe | [] | [
690,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
688,
1
] |
Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | AffineMap.differentiableOn | [] | [
55,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
11
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | DoubleCentralizer.int_cast_snd | [] | [
319,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | ZFSet.insert_nonempty | [] | [
930,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
929,
1
] |
Std/Data/List/Lemmas.lean | List.map_filterMap_of_inv | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → Option β\ng : β → α\nH : ∀ (x : α), Option.map g (f x) = some x\nl : List α\n⊢ map g (filterMap f l) = l",
"tactic": "simp only [map_filterMap, H, filterMap_some]"
}
] | [
1216,
96
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1215,
1
] |
Mathlib/Order/Cover.lean | not_covby_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.26802\ninst✝ : LT α\na b : α\nh : a < b\n⊢ ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b",
"tactic": "simp_rw [Covby, h, true_and_iff, not_forall, exists_prop, not_not]"
}
] | [
231,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
230,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | exists_lt_of_lt_csSup | [
{
"state_after": "α : Type u_1\nβ : Type ?u.67098\nγ : Type ?u.67101\nι : Sort ?u.67104\ninst✝ : ConditionallyCompleteLinearOrder α\ns t : Set α\na b : α\nhs : Set.Nonempty s\nhb : ∀ (a : α), a ∈ s → a ≤ b\n⊢ sSup s ≤ b",
"state_before": "α : Type u_1\nβ : Type ?u.67098\nγ : Type ?u.67101\nι : Sort ?u.67104... | [
940,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
938,
1
] |
Mathlib/Data/MvPolynomial/Supported.lean | MvPolynomial.X_mem_supported | [
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nτ : Type ?u.128582\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\ninst✝ : Nontrivial R\ni : σ\n⊢ X i ∈ supported R s ↔ i ∈ s",
"tactic": "simp [mem_supported]"
}
] | [
121,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.add_comm_equiv | [] | [
1643,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1642,
1
] |
Mathlib/Analysis/Seminorm.lean | Seminorm.neg_ball | [
{
"state_after": "case h\nR : Type ?u.1261867\nR' : Type ?u.1261870\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1261876\n𝕜₃ : Type ?u.1261879\n𝕝 : Type ?u.1261882\nE : Type u_2\nE₂ : Type ?u.1261888\nE₃ : Type ?u.1261891\nF : Type ?u.1261894\nG : Type ?u.1261897\nι : Type ?u.1261900\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCo... | [
1016,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1014,
1
] |
Mathlib/FieldTheory/RatFunc.lean | RatFunc.denom_add_dvd | [
{
"state_after": "K : Type u\ninst✝ : Field K\nx y : RatFunc K\n⊢ ∃ p, x + y = ↑(algebraMap K[X] (RatFunc K)) p / ↑(algebraMap K[X] (RatFunc K)) (denom x * denom y)",
"state_before": "K : Type u\ninst✝ : Field K\nx y : RatFunc K\n⊢ denom (x + y) ∣ denom x * denom y",
"tactic": "rw [denom_dvd (mul_ne_zer... | [
1337,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1331,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean | edist_le_Ico_sum_edist | [
{
"state_after": "case base\nα : Type u\nβ : Type v\nX : Type ?u.8135\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nm n : ℕ\n⊢ edist (f m) (f m) ≤ ∑ i in Finset.Ico m m, edist (f i) (f (i + 1))\n\ncase succ\nα : Type u\nβ : Type v\nX : Type ?u.8135\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nm n n✝ : ℕ\nhn✝ : m ≤ n✝\... | [
138,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | aemeasurable_id | [] | [
718,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
717,
1
] |
Mathlib/Data/Sym/Basic.lean | Sym.eq_replicate_of_subsingleton | [] | [
313,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean | HasStrictDerivAt.sum | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter �... | [
172,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
170,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.countp_False | [] | [
2287,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2286,
1
] |
Mathlib/Topology/Order/Basic.lean | IsLUB.mem_of_isClosed | [] | [
2088,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2086,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | mem_comp_of_mem_ball | [
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.55323\ninst✝ : UniformSpace α\nV W : Set (β × β)\nx y z : β\nhV : SymmetricRel V\nhx : z ∈ ball x V\nhy : y ∈ ball z W\n⊢ (x, y) ∈ V ○ W",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.55323\... | [
667,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
664,
1
] |
Mathlib/Algebra/Module/LinearMap.lean | LinearMap.toAddMonoidHom_injective | [] | [
483,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
481,
1
] |
Mathlib/Data/Dfinsupp/Interval.lean | Dfinsupp.mem_pi | [] | [
151,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean | mem_chart_target | [] | [
545,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
544,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.radical_inf | [] | [
930,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
926,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean | PFunctor.Approx.truncate_eq_of_agree | [
{
"state_after": "case zero.continue.intro\nF : PFunctor\na✝¹ : F.A\na✝ : B F a✝¹ → CofixA F 0\nh : Agree CofixA.continue (CofixA.intro a✝¹ a✝)\n⊢ truncate (CofixA.intro a✝¹ a✝) = CofixA.continue\n\ncase succ.intro.intro\nF : PFunctor\nn✝ : ℕ\nn_ih✝ : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → tr... | [
118,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
] |
Mathlib/Topology/Algebra/FilterBasis.lean | GroupFilterBasis.mem_nhds_one | [
{
"state_after": "G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\nU : Set G\nhU : U ∈ B\n⊢ ∃ i, i ∈ B ∧ id i ⊆ U",
"state_before": "G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\nU : Set G\nhU : U ∈ B\n⊢ U ∈ 𝓝 1",
"tactic": "rw [B.nhds_one_hasBasis.mem_iff]"
},
{
"state_after":... | [
236,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
233,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.coe_sub | [
{
"state_after": "case mk\nn m : ℕ\nb : Fin n\nval✝ : ℕ\nisLt✝ : val✝ < n\n⊢ ↑({ val := val✝, isLt := isLt✝ } - b) = (↑{ val := val✝, isLt := isLt✝ } + (n - ↑b)) % n",
"state_before": "n m : ℕ\na b : Fin n\n⊢ ↑(a - b) = (↑a + (n - ↑b)) % n",
"tactic": "cases a"
},
{
"state_after": "case mk.mk\nn... | [
1932,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1931,
11
] |
Mathlib/CategoryTheory/StructuredArrow.lean | CategoryTheory.StructuredArrow.mono_of_mono_right | [] | [
156,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Logic/Function/Iterate.lean | Function.iterate_comm | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nf✝ : α → α\nm✝ n✝ : ℕ\na : α\nf : α → α\nm n : ℕ\n⊢ f^[n * m] = f^[m * n]",
"tactic": "rw [Nat.mul_comm]"
}
] | [
220,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
] |
src/lean/Init/Data/Nat/Basic.lean | Nat.sub_add_cancel | [
{
"state_after": "no goals",
"state_before": "n m : Nat\nh : m ≤ n\n⊢ n - m + m = n",
"tactic": "rw [Nat.add_comm, Nat.add_sub_of_le h]"
}
] | [
584,
41
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
583,
11
] |
src/lean/Init/Data/Nat/Basic.lean | Nat.pos_pow_of_pos | [] | [
495,
43
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
494,
1
] |
Mathlib/Data/Polynomial/Coeff.lean | Polynomial.coeff_C_mul_X_pow | [
{
"state_after": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nx : R\nk n : ℕ\n⊢ (if k = n then x else 0) = if n = k then x else 0",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nx : R\nk n : ℕ\n⊢ coeff (↑C x * X ^ k) n = if n =... | [
159,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Order/JordanHolder.lean | CompositionSeries.coeFn_mk | [] | [
166,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Mathlib/Analysis/Calculus/Deriv/Support.lean | support_deriv_subset | [
{
"state_after": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\n⊢ x ∈ support (deriv f) → x ∈ tsupport f",
"state_before": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E... | [
46,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/GroupTheory/GroupAction/Pi.lean | Pi.single_smul' | [] | [
193,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
] |
Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | QuaternionGroup.xa_sq | [
{
"state_after": "no goals",
"state_before": "n : ℕ\ni : ZMod (2 * n)\n⊢ xa i ^ 2 = a ↑n",
"tactic": "simp [sq]"
}
] | [
200,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
200,
1
] |
Mathlib/Order/UpperLower/Basic.lean | not_bddBelow_Iic | [] | [
352,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
351,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean | pos_iff_neg_of_mul_neg | [] | [
1208,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1207,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean | smul_zpow | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : Group β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\np : ℤ\n⊢ (c • x) ^ p = c ^ p • x ^ p",
"tactic": "cases p <;>\nsimp [smul_pow, smul_inv]"... | [
131,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/Algebra/Lie/Solvable.lean | LieAlgebra.derivedSeries_of_bot_eq_bot | [
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nk : ℕ\n⊢ D k ⊥ ≤ ⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst... | [
133,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
132,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean | MeasurableEquiv.image_eq_preimage | [] | [
1300,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1299,
1
] |
Mathlib/Data/List/Pairwise.lean | List.pwFilter_cons_of_neg | [] | [
369,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
367,
1
] |
Mathlib/Algebra/MonoidAlgebra/Division.lean | AddMonoidAlgebra.modOf_add_divOf | [
{
"state_after": "no goals",
"state_before": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\n⊢ x %ᵒᶠ g + of' k G g * (x /ᵒᶠ g) = x",
"tactic": "rw [add_comm, divOf_add_modOf]"
}
] | [
193,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/CategoryTheory/Monoidal/Functor.lean | CategoryTheory.MonoidalFunctor.εIso_hom | [] | [
256,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
255,
1
] |
Mathlib/Data/Set/Basic.lean | Set.mem_singleton_of_eq | [] | [
1290,
4
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1289,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.coeFn_compMeasurable | [
{
"state_after": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.390095\ninst✝¹⁰ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : TopologicalSpace γ\ninst✝⁷ : TopologicalSpace δ\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : PseudoMetrizableSpace β\ninst✝⁴ : BorelSpace β\ninst✝³ : Measu... | [
263,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
260,
1
] |
Mathlib/LinearAlgebra/ProjectiveSpace/Basic.lean | Projectivization.map_mk | [] | [
207,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
] |
Mathlib/CategoryTheory/Monoidal/Opposite.lean | CategoryTheory.op_tensorUnit | [] | [
193,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | ZFSet.mk_out | [] | [
689,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
688,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean | Set.neg_smul_set | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.218829\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.218838\ninst✝² : Ring α\ninst✝¹ : AddCommGroup β\ninst✝ : Module α β\na : α\ns : Set α\nt : Set β\n⊢ -a • t = -(a • t)",
"tactic": "simp_rw [← image_smul, ← image_neg, image_image, neg_smul]"
}
] | [
1107,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1106,
1
] |
Mathlib/Control/Traversable/Basic.lean | ApplicativeTransformation.preserves_map | [
{
"state_after": "no goals",
"state_before": "F : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β : Type u\nx : α → β\ny : F α\n⊢ (fun {α} => app η α) (x <$> y) = x <$> (fu... | [
154,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
153,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.mul_le_max_of_aleph0_le_right | [
{
"state_after": "no goals",
"state_before": "a b : Cardinal\nh : ℵ₀ ≤ b\n⊢ a * b ≤ max a b",
"tactic": "simpa only [mul_comm b, max_comm b] using mul_le_max_of_aleph0_le_left h"
}
] | [
624,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
623,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | Real.tan_ne_zero_iff | [
{
"state_after": "θ : ℝ\n⊢ (∀ (k : ℤ), ↑θ ≠ ↑k * ↑π / 2) ↔ ∀ (k : ℤ), θ ≠ ↑k * π / 2",
"state_before": "θ : ℝ\n⊢ tan θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ ↑k * π / 2",
"tactic": "rw [← Complex.ofReal_ne_zero, Complex.ofReal_tan, Complex.tan_ne_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "θ :... | [
52,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.isConj_iff_cycleType_eq | [
{
"state_after": "case intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ π : Perm α\nh : IsConj σ (π * σ * π⁻¹)\n⊢ cycleType σ = cycleType (π * σ * π⁻¹)",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nh : IsConj σ τ\n⊢ cycleType σ = cycleType τ",
... | [
272,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
269,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.I_mul_I | [
{
"state_after": "no goals",
"state_before": "⊢ (I * I).re = (-1).re ∧ (I * I).im = (-1).im",
"tactic": "simp"
}
] | [
314,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
313,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean | Cubic.degree_of_c_ne_zero' | [] | [
346,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
345,
1
] |
Mathlib/CategoryTheory/Category/Basic.lean | CategoryTheory.eq_of_comp_right_eq' | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y Z : C\nf g : Y ⟶ Z\nw : (fun {X} h => h ≫ f) = fun {X} h => h ≫ g\nX : C\nh : X ⟶ Y\n⊢ h ≫ f = h ≫ g",
"tactic": "convert congr_fun (congr_fun w X) h"
}
] | [
229,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
227,
1
] |
Mathlib/Logic/Basic.lean | or_iff_not_imp_right | [] | [
358,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
358,
1
] |
Mathlib/Data/Matrix/Notation.lean | Matrix.cons_transpose | [
{
"state_after": "case a.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nv : n' → α\nA : Matrix (Fin m) n' α\ni : n'\nj : Fin (Nat.succ m)\n⊢ (↑of (vecCons v A))ᵀ i j = ↑of (fun i => vecCons (v i) (Aᵀ i)) i j",
"state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\n... | [
208,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
] |
Mathlib/CategoryTheory/Closed/Monoidal.lean | CategoryTheory.MonoidalClosed.id_tensor_pre_app_comp_ev | [] | [
259,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
] |
Mathlib/Data/Polynomial/Derivative.lean | Polynomial.derivative_nat_cast | [
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Semiring R\nn : ℕ\n⊢ ↑derivative (↑C ↑n) = 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Semiring R\nn : ℕ\n⊢ ↑derivative ↑n = 0",
"tactic": ... | [
233,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Tower.lean | Subalgebra.mem_restrictScalars | [] | [
117,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineIsometryEquiv.toAffineEquiv_injective | [] | [
358,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
357,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean | MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_negative | [
{
"state_after": "α : Type u_1\nβ : Type ?u.67068\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict (... | [
355,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
348,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.exp_im | [
{
"state_after": "x y : ℂ\n⊢ (exp ↑x.re * (cos ↑x.im + sin ↑x.im * I)).im = Real.exp x.re * Real.sin x.im",
"state_before": "x y : ℂ\n⊢ (exp x).im = Real.exp x.re * Real.sin x.im",
"tactic": "rw [exp_eq_exp_re_mul_sin_add_cos]"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ (exp... | [
1100,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1098,
1
] |
Mathlib/RingTheory/Finiteness.lean | Submodule.fg_iSup | [
{
"state_after": "case intro\nR : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_1\ninst✝ : Finite ι\nN : ι → Submodule R M\nh : ∀ (i : ι), FG (N i)\nval✝ : Fintype ι\n⊢ FG (iSup N)",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\nin... | [
200,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.constantCoeff_inv | [
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type ?u.1677822\nk : Type u_2\ninst✝ : Field k\nφ : MvPowerSeries σ k\n⊢ ↑(constantCoeff σ k) φ⁻¹ = (↑(constantCoeff σ k) φ)⁻¹",
"tactic": "rw [← coeff_zero_eq_constantCoeff_apply, coeff_inv, if_pos rfl]"
}
] | [
952,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
950,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean | Fin.partialProd_succ' | [
{
"state_after": "α : Type u_1\nβ : Type ?u.93394\ninst✝ : Monoid α\nn : ℕ\nf : Fin (n + 1) → α\nj : Fin (n + 1)\n⊢ f 0 * List.prod (List.take (↑j) (List.ofFn fun i => f (succ i))) =\n f 0 * List.prod (List.take (↑j) (List.ofFn (tail f)))",
"state_before": "α : Type u_1\nβ : Type ?u.93394\ninst✝ : Monoid... | [
243,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
240,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousOn.prod_mapL | [] | [
1086,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1083,
1
] |
Mathlib/Order/Filter/NAry.lean | Filter.map₂_mk_eq_prod | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.19006\nβ : Type u_2\nβ' : Type ?u.19012\nγ : Type ?u.19015\nγ' : Type ?u.19018\nδ : Type ?u.19021\nδ' : Type ?u.19024\nε : Type ?u.19027\nε' : Type ?u.19030\nm : α → β → γ\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\n... | [
80,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
] |
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