file_path stringlengths 11 79 | full_name stringlengths 2 100 | traced_tactics list | end list | commit stringclasses 4
values | url stringclasses 4
values | start list |
|---|---|---|---|---|---|---|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | ciInf_unique | [] | [
848,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
847,
1
] |
Mathlib/CategoryTheory/Conj.lean | CategoryTheory.Functor.map_conjAut | [
{
"state_after": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u₁\ninst✝ : Category D\nF✝ F : C ⥤ D\nX Y : C\nα : X ≅ Y\nf : Aut X\n⊢ (F.mapIso (↑(Iso.conjAut α) f)).hom = (↑(Iso.conjAut (F.mapIso α)) (F.mapIso f)).hom",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u₁\ninst✝ : Category D\... | [
191,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
189,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snorm_exponent_zero | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.909869\nF : Type u_2\nG : Type ?u.909875\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\n⊢ snorm f 0 μ = 0",
"tactic... | [
179,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
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Mathlib/Analysis/Calculus/Inverse.lean | ApproximatesLinearOn.lipschitz_sub | [] | [
163,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean | UniqueFactorizationMonoid.dvd_of_dvd_mul_left_of_no_prime_factors | [
{
"state_after": "case refine'_1\nα : Type ?u.940597\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b c : R\nha : a ≠ 0\n⊢ (∀ {d : R}, d ∣ a → d ∣ 0 → ¬Prime d) → a ∣ b * 0 → a ∣ b\n\ncase refine'_2\nα : Type ?u.940597\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero ... | [
883,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
868,
1
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Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.coe_toEquiv_symm | [] | [
279,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
278,
1
] |
Mathlib/Logic/Basic.lean | exists_unique_prop | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.22665\nα : Sort ?u.22670\nκ : ι → Sort ?u.22667\np✝ q✝ : α → Prop\np q : Prop\n⊢ (∃! x, q) ↔ p ∧ q",
"tactic": "simp"
}
] | [
860,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
860,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.range_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.1250249\nR₂ : Type u_2\nR₃ : Type ?u.1250255\nR₄ : Type ?u.1250258\nS : Type ?u.1250261\nK : Type ?u.1250264\nK₂ : Type ?u.1250267\nM : Type u_4\nM' : Type ?u.1250273\nM₁ : Type ?u.1250276\nM₂ : Type u_3\nM₃ : Type ?u.1250282\nM₄ : Typ... | [
1404,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1403,
1
] |
Mathlib/Order/BooleanAlgebra.lean | sdiff_sdiff_right_self | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.29848\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x \\ (x \\ y) = x ⊓ y",
"tactic": "rw [sdiff_sdiff_right, inf_idem, sdiff_self, bot_sup_eq]"
}
] | [
381,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
380,
1
] |
Mathlib/Data/Real/NNReal.lean | Real.toNNReal_inv | [
{
"state_after": "case inl\nx : ℝ\nhx : 0 ≤ x\n⊢ toNNReal x⁻¹ = (toNNReal x)⁻¹\n\ncase inr\nx : ℝ\nhx : x ≤ 0\n⊢ toNNReal x⁻¹ = (toNNReal x)⁻¹",
"state_before": "x : ℝ\n⊢ toNNReal x⁻¹ = (toNNReal x)⁻¹",
"tactic": "cases' le_total 0 x with hx hx"
},
{
"state_after": "case inl\nx : ℝ\nhx : 0 ≤ x\n... | [
889,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
885,
1
] |
Mathlib/RingTheory/ChainOfDivisors.lean | DivisorChain.second_of_chain_is_irreducible | [
{
"state_after": "case zero\nM : Type u_1\ninst✝ : CancelCommMonoidWithZero M\nq : Associates M\nhq : q ≠ 0\nhn : Nat.zero ≠ 0\nc : Fin (Nat.zero + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\n⊢ Irreducible (c 1)\n\ncase succ\nM : Type u_1\ninst✝ : CancelCommMonoidWithZ... | [
112,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/Data/TypeVec.lean | TypeVec.fst_diag | [
{
"state_after": "case h\nn : ℕ\nα : TypeVec n\ni : Fin2 n\n⊢ (prod.fst ⊚ prod.diag) i = id i",
"state_before": "n : ℕ\nα : TypeVec n\n⊢ prod.fst ⊚ prod.diag = id",
"tactic": "funext i"
},
{
"state_after": "case h.fz\nn n✝ : ℕ\nα : TypeVec (succ n✝)\n⊢ (prod.fst ⊚ prod.diag) Fin2.fz = id Fin2.fz... | [
587,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
584,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.succ_aleph0 | [
{
"state_after": "no goals",
"state_before": "⊢ succ ℵ₀ = aleph 1",
"tactic": "rw [← aleph_zero, ← aleph_succ, Ordinal.succ_zero]"
}
] | [
343,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | abs_real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul | [] | [
1568,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1566,
1
] |
Mathlib/Order/CompleteLattice.lean | sInf_Prop_eq | [] | [
1727,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1726,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | lebesgue_number_of_compact_open | [
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.194991\ninst✝ : UniformSpace α\nK U : Set α\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\nW : ↑K → Set (α × α) := fun k => Classical.choose (_ : ∃ V, V ∈ 𝓤 α ∧ IsOpen V ∧ UniformSpace.ball (↑k) V ⊆ U)\n⊢ ∃ V, V ∈ 𝓤 α ∧ IsOpen V... | [
1890,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1871,
1
] |
Mathlib/Data/Real/Hyperreal.lean | Hyperreal.infinite_mul_of_infinite_not_infinitesimal | [] | [
891,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
882,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | nnnorm_toDual | [] | [
2329,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2328,
1
] |
Mathlib/RingTheory/Polynomial/Content.lean | Polynomial.eq_C_content_mul_primPart | [
{
"state_after": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : p = 0\n⊢ p = ↑C (content p) * primPart p\n\ncase neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ¬p = 0\n⊢ p = ↑C (content p)... | [
258,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_iff_exists_le_nhds | [] | [
415,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
1
] |
Mathlib/Init/Propext.lean | eq_true_intro | [] | [
26,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
25,
1
] |
Mathlib/RingTheory/Multiplicity.lean | multiplicity.finite_pow | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np a : α\nhp : Prime p\nx✝ : Finite p a\n⊢ ¬p ^ (0 + 1) ∣ a ^ 0",
"tactic": "simp [mt isUnit_iff_dvd_one.2 hp.2.1]"
},
{
"state_after": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np a : α\nhp : ... | [
523,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.mul_right_iff | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\n⊢ IsCoprime x (y * z) ↔ IsCoprime x y ∧ IsCoprime x z",
"tactic": "rw [isCoprime_comm, IsCoprime.mul_left_iff, isCoprime_comm, @isCoprime_comm _ _ z]"
}
] | [
154,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
153,
1
] |
Std/Data/List/Lemmas.lean | List.isInfix.length_le | [] | [
1632,
22
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1631,
1
] |
Mathlib/CategoryTheory/Sums/Basic.lean | CategoryTheory.sum_comp_inl | [] | [
69,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.preimage_kernImage | [] | [
205,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
11
] |
Mathlib/Analysis/Calculus/ExtendDeriv.lean | hasDerivAt_of_hasDerivAt_of_ne' | [
{
"state_after": "case inl\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.151621\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : ℝ → E\ny : ℝ\nf_diff : ∀ (y_1 : ℝ), y_1 ≠ y → HasDerivAt f (g y_1) y_1\nhf : ContinuousAt f y\nhg : ContinuousAt g y\n⊢ HasDerivAt ... | [
222,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.parallel_comm | [] | [
1730,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1729,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.div_pos | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.233480\nβ : Type ?u.233483\nγ : Type ?u.233486\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nb c : Ordinal\nh : c ≠ 0\n⊢ 0 < b / c ↔ c ≤ b",
"tactic": "simp [lt_div h]"
}
] | [
905,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
905,
1
] |
Mathlib/Data/List/Perm.lean | List.Perm.of_eq | [] | [
83,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
1
] |
Mathlib/Analysis/SpecialFunctions/Sqrt.lean | deriv_sqrt | [] | [
108,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | LiouvilleWith.exists_pos | [
{
"state_after": "case intro\np q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\nC : ℝ\nhC : ∃ᶠ (n : ℕ) in atTop, ∃ m, x ≠ ↑m / ↑n ∧ abs (x - ↑m / ↑n) < C / ↑n ^ p\n⊢ ∃ C _h₀, ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ abs (x - ↑m / ↑n) < C / ↑n ^ p",
"state_before": "p q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\nh : LiouvilleWith... | [
86,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
] |
Mathlib/Algebra/Regular/SMul.lean | IsSMulRegular.smul_iff | [] | [
88,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/Topology/Instances/EReal.lean | EReal.openEmbedding_coe | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.2194\ninst✝ : TopologicalSpace α\n⊢ IsOpen (range Real.toEReal)",
"tactic": "simp only [range_coe_eq_Ioo, isOpen_Ioo]"
}
] | [
62,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
] |
Mathlib/Data/Set/Sups.lean | Set.sups_union_right | [] | [
165,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Mathlib/LinearAlgebra/Prod.lean | LinearMap.mem_graph_iff | [] | [
1012,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1011,
1
] |
Mathlib/Algebra/Quandle.lean | Rack.ad_conj | [
{
"state_after": "R✝ : Type ?u.26283\ninst✝¹ : Rack R✝\nR : Type u_1\ninst✝ : Rack R\nx y : R\n⊢ act' (x ◃ y) * act' x = act' x * act' y",
"state_before": "R✝ : Type ?u.26283\ninst✝¹ : Rack R✝\nR : Type u_1\ninst✝ : Rack R\nx y : R\n⊢ act' (x ◃ y) = act' x * act' y * (act' x)⁻¹",
"tactic": "rw [eq_mul_i... | [
256,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.liftHom_eq_algHom | [
{
"state_after": "R : Type u\nS : Type v\nK : Type w\ninst✝² : CommRing R\nf✝ : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh : eval₂ i a f✝ = 0\ninst✝ : Algebra R S\nf : R[X]\nϕ : AdjoinRoot f →ₐ[R] S\n⊢ AlgHom.equalizer ϕ (liftHom f (↑ϕ (root f)) (_ : ↑(aeval (↑ϕ (root f))) f = 0)) = ⊤",
"state_before"... | [
330,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
325,
1
] |
Mathlib/Combinatorics/Derangements/Finite.lean | numDerangements_one | [] | [
83,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean | SimpleGraph.interedges_disjoint_right | [] | [
354,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.mapRange.addMonoidHom_id | [] | [
221,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean | Ideal.span_univ | [] | [
129,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | BoxIntegral.Box.injective_splitCenterBox | [] | [
82,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/Tactic/Ring/Basic.lean | Mathlib.Tactic.Ring.mul_exp_pos | [] | [
609,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
608,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | div_mem_comm_iff | [] | [
165,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean | MeasureTheory.MeasurePreserving.integrableOn_image | [] | [
257,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean | circleIntegral.integral_smul | [
{
"state_after": "no goals",
"state_before": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\ninst✝³ : CompleteSpace E\n𝕜 : Type u_1\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : SMulCommClass 𝕜 ℂ E\na : 𝕜\nf : ℂ → E\nc : ℂ\nR : ℝ\n⊢ (∮ (z : ℂ) in C(c, R), a • f z) = a • ... | [
447,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
445,
1
] |
Mathlib/Algebra/Homology/Exact.lean | CategoryTheory.comp_eq_zero_of_image_eq_kernel | [
{
"state_after": "V : Type u\ninst✝³ : Category V\ninst✝² : HasImages V\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasKernels V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\np : imageSubobject f = kernelSubobject g\n⊢ Subobject.arrow (imageSubobject f) ≫ g = 0",
"state_before": "V : Type u\ninst✝³ : Category V\ninst✝² : Has... | [
145,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | MeasureTheory.integrableOn_Ioi_deriv_of_nonneg | [
{
"state_after": "case refine_1\nE : Type ?u.153019\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : ∀ (x : ℝ), x ∈ Ioi a → HasDerivAt g (g' x) x\ng'pos : ∀ (x : ℝ), x ∈ Ioi a → 0 ≤ g' x\nhg ... | [
724,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
702,
1
] |
Mathlib/Init/CcLemmas.lean | eq_true_of_and_eq_true_left | [] | [
99,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
1
] |
Mathlib/Data/Polynomial/Basic.lean | Polynomial.support_C_mul_X | [
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\nc : R\nh : c ≠ 0\n⊢ support (↑C c * X) = {1}",
"tactic": "rw [C_mul_X_eq_monomial, support_monomial 1 h]"
}
] | [
845,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
844,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.I_pow_bit1 | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ I ^ bit1 n = (-1) ^ n * I",
"tactic": "rw [pow_bit1', Complex.I_mul_I]"
}
] | [
476,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
476,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.insert_erase_subset | [] | [
1999,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1998,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.L1.SimpleFunc.setToL1S_neg | [
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.671027\nG : Type ?u.671030\n𝕜 : Type ?u.671033\np : ℝ≥0∞\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : NormedSpace ℝ F'\ninst✝² ... | [
773,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
767,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean | HasDerivWithinAt.nhdsWithin | [] | [
419,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
417,
1
] |
Mathlib/SetTheory/Ordinal/Principal.lean | Ordinal.principal_one_iff | [
{
"state_after": "case refine'_1\nop : Ordinal → Ordinal → Ordinal\nh : Principal op 1\n⊢ op 0 0 = 0\n\ncase refine'_2\nop : Ordinal → Ordinal → Ordinal\nh : op 0 0 = 0\na b : Ordinal\nha : a < 1\nhb : b < 1\n⊢ op a b < 1",
"state_before": "op : Ordinal → Ordinal → Ordinal\n⊢ Principal op 1 ↔ op 0 0 = 0",
... | [
69,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
] |
Mathlib/Data/Complex/Exponential.lean | Real.exp_bound | [
{
"state_after": "x : ℝ\nhx : abs' x ≤ 1\nn : ℕ\nhn : 0 < n\nhxc : ↑Complex.abs ↑x ≤ 1\n⊢ abs' (exp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤ abs' x ^ n * (↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n))",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\nn : ℕ\nhn : 0 < n\n⊢ abs' (exp x - ∑ m in range n, x ^ m /... | [
1709,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1703,
8
] |
Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | TopCat.Sheaf.interUnionPullbackCone_pt | [] | [
327,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
326,
1
] |
Mathlib/Data/Set/Basic.lean | Set.eq_empty_iff_forall_not_mem | [] | [
581,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
580,
1
] |
Mathlib/Data/Fin/Tuple/Monotone.lean | antitone_vecCons | [] | [
59,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
1
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean | Matrix.cramer_submatrix_equiv | [
{
"state_after": "case h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA✝ : Matrix n n α\nb✝ : n → α\nA : Matrix m m α\ne : n ≃ m\nb : n → α\ni : n\n⊢ ↑(cramer (submatrix A ↑e ↑e)) b i = (↑(cramer A) (b ∘ ↑e.symm)... | [
181,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
Mathlib/Data/Real/Irrational.lean | irrational_div_rat_iff | [
{
"state_after": "no goals",
"state_before": "q : ℚ\nm : ℤ\nn : ℕ\nx : ℝ\n⊢ Irrational (x / ↑q) ↔ q ≠ 0 ∧ Irrational x",
"tactic": "rw [div_eq_mul_inv, ← cast_inv, irrational_mul_rat_iff, Ne.def, inv_eq_zero]"
}
] | [
639,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
638,
1
] |
Mathlib/Data/Bundle.lean | Bundle.Pullback.proj_lift | [] | [
150,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean | List.prod_map_mul | [] | [
127,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean | Subsemigroup.closure_induction' | [
{
"state_after": "M : Type u_1\nN : Type ?u.16694\nA : Type ?u.16697\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\ns : Set M\np : (x : M) → x ∈ closure s → Prop\nHs : ∀ (x : M) (h : x ∈ s), p x (_ : x ∈ ↑(closure s))\nHmul : ∀ (x : M) (hx : x ∈ closure s) (y : M) (hy : y ∈ closure s... | [
372,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
365,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.val_le_iff_val_subset | [] | [
3222,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3221,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.ediam_univ_of_noncompact | [] | [
2668,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2666,
1
] |
src/lean/Init/Data/Nat/Basic.lean | Nat.lt_of_lt_of_eq | [] | [
263,
23
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
262,
11
] |
Mathlib/Analysis/Convex/Cone/Basic.lean | ConvexCone.positive_le_strictlyPositive | [] | [
621,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
621,
1
] |
Mathlib/Topology/DenseEmbedding.lean | DenseInducing.extend_eq_of_tendsto | [] | [
153,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/Data/Set/Basic.lean | Set.Nonempty.subset_singleton_iff | [] | [
1501,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1500,
1
] |
Mathlib/Order/UpperLower/Basic.lean | isLowerSet_setOf | [] | [
282,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
281,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.equivOfEq_trans | [] | [
1099,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1098,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | CategoryTheory.Limits.biprod.mapBiprod_hom_desc | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nX Y : C\ninst✝¹ : HasBinaryBiproduct X Y\ninst✝ : PreservesBinaryBiproduct X Y F\nW : C\nf... | [
468,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
466,
1
] |
Mathlib/Analysis/NormedSpace/Ray.lean | SameRay.norm_sub | [
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.4436\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nu : E\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nh : SameRay ℝ (a • u) (b • u)\n⊢ ‖a • u - b • u‖ = abs (‖... | [
50,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean | MeasureTheory.set_integral_congr_ae₀ | [] | [
77,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.div_add_mod' | [
{
"state_after": "m✝ n k✝ m k : ℕ\n⊢ k * (m / k) + m % k = m",
"state_before": "m✝ n k✝ m k : ℕ\n⊢ m / k * k + m % k = m",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "m✝ n k✝ m k : ℕ\n⊢ k * (m / k) + m % k = m",
"tactic": "exact div_add_mod _ _"
}
] | [
713,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
711,
1
] |
Mathlib/Data/List/Basic.lean | List.tail_replicate | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.19246\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\n⊢ tail (replicate n a) = replicate (n - 1) a",
"tactic": "cases n <;> rfl"
}
] | [
462,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
461,
9
] |
Mathlib/Data/Set/Sups.lean | Set.image_inf_prod | [
{
"state_after": "α : Type u_1\ninst✝ : SemilatticeInf α\ns✝ s₁ s₂ t✝ t₁ t₂ u v : Set α\na b c : α\ns t : Set α\nthis : (uncurry fun x x_1 => x ⊓ x_1) '' s ×ˢ t = image2 (fun x x_1 => x ⊓ x_1) s t\n⊢ (uncurry fun x x_1 => x ⊓ x_1) '' s ×ˢ t = s ⊼ t",
"state_before": "α : Type u_1\ninst✝ : SemilatticeInf α\n... | [
346,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
342,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.sub_lt_of_lt_add | [] | [
1169,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1168,
11
] |
Mathlib/Data/List/BigOperators/Basic.lean | List.alternatingProd_cons_cons | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.162207\nα : Type u_1\nM : Type ?u.162213\nN : Type ?u.162216\nP : Type ?u.162219\nM₀ : Type ?u.162222\nG : Type ?u.162225\nR : Type ?u.162228\ninst✝ : DivInvMonoid α\na b : α\nl : List α\n⊢ alternatingProd (a :: b :: l) = a / b * alternatingProd l",
... | [
646,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
644,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.map_add_left_Ioc | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map ((fun x x_1 => x + x_1) c) (Ioc a b) = Ioc (c + a) (c + b)",
"tactic": "classical rw [Ioc, Ioc, ← Finset.image_add_left_Ioc, Finset.... | [
301,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
299,
1
] |
Std/Data/Int/Lemmas.lean | Int.ne_of_gt | [] | [
631,
84
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
631,
11
] |
Mathlib/Analysis/BoxIntegral/Partition/Split.lean | BoxIntegral.Prepartition.iUnion_splitMany | [] | [
271,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
270,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.leRecOn_succ | [
{
"state_after": "no goals",
"state_before": "m✝ n✝ k : ℕ\nC : ℕ → Sort u\nn m : ℕ\nh1 : n ≤ m\nh2 : n ≤ m + 1\nnext : {k : ℕ} → C k → C (k + 1)\nx : C n\n⊢ leRecOn h2 next x = next (leRecOn h1 next x)",
"tactic": "conv =>\n lhs\n rw [leRecOn, Or.by_cases, dif_pos h1]"
}
] | [
411,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
407,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | Subring.centralizer_univ | [] | [
884,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
883,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.smul_sSup | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.158034\nβ : Type ?u.158037\nγ : Type ?u.158040\na b c✝ d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns✝ : Set ℝ≥0∞\nR : Type u_1\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ns : Set ℝ≥0∞\nc : R\n⊢ c • sSup s = ⨆ (i : ℝ≥0∞) (_ : i ∈ s), c ... | [
668,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
665,
1
] |
Mathlib/Data/QPF/Univariate/Basic.lean | Qpf.Fix.ind | [
{
"state_after": "case mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\np : Fix F → Prop\nh : ∀ (x : F (Fix F)), Liftp p x → p (mk x)\n⊢ ∀ (a : PFunctor.W (P F)), p (Quot.mk Setoid.r a)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\np : Fix F → Prop\nh : ∀ (x : F (Fix F)), Liftp... | [
360,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
351,
1
] |
Mathlib/Data/Finset/Sort.lean | Finset.sort_nodup | [
{
"state_after": "α : Type u_1\nβ : Type ?u.1443\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\ninst✝¹ : IsAntisymm α r\ninst✝ : IsTotal α r\ns : Finset α\n⊢ Nodup s.val",
"state_before": "α : Type u_1\nβ : Type ?u.1443\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\nins... | [
52,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.cast_castAdd_left | [
{
"state_after": "case h\nn✝ m✝ n n' m : ℕ\ni : Fin n'\nh : n' + m = n + m\n⊢ ↑(↑(cast h) (↑(castAdd m) i)) = ↑(↑(castAdd m) (↑(cast (_ : n' = n)) i))",
"state_before": "n✝ m✝ n n' m : ℕ\ni : Fin n'\nh : n' + m = n + m\n⊢ ↑(cast h) (↑(castAdd m) i) = ↑(castAdd m) (↑(cast (_ : n' = n)) i)",
"tactic": "ex... | [
1172,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1169,
1
] |
Mathlib/Analysis/Convex/Independent.lean | Convex.convexIndependent_extremePoints | [] | [
209,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean | DifferentiableAt.sub_const | [] | [
560,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
558,
1
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean | RingSubgroupsBasis.nonarchimedean | [
{
"state_after": "A : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\nthis : TopologicalSpace A := topology hB\n⊢ NonarchimedeanRing A",
"state_before": "A : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB ... | [
206,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean | Ordinal.type_lift_preimage_aux | [] | [
686,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
683,
1
] |
Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.coe_int_mem | [] | [
248,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
247,
11
] |
Mathlib/SetTheory/ZFC/Basic.lean | arity_succ | [] | [
84,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | CategoryTheory.Limits.prod.leftUnitor_inv_naturality | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX Y : C\ninst✝¹ : HasTerminal C\ninst✝ : HasBinaryProducts C\nf : X ⟶ Y\n⊢ (leftUnitor X).inv ≫ map (𝟙 (⊤_ C)) f = f ≫ (leftUnitor Y).inv",
"tactic": "rw [Iso.inv_comp_eq, ← Category.assoc, Iso.eq_comp_inv, prod.leftUnitor_h... | [
1083,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1081,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean | Subsemigroup.le_prod_iff | [
{
"state_after": "case mp\nM : Type u_1\nN : Type u_2\nP : Type ?u.58318\nσ : Type ?u.58321\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS s : Subsemigroup M\nt : Subsemigroup N\nu : Subsemigroup (M × N)\n⊢ u ≤ prod s t → map (fst M N) u ≤ s ∧ map (snd M N) u ≤ t\n\ncase mpr\nM : Type u_1\nN : Type u_2\nP : ... | [
739,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
729,
1
] |
Std/Logic.lean | and_assoc | [] | [
178,
75
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
177,
1
] |
Mathlib/Algebra/Category/Ring/Basic.lean | SemiRingCat.coe_of | [] | [
97,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
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