file_path stringlengths 11 79 | full_name stringlengths 2 100 | traced_tactics list | end list | commit stringclasses 4
values | url stringclasses 4
values | start list |
|---|---|---|---|---|---|---|
Mathlib/Data/Polynomial/RingDivision.lean | Polynomial.degree_pos_of_aeval_root | [] | [
65,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.any_singleton | [
{
"state_after": "case inr.inl\nα : Type u_1\nP : α → Prop\nx : α\nh : P x\n⊢ P x",
"state_before": "α : Type u_1\nP : α → Prop\nx : α\n⊢ Any P {x} → P x",
"tactic": "rintro (⟨⟨⟩⟩ | h | ⟨⟨⟩⟩)"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nP : α → Prop\nx : α\nh... | [
469,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
468,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Finset.exists_cycleOn | [
{
"state_after": "ι : Type ?u.3070884\nα : Type u_1\nβ : Type ?u.3070890\ninst✝² inst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset α\n⊢ IsCycleOn (List.formPerm (toList s)) ↑s",
"state_before": "ι : Type ?u.3070884\nα : Type u_1\nβ : Type ?u.3070890\ninst✝² inst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns :... | [
1879,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1873,
1
] |
Mathlib/GroupTheory/Perm/Support.lean | Equiv.Perm.card_support_le_one | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\n⊢ card (support f) ≤ 1 ↔ f = 1",
"tactic": "rw [le_iff_lt_or_eq, Nat.lt_succ_iff, le_zero_iff, card_support_eq_zero, or_iff_not_imp_right,\n imp_iff_right f.card_support_ne_one]"
}
... | [
601,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
599,
1
] |
Std/Data/List/Lemmas.lean | List.mem_cons_self | [] | [
66,
68
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
66,
1
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Mathlib/Order/Heyting/Regular.lean | Heyting.Regular.coe_sup | [] | [
224,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
223,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean | fderivWithin_sum | [] | [
383,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
380,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | Real.arcsin_neg | [
{
"state_after": "case inl\nx : ℝ\nhx₁ : x ≤ -1\n⊢ arcsin (-x) = -arcsin x\n\ncase inr\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ arcsin (-x) = -arcsin x",
"state_before": "x : ℝ\n⊢ arcsin (-x) = -arcsin x",
"tactic": "cases' le_total x (-1) with hx₁ hx₁"
},
{
"state_after": "case inr.inl\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂... | [
145,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean | Polynomial.frange_ofSubring | [
{
"state_after": "R : Type u\nS : Type ?u.109024\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\nT : Subring R\np : { x // x ∈ T }[X]\ni : R\nhi : i ∈ ↑(frange (ofSubring T p))\n⊢ i ∈ ↑T",
"state_before": "R : Type u\nS : Type ?u.109024\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\nT :... | [
437,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
432,
1
] |
Mathlib/LinearAlgebra/Determinant.lean | LinearMap.bot_lt_ker_of_det_eq_zero | [
{
"state_after": "R : Type ?u.1194769\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.1195360\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type ?u.1195902\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA : Type ?u.1196380\ninst✝⁴ : CommRing... | [
364,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
359,
1
] |
Mathlib/GroupTheory/Finiteness.lean | Group.rank_congr | [] | [
426,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
424,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | NNReal.rpow_left_injective | [
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ≠ 0\ny z : ℝ≥0\nhyz : (fun y => y ^ x) y = (fun y => y ^ x) z\n⊢ y = z",
"tactic": "simpa only [rpow_inv_rpow_self hx] using congr_arg (fun y => y ^ (1 / x)) hyz"
}
] | [
243,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
242,
1
] |
Mathlib/Data/Finset/Sups.lean | Finset.Nonempty.sups | [] | [
123,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
11
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean | CommGroupCat.epi_iff_surjective | [
{
"state_after": "no goals",
"state_before": "A B : CommGroupCat\nf : A ⟶ B\n⊢ Epi f ↔ Function.Surjective ↑f",
"tactic": "rw [epi_iff_range_eq_top, MonoidHom.range_top_iff_surjective]"
}
] | [
454,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
453,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean | lcm_dvd | [] | [
724,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
723,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean | Submodule.finrank_sup_add_finrank_inf_eq | [
{
"state_after": "K : Type u\nV : Type v\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ns t : Submodule K V\ninst✝¹ : FiniteDimensional K { x // x ∈ s }\ninst✝ : FiniteDimensional K { x // x ∈ t }\nkey :\n Module.rank K { x // x ∈ s ⊔ t } + Module.rank K { x // x ∈ s ⊓ t } =\n Modul... | [
782,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
776,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.div_le_div_right | [
{
"state_after": "no goals",
"state_before": "m✝ n✝ k✝ n m : ℕ\nh : n ≤ m\nk : ℕ\nk0 : k = 0\n⊢ n / k ≤ m / k",
"tactic": "simp [k0]"
}
] | [
625,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
623,
11
] |
Mathlib/Algebra/Module/LocalizedModule.lean | IsLocalizedModule.mk'_eq_iff | [
{
"state_after": "no goals",
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1023,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1022,
1
] |
Mathlib/Order/OrderIsoNat.lean | exists_increasing_or_nonincreasing_subseq' | [
{
"state_after": "α : Type ?u.41863\nr : α → α → Prop\nf : ℕ → α\nbad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}\n⊢ ∃ g, (∀ (n : ℕ), r (f (↑g n)) (f (↑g (n + 1)))) ∨ ∀ (m n : ℕ), m < n → ¬r (f (↑g m)) (f (↑g n))",
"state_before": "α : Type ?u.41863\nr : α → α → Prop\nf : ℕ → α\n⊢ ∃ g, (∀ (n : ℕ), r ... | [
201,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.rel_zero_right | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.442937\nδ : Type ?u.442940\nr : α → β → Prop\np : γ → δ → Prop\na : Multiset α\n⊢ (a = 0 ∧ 0 = 0 ∨\n Exists fun {a_1} =>\n Exists fun {b} => Exists fun {as} => Exists fun {bs} => r a_1 b ∧ Rel r as bs ∧ a = a_1 ::ₘ as ∧ 0 = b ::ₘ bs) ↔\n a... | [
2717,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2717,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean | ContDiff.norm_sq | [] | [
163,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
] |
Mathlib/Algebra/Group/Units.lean | divp_eq_iff_mul_eq | [
{
"state_after": "α : Type u\ninst✝ : Monoid α\na b c x : α\nu : αˣ\ny : α\n⊢ x = y * ↑u ↔ y * ↑u = x",
"state_before": "α : Type u\ninst✝ : Monoid α\na b c x : α\nu : αˣ\ny : α\n⊢ x /ₚ u * ↑u = y * ↑u ↔ y * ↑u = x",
"tactic": "rw [divp_mul_cancel]"
},
{
"state_after": "no goals",
"state_bef... | [
490,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
489,
1
] |
Mathlib/Algebra/Hom/Group.lean | OneHom.coe_copy_eq | [] | [
824,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
822,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | mapsTo_extChartAt | [] | [
1064,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1062,
1
] |
Mathlib/Algebra/MonoidAlgebra/Grading.lean | AddMonoidAlgebra.gradeBy_id | [
{
"state_after": "no goals",
"state_before": "M : Type u_1\nι : Type ?u.5622\nR : Type u_2\ninst✝¹ : DecidableEq M\ninst✝ : CommSemiring R\n⊢ gradeBy R id = grade R",
"tactic": "rfl"
}
] | [
62,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
62,
1
] |
Mathlib/Topology/Order/Basic.lean | mem_nhdsWithin_Iic_iff_exists_Ioc_subset | [] | [
1829,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1826,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.cosh_ofReal_im | [
{
"state_after": "no goals",
"state_before": "x✝ y : ℂ\nx : ℝ\n⊢ (cosh ↑x).im = 0",
"tactic": "rw [← ofReal_cosh_ofReal_re, ofReal_im]"
}
] | [
684,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
684,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean | Left.one_lt_mul | [] | [
746,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
744,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean | Zsqrtd.mul_im | [] | [
179,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
1
] |
Mathlib/Algebra/Quaternion.lean | Quaternion.mul_imJ | [
{
"state_after": "no goals",
"state_before": "S : Type ?u.421024\nT : Type ?u.421027\nR : Type u_1\ninst✝ : CommRing R\nr x y z : R\na b c : ℍ[R]\n⊢ a.re * b.imJ + -1 * a.imI * b.imK + a.imJ * b.re - -1 * a.imK * b.imI =\n a.re * b.imJ - a.imI * b.imK + a.imJ * b.re + a.imK * b.imI",
"tactic": "simp ... | [
959,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
958,
1
] |
Mathlib/Topology/MetricSpace/MetrizableUniformity.lean | PseudoMetricSpace.le_two_mul_dist_ofPreNNDist | [
{
"state_after": "X : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nx y : X\n⊢ d x y ≤ ⨅ (i : List X), 2 * sum (zipWith d (x :: i) (i ++ [y]))",
"state_before": "X : Type u... | [
180,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | two_mul_le_add_sq | [] | [
761,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
760,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.reClm_apply | [] | [
895,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
894,
1
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.Theory.model_iff | [] | [
817,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
816,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineIsometryEquiv.map_eq_iff | [] | [
652,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
651,
1
] |
Mathlib/Algebra/Group/Prod.lean | MulHom.coprod_apply | [] | [
427,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
426,
1
] |
Mathlib/CategoryTheory/Equivalence.lean | CategoryTheory.Equivalence.counitInv_app_functor | [
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nX : C\n⊢ e.functor.map ((unit e).app X) = (counitInv e).app (e.functor.obj X)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nX : C\n⊢ (counitInv e).app (e.functor... | [
171,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/Algebra/Associated.lean | Associates.mk_ne_zero | [] | [
969,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
968,
1
] |
Std/Data/String/Lemmas.lean | String.utf8Len_nil | [] | [
47,
52
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
47,
9
] |
Mathlib/Data/Real/Pointwise.lean | Real.smul_iSup_of_nonpos | [] | [
90,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean | EMetric.cauchy_iff | [
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.28215\ninst✝ : PseudoEMetricSpace α\nf : Filter α\n⊢ Cauchy f ↔ NeBot f ∧ ∀ (ε : ℝ≥0∞), ε > 0 → ∃ t, t ∈ f ∧ ∀ (x : α), x ∈ t → ∀ (y : α), y ∈ t → edist x y < ε",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.28215\ninst✝ : PseudoEMetricSpace α\nf... | [
319,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
11
] |
Mathlib/FieldTheory/Separable.lean | Polynomial.separable_C | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nr : R\n⊢ Separable (↑C r) ↔ IsUnit r",
"tactic": "rw [separable_def, derivative_C, isCoprime_zero_right, isUnit_C]"
}
] | [
81,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean | CategoryTheory.Presieve.IsSheafFor.hom_ext | [] | [
576,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
573,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | intervalIntegral.integral_hasDerivAt_left | [] | [
793,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
790,
1
] |
Mathlib/Data/Nat/Order/Basic.lean | Nat.add_eq_three_iff | [
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\n⊢ m + n = 3 ↔ m = 0 ∧ n = 3 ∨ m = 1 ∧ n = 2 ∨ m = 2 ∧ n = 1 ∨ m = 3 ∧ n = 0",
"tactic": "cases n <;>\nsimp [(succ_ne_zero 1).symm, succ_eq_add_one, (show 3 = Nat.succ 2 from rfl),\n ← add_assoc, succ_inj', add_eq_two_iff]"
}
] | [
209,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
] |
Mathlib/Topology/Basic.lean | ContinuousAt.comp_of_eq | [
{
"state_after": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.168296\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ng : β → γ\nf : α → β\nx : α\nhf : ContinuousAt f x\nhg : ContinuousAt g (f x)\n⊢ ContinuousAt (g ∘ f) x",
"state_before": "α : Type u_3\nβ : T... | [
1657,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1656,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean | tendstoUniformlyOn_univ | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\n⊢ TendstoUniformlyOn F f p univ ↔ TendstoUniformly F f p",
"tactic": "simp [TendstoUniformlyOn, Ten... | [
141,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
140,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Ico_subset_Iio_self | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.55946\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\ninst✝ : LocallyFiniteOrderBot α\n⊢ Ico a b ⊆ Iio b",
"tactic": "simpa [← coe_subset] using Set.Ico_subset_Iio_self"
}
] | [
435,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
434,
1
] |
Mathlib/Data/Set/NAry.lean | Set.image2_singleton | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.27806\nβ : Type u_3\nβ' : Type ?u.27812\nγ : Type u_1\nγ' : Type ?u.27818\nδ : Type ?u.27821\nδ' : Type ?u.27824\nε : Type ?u.27827\nε' : Type ?u.27830\nζ : Type ?u.27833\nζ' : Type ?u.27836\nν : Type ?u.27839\nf f' : α → β → γ\ng g' :... | [
200,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
200,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.eq_zero_of_nonpos | [] | [
156,
50
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
154,
11
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_zero | [] | [
131,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.disjoint_insert_left | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.121015\nγ : Type ?u.121018\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\n⊢ _root_.Disjoint (insert a s) t ↔ ¬a ∈ t ∧ _root_.Disjoint s t",
"tactic": "simp only [disjoint_left, mem_insert, or_imp, forall_and, forall_eq]"
}
] | [
1273,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1272,
1
] |
Mathlib/Data/Num/Lemmas.lean | PosNum.divMod_to_nat | [
{
"state_after": "d n : PosNum\n⊢ ↑(divMod d n).snd + ↑d * ↑(divMod d n).fst = ↑n ∧ ↑(divMod d n).snd < ↑d",
"state_before": "d n : PosNum\n⊢ ↑n / ↑d = ↑(divMod d n).fst ∧ ↑n % ↑d = ↑(divMod d n).snd",
"tactic": "rw [Nat.div_mod_unique (PosNum.cast_pos _)]"
},
{
"state_after": "case one\nd : Pos... | [
1625,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1602,
1
] |
Mathlib/Init/CcLemmas.lean | if_eq_of_eq_false | [] | [
90,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean | AffineIndependent.subtype | [] | [
316,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
11
] |
Mathlib/Analysis/NormedSpace/Star/Basic.lean | CstarRing.norm_coe_unitary | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.74050\nE : Type u_1\nα : Type ?u.74056\ninst✝³ : NormedRing E\ninst✝² : StarRing E\ninst✝¹ : CstarRing E\ninst✝ : Nontrivial E\nU : { x // x ∈ unitary E }\n⊢ ‖↑U‖ = 1",
"tactic": "rw [← sq_eq_sq (norm_nonneg _) zero_le_one, one_pow 2, sq, ← Csta... | [
220,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_inf | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.104778\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nH₁ H₂ : Subgraph G\n⊢ ∀ (x y : V),\n Quotient.mk (Sym2.Rel.setoid V) (x, y) ∈ edgeSet (H₁ ⊓ H₂) ↔\n Quotient.mk (Sym2.Rel.setoid V) (x, y) ∈ edgeSet H₁ ∩ edgeSet ... | [
549,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
548,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | Submodule.isOrtho_iff_inner_eq | [] | [
275,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean | UniqueDiffOn.univ_pi | [] | [
388,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
385,
1
] |
Mathlib/Data/List/Indexes.lean | List.mapIdxGo_length | [
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\n⊢ ∀ (arr : Array β), length (mapIdx.go f l arr) = length l + Array.size arr",
"state_before": "α : Type u\nβ : Type v\n⊢ ∀ (f : ℕ → α → β) (l : List α) (arr : Array β), length (mapIdx.go f l arr) = length l + Array.size arr",
"tactic":... | [
128,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
1
] |
Mathlib/Algebra/GroupWithZero/Power.lean | Commute.self_zpow₀ | [] | [
140,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Mathlib/Topology/LocallyFinite.lean | LocallyFinite.continuous' | [] | [
115,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
11
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.ball_subset_ball | [] | [
584,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
583,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean | Pell.xn_modEq_x2n_add_lem | [
{
"state_after": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh1 : Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (Pell.d a1 * yn a1 n * yn a1 n + 1) * xn a1 j\n⊢ xn a1 n ∣ Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j",
"state_before": "a : ℕ\na1 : 1 < a\nn j : ℕ\n⊢ xn a1 n ∣ Pell.d a1 * yn a1 n * (yn a1 n * ... | [
597,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
590,
1
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean | SimplexCategory.δ_comp_δ_self | [] | [
248,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
247,
1
] |
Mathlib/Data/Set/Basic.lean | Set.default_coe_singleton | [] | [
1374,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1373,
1
] |
Mathlib/RingTheory/RingInvo.lean | RingInvo.map_eq_zero_iff | [] | [
116,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean | ne_zero_of_one_div_ne_zero | [
{
"state_after": "α : Type ?u.28943\nM₀ : Type ?u.28946\nG₀ : Type u_1\nM₀' : Type ?u.28952\nG₀' : Type ?u.28955\nF : Type ?u.28958\nF' : Type ?u.28961\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : 0 ≠ 0\nha : a = 0\n⊢ False",
"state_before": "α : Type ?u.28943\nM₀ : Type ?u.28946\nG₀ : Type u_1\nM₀' : Type... | [
417,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
415,
1
] |
Mathlib/Data/Polynomial/Laurent.lean | LaurentPolynomial.degree_C | [
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\na : R\na0 : a ≠ 0\n⊢ degree (↑C a * T 0) = 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\na : R\na0 : a ≠ 0\n⊢ degree (↑C a) = 0",
"tactic": "rw [← mul_one (C a), ← T_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type ... | [
531,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
529,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean | measurable_of_measurable_on_compl_singleton | [] | [
623,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
621,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.sum_append | [
{
"state_after": "no goals",
"state_before": "l₁ l₂ : List Nat\n⊢ Nat.sum (l₁ ++ l₂) = Nat.sum l₁ + Nat.sum l₂",
"tactic": "induction l₁ <;> simp [*, Nat.add_assoc]"
}
] | [
834,
43
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
833,
9
] |
Mathlib/Analysis/Complex/Basic.lean | Complex.nndist_conj_comm | [] | [
344,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
] |
Mathlib/Algebra/Order/Module.lean | smul_add_smul_le_smul_add_smul' | [
{
"state_after": "k : Type u_2\nM : Type u_1\nN : Type ?u.63738\ninst✝⁴ : OrderedRing k\ninst✝³ : OrderedAddCommGroup M\ninst✝² : Module k M\ninst✝¹ : OrderedSMul k M\na✝ b✝ : M\nc✝ : k\ninst✝ : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : k\nc d : M\nhba : b ≤ a\nhdc : d ≤ c\n⊢ b •... | [
132,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.Equiv.theory_model | [] | [
982,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
981,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.inter_self | [] | [
1630,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1629,
1
] |
Mathlib/Logic/Equiv/Basic.lean | Equiv.subtypePreimage_symm_apply_coe_neg | [] | [
649,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
647,
1
] |
Mathlib/Topology/Instances/Matrix.lean | Summable.matrix_diag | [] | [
376,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
374,
1
] |
Mathlib/Logic/Equiv/Functor.lean | Functor.mapEquiv_symm_apply | [] | [
56,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | min_lt_max_of_mul_lt_mul | [
{
"state_after": "β : Type ?u.146910\nA : Type ?u.146913\nG : Type ?u.146916\nM : Type u_1\nR : Type ?u.146922\ninst✝³ : Monoid M\ninst✝² : LinearOrder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d... | [
299,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/Analysis/Convex/Join.lean | mem_convexJoin | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.1646\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns t s₁ s₂ t₁ t₂ u : Set E\nx y : E\n⊢ x ∈ convexJoin 𝕜 s t ↔ ∃ a, a ∈ s ∧ ∃ b, b ∈ t ∧ x ∈ segment 𝕜 a b",
"tactic": "simp [convexJoi... | [
42,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/Order/Bounds/Basic.lean | isGreatest_singleton | [] | [
617,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
616,
1
] |
Mathlib/Algebra/Homology/Additive.lean | HomologicalComplex.zero_f_apply | [] | [
69,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
68,
1
] |
Mathlib/LinearAlgebra/Quotient.lean | Submodule.liftQ_mkQ | [
{
"state_after": "case h\nR : Type u_1\nM : Type u_3\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ ker f\nx✝ : M\n⊢ ↑... | [
360,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
360,
1
] |
Mathlib/Topology/Instances/Rat.lean | Rat.uniformContinuous_add | [
{
"state_after": "no goals",
"state_before": "⊢ UniformContinuous fun x => ↑x.fst + ↑x.snd",
"tactic": "exact Real.uniformContinuous_add.comp\n (Rat.uniformContinuous_coe_real.prod_map Rat.uniformContinuous_coe_real)"
}
] | [
95,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
] |
Mathlib/Data/Finset/NAry.lean | Finset.image₂_image_left_anticomm | [] | [
498,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
495,
1
] |
Mathlib/Data/Real/Irrational.lean | Irrational.mul_rat | [
{
"state_after": "no goals",
"state_before": "q✝ : ℚ\nx y : ℝ\nh : Irrational x\nq : ℚ\nhq : q ≠ 0\n⊢ Irrational (x * ↑q * ↑q⁻¹)",
"tactic": "rwa [mul_assoc, ← cast_mul, mul_inv_cancel hq, cast_one, mul_one]"
}
] | [
344,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean | MeasureTheory.NullMeasurableSet.union | [] | [
189,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
11
] |
Mathlib/GroupTheory/FreeGroup.lean | FreeGroup.Red.red_iff_irreducible | [
{
"state_after": "case h\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx1 : α\nb1 : Bool\nx2 : α\nb2 : Bool\nh : (x1, b1) ≠ (x2, b2)\n⊢ ∀ (b : List (α × Bool)), ¬Step [(x1, !b1), (x2, b2)] b",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx1 : α\nb1 : Bool\nx2 : α\nb2 : Bool\nh : (x1, b1) ≠ ... | [
355,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
347,
1
] |
Std/Data/Int/DivMod.lean | Int.dvd_iff_emod_eq_zero | [] | [
688,
45
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
687,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | ENNReal.rpow_mul | [
{
"state_after": "case none\ny z : ℝ\n⊢ none ^ (y * z) = (none ^ y) ^ z\n\ncase some\ny z : ℝ\nx : ℝ≥0\n⊢ Option.some x ^ (y * z) = (Option.some x ^ y) ^ z",
"state_before": "x : ℝ≥0∞\ny z : ℝ\n⊢ x ^ (y * z) = (x ^ y) ^ z",
"tactic": "cases' x with x"
},
{
"state_after": "no goals",
"state_b... | [
467,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
455,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.map_sub_atTop_eq_nat | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.343771\nι' : Type ?u.343774\nα : Type ?u.343777\nβ : Type ?u.343780\nγ : Type ?u.343783\nk b : ℕ\nx✝ : b ≥ 0\n⊢ b ≤ (fun a => a + k) b - k",
"tactic": "rw [add_tsub_cancel_right]"
}
] | [
1646,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1644,
1
] |
Mathlib/Algebra/Quaternion.lean | Quaternion.im_re | [] | [
805,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
805,
9
] |
Mathlib/Algebra/Order/Interval.lean | NonemptyInterval.one_mem_one | [] | [
125,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/Data/Rat/Order.lean | Rat.nonneg_iff_zero_le | [
{
"state_after": "a✝ b c a : ℚ\n⊢ Rat.Nonneg a ↔ Rat.Nonneg (a - 0)",
"state_before": "a✝ b c a : ℚ\n⊢ Rat.Nonneg a ↔ 0 ≤ a",
"tactic": "rw [Rat.le_iff_Nonneg]"
},
{
"state_after": "a✝ b c a : ℚ\n⊢ Rat.Nonneg a ↔ Rat.Nonneg (a - 0)",
"state_before": "a✝ b c a : ℚ\n⊢ Rat.Nonneg a ↔ Rat.Nonneg... | [
241,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
238,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean | LocallyConstant.ofClopen_fiber_one | [
{
"state_after": "case h\nX✝ : Type ?u.21432\nY : Type ?u.21435\nZ : Type ?u.21438\nα : Type ?u.21441\ninst✝² : TopologicalSpace X✝\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nU : Set X\ninst✝ : (x : X) → Decidable (x ∈ U)\nhU : IsClopen U\nx✝ : X\n⊢ x✝ ∈ ↑(ofClopen hU) ⁻¹' {1} ↔ x✝ ∈ Uᶜ",
"state_before": "... | [
368,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
364,
1
] |
Mathlib/Order/Filter/Lift.lean | Filter.comap_lift_eq | [
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Sort ?u.14103\nf f₁ f₂ : Filter α\ng g₁ g₂ : Set α → Filter β\nm : γ → β\n⊢ (⨅ (i : Set α) (_ : i ∈ f), comap m (g i)) = ⨅ (s : Set α) (_ : s ∈ f), (comap m ∘ g) s",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Sort ?u.14103\n... | [
119,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
118,
1
] |
Std/Logic.lean | and_or_left | [] | [
322,
97
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
321,
1
] |
Mathlib/ModelTheory/Satisfiability.lean | FirstOrder.Language.exists_elementaryEmbedding_card_eq_of_le | [
{
"state_after": "case intro.intro\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nM : Type w'\ninst✝¹ : Structure L M\ninst✝ : Nonempty M\nκ : Cardinal\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nh3 : lift κ ≤ lift (#M)\nS : ElementarySubstructure L M\nleft✝ : ∅ ⊆ ↑S\nhS : lift (#{ x // x ∈ S }) = lift κ\n⊢ ∃ N... | [
227,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
1
] |
Mathlib/Analysis/Convex/Extreme.lean | mem_extremePoints | [] | [
139,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/Computability/EpsilonNFA.lean | εNFA.step_zero | [] | [
235,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
234,
1
] |
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