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/Algebra/Ring/Prod.lean
|
NonUnitalRingHom.coe_prodMap
|
[] |
[
176,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/MeasureTheory/Group/Prod.lean
|
MeasureTheory.measurePreserving_prod_mul_swap_right
|
[] |
[
380,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.range_nonempty_iff_nonempty
|
[] |
[
740,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
739,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Int.cast_finsupp_sum
|
[] |
[
413,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
|
Matrix.SpecialLinearGroup.coe_mul
|
[] |
[
146,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Algebra/Group/Commute.lean
|
Commute.isUnit_mul_iff
|
[] |
[
287,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.rpow_le_rpow_iff
|
[] |
[
574,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
573,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.mul_empty
|
[] |
[
362,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Order/Filter/Cofinite.lean
|
Function.Surjective.le_map_cofinite
|
[] |
[
193,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
pow_bit1_neg_iff
|
[] |
[
716,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
nndist_eq_nnnorm_div
|
[] |
[
906,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
905,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snormEssSup_const_smul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.5974976\nF : Type u_2\nG : Type ?u.5974982\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nc : 𝕜\nf : α → F\n⊢ snormEssSup (c • f) μ = ↑‖c‖₊ * snormEssSup f μ",
"tactic": "simp_rw [snormEssSup, Pi.smul_apply, nnnorm_smul, ENNReal.coe_mul, ENNReal.essSup_const_mul]"
}
] |
[
1565,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1563,
1
] |
Mathlib/Logic/Basic.lean
|
forall_of_ball
|
[] |
[
1065,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1065,
1
] |
Mathlib/Algebra/Order/AbsoluteValue.lean
|
AbsoluteValue.map_sub
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na b : R\n⊢ ↑abv (a - b) = ↑abv (b - a)",
"tactic": "rw [← neg_sub, abv.map_neg]"
}
] |
[
244,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
11
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.div_singleton
|
[] |
[
665,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/Data/Set/Intervals/SurjOn.lean
|
surjOn_Ioc_of_monotone_surjective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\n⊢ SurjOn f (Ioc a b) (Ioc (f a) (f b))",
"tactic": "simpa using surjOn_Ico_of_monotone_surjective h_mono.dual h_surj (toDual b) (toDual a)"
}
] |
[
53,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean
|
CategoryTheory.CostructuredArrow.right_eq_id
|
[] |
[
335,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.range_finite
|
[] |
[
379,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Algebra/Star/StarAlgHom.lean
|
NonUnitalStarAlgHom.fst_prod
|
[
{
"state_after": "case h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁹ : Monoid R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : DistribMulAction R A\ninst✝⁶ : Star A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : DistribMulAction R B\ninst✝³ : Star B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : DistribMulAction R C\ninst✝ : Star C\nf : A →⋆ₙₐ[R] B\ng : A →⋆ₙₐ[R] C\nx✝ : A\n⊢ ↑(comp (fst R B C) (prod f g)) x✝ = ↑f x✝",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁹ : Monoid R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : DistribMulAction R A\ninst✝⁶ : Star A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : DistribMulAction R B\ninst✝³ : Star B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : DistribMulAction R C\ninst✝ : Star C\nf : A →⋆ₙₐ[R] B\ng : A →⋆ₙₐ[R] C\n⊢ comp (fst R B C) (prod f g) = f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁹ : Monoid R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : DistribMulAction R A\ninst✝⁶ : Star A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : DistribMulAction R B\ninst✝³ : Star B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : DistribMulAction R C\ninst✝ : Star C\nf : A →⋆ₙₐ[R] B\ng : A →⋆ₙₐ[R] C\nx✝ : A\n⊢ ↑(comp (fst R B C) (prod f g)) x✝ = ↑f x✝",
"tactic": "rfl"
}
] |
[
534,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
star_id_of_comm
|
[] |
[
249,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Data/Finsupp/Multiset.lean
|
Multiset.toFinsupp_strictMono
|
[] |
[
239,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Topology/Algebra/StarSubalgebra.lean
|
StarSubalgebra.isClosed_topologicalClosure
|
[] |
[
92,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.mul_div_root_cancel
|
[] |
[
428,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/Init/Algebra/Order.lean
|
compare_lt_iff_lt
|
[
{
"state_after": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.lt ↔ a < b",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ compare a b = Ordering.lt ↔ a < b",
"tactic": "rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.lt ↔ a < b",
"tactic": "split_ifs <;> simp only [*, lt_irrefl]"
}
] |
[
426,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.pmap_eq_map
|
[] |
[
1514,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1512,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.apply_eq_of_preconnectedSpace
|
[] |
[
389,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.subtype_injective
|
[
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nP✝ : Sylow p G\nK : Type ?u.48738\ninst✝ : Group K\nϕ : K →* G\nN : Subgroup G\nP Q : Sylow p G\nhP : ↑P ≤ N\nhQ : ↑Q ≤ N\nh✝ : Sylow.subtype P hP = Sylow.subtype Q hQ\nh : ∀ (x : { x // x ∈ N }), x ∈ Sylow.subtype P hP ↔ x ∈ Sylow.subtype Q hQ\n⊢ ∀ (x : G), x ∈ P ↔ x ∈ Q",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nP✝ : Sylow p G\nK : Type ?u.48738\ninst✝ : Group K\nϕ : K →* G\nN : Subgroup G\nP Q : Sylow p G\nhP : ↑P ≤ N\nhQ : ↑Q ≤ N\nh : Sylow.subtype P hP = Sylow.subtype Q hQ\n⊢ P = Q",
"tactic": "rw [SetLike.ext_iff] at h⊢"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nP✝ : Sylow p G\nK : Type ?u.48738\ninst✝ : Group K\nϕ : K →* G\nN : Subgroup G\nP Q : Sylow p G\nhP : ↑P ≤ N\nhQ : ↑Q ≤ N\nh✝ : Sylow.subtype P hP = Sylow.subtype Q hQ\nh : ∀ (x : { x // x ∈ N }), x ∈ Sylow.subtype P hP ↔ x ∈ Sylow.subtype Q hQ\n⊢ ∀ (x : G), x ∈ P ↔ x ∈ Q",
"tactic": "exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩"
}
] |
[
143,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.pos
|
[] |
[
41,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_restrict_eq_bot
|
[] |
[
2911,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2910,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_zero
|
[
{
"state_after": "no goals",
"state_before": "⊢ arg 0 = 0",
"tactic": "simp [arg, le_refl]"
}
] |
[
123,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Data/Matrix/Rank.lean
|
Matrix.rank_reindex
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.175068\nm : Type u_1\nn : Type u_2\no : Type ?u.175077\nR : Type u_3\nm_fin : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : CommRing R\ninst✝ : Fintype m\ne₁ e₂ : m ≃ n\nA : Matrix m m R\n⊢ rank (↑(reindex e₁ e₂) A) = rank A",
"tactic": "rw [rank, rank, mulVecLin_reindex, LinearMap.range_comp, LinearMap.range_comp,\n LinearEquiv.range, Submodule.map_top, LinearEquiv.finrank_map_eq]"
}
] |
[
119,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.isCompact_iff_isClosed_bounded
|
[] |
[
2530,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2528,
1
] |
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
Asymptotics.SuperpolynomialDecay.mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : CommSemiring β\ninst✝ : ContinuousMul β\nhf : SuperpolynomialDecay l k f\nhg : SuperpolynomialDecay l k g\nz : ℕ\n⊢ Tendsto (fun a => k a ^ z * (f * g) a) l (𝓝 0)",
"tactic": "simpa only [mul_assoc, one_mul, MulZeroClass.mul_zero, pow_zero] using (hf z).mul (hg 0)"
}
] |
[
94,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Data/List/Basic.lean
|
List.getD_singleton_default_eq
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.502615\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✝ n : ℕ\n⊢ getD [d] n d = d",
"tactic": "cases n <;> simp"
}
] |
[
4417,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4417,
1
] |
Std/Data/String/Lemmas.lean
|
String.get'_eq
|
[] |
[
309,
81
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
309,
9
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.limit.pre_post
|
[
{
"state_after": "J : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF✝ : J ⥤ C\nD : Type u'\ninst✝³ : Category D\nE : K ⥤ J\nF : J ⥤ C\nG : C ⥤ D\ninst✝² : HasLimit F\ninst✝¹ : HasLimit (E ⋙ F)\ninst✝ : HasLimit (F ⋙ G)\nh : HasLimit ((E ⋙ F) ⋙ G)\nthis : HasLimit (E ⋙ F ⋙ G)\n⊢ G.map (pre F E) ≫ post (E ⋙ F) G = post F G ≫ pre (F ⋙ G) E",
"state_before": "J : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF✝ : J ⥤ C\nD : Type u'\ninst✝³ : Category D\nE : K ⥤ J\nF : J ⥤ C\nG : C ⥤ D\ninst✝² : HasLimit F\ninst✝¹ : HasLimit (E ⋙ F)\ninst✝ : HasLimit (F ⋙ G)\nh : HasLimit ((E ⋙ F) ⋙ G)\n⊢ G.map (pre F E) ≫ post (E ⋙ F) G = post F G ≫ pre (F ⋙ G) E",
"tactic": "haveI : HasLimit (E ⋙ F ⋙ G) := h"
},
{
"state_after": "case w\nJ : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF✝ : J ⥤ C\nD : Type u'\ninst✝³ : Category D\nE : K ⥤ J\nF : J ⥤ C\nG : C ⥤ D\ninst✝² : HasLimit F\ninst✝¹ : HasLimit (E ⋙ F)\ninst✝ : HasLimit (F ⋙ G)\nh : HasLimit ((E ⋙ F) ⋙ G)\nthis : HasLimit (E ⋙ F ⋙ G)\nj✝ : K\n⊢ (G.map (pre F E) ≫ post (E ⋙ F) G) ≫ π ((E ⋙ F) ⋙ G) j✝ = (post F G ≫ pre (F ⋙ G) E) ≫ π ((E ⋙ F) ⋙ G) j✝",
"state_before": "J : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF✝ : J ⥤ C\nD : Type u'\ninst✝³ : Category D\nE : K ⥤ J\nF : J ⥤ C\nG : C ⥤ D\ninst✝² : HasLimit F\ninst✝¹ : HasLimit (E ⋙ F)\ninst✝ : HasLimit (F ⋙ G)\nh : HasLimit ((E ⋙ F) ⋙ G)\nthis : HasLimit (E ⋙ F ⋙ G)\n⊢ G.map (pre F E) ≫ post (E ⋙ F) G = post F G ≫ pre (F ⋙ G) E",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w\nJ : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF✝ : J ⥤ C\nD : Type u'\ninst✝³ : Category D\nE : K ⥤ J\nF : J ⥤ C\nG : C ⥤ D\ninst✝² : HasLimit F\ninst✝¹ : HasLimit (E ⋙ F)\ninst✝ : HasLimit (F ⋙ G)\nh : HasLimit ((E ⋙ F) ⋙ G)\nthis : HasLimit (E ⋙ F ⋙ G)\nj✝ : K\n⊢ (G.map (pre F E) ≫ post (E ⋙ F) G) ≫ π ((E ⋙ F) ⋙ G) j✝ = (post F G ≫ pre (F ⋙ G) E) ≫ π ((E ⋙ F) ⋙ G) j✝",
"tactic": "erw [assoc, limit.post_π, ← G.map_comp, limit.pre_π, assoc, limit.pre_π, limit.post_π]"
}
] |
[
496,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.Pairwise.forall
|
[] |
[
115,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Order/MinMax.lean
|
max_le_max
|
[] |
[
76,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/Down.lean
|
Down.mem_compression_of_insert_mem_compression
|
[
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\nha : a ∈ s\n⊢ s ∈ 𝓓 a 𝒜\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\nha : ¬a ∈ s\n⊢ s ∈ 𝓓 a 𝒜",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\n⊢ s ∈ 𝓓 a 𝒜",
"tactic": "by_cases ha : a ∈ s"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\nha : a ∈ s\n⊢ s ∈ 𝓓 a 𝒜",
"tactic": "rwa [insert_eq_of_mem ha] at h"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\nha : ¬a ∈ s\n⊢ erase (insert a s) a ∈ 𝓓 a 𝒜",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\nha : ¬a ∈ s\n⊢ s ∈ 𝓓 a 𝒜",
"tactic": "rw [← erase_insert ha]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\nha : ¬a ∈ s\n⊢ erase (insert a s) a ∈ 𝓓 a 𝒜",
"tactic": "exact erase_mem_compression_of_mem_compression h"
}
] |
[
194,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
orderOf_eq_orderOf_iff
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y✝ : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : AddMonoid A\nH : Type u_1\ninst✝ : Monoid H\ny : H\n⊢ orderOf x = orderOf y ↔ ∀ (n : ℕ), x ^ n = 1 ↔ y ^ n = 1",
"tactic": "simp_rw [← isPeriodicPt_mul_iff_pow_eq_one, ← minimalPeriod_eq_minimalPeriod_iff, orderOf]"
}
] |
[
328,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
326,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
isLowerSet_preimage_ofDual_iff
|
[] |
[
175,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
CategoryTheory.Limits.IsBilimit.binary_total
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Preadditive C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nb : BinaryBicone X Y\ni : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n⊢ (b.fst ≫ b.inl + b.snd ≫ b.inr) ≫ (BinaryBicone.toCone b).π.app j = 𝟙 b.pt ≫ (BinaryBicone.toCone b).π.app j",
"tactic": "rcases j with ⟨⟨⟩⟩ <;> simp"
}
] |
[
324,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
src/lean/Init/SimpLemmas.lean
|
true_and
|
[] |
[
84,
88
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
84,
9
] |
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
PhragmenLindelof.isBigO_sub_exp_exp
|
[
{
"state_after": "E : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nhBf : ∃ c, c < a ∧ ∃ B, f =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nhBg : ∃ c, c < a ∧ ∃ B, g =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\n⊢ ∃ c, c < a ∧ ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))",
"state_before": "E : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nhBf : ∃ c, c < a ∧ ∃ B, f =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nhBg : ∃ c, c < a ∧ ∃ B, g =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\n⊢ ∃ c, c < a ∧ ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))",
"tactic": "have : ∀ {c₁ c₂ B₁ B₂}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ z,\n ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖ := fun hc hB₀ hB z ↦ by\n rw [Real.norm_eq_abs, Real.norm_eq_abs, Real.abs_exp, Real.abs_exp, Real.exp_le_exp]\n exact\n mul_le_mul hB (Real.exp_le_exp.2 <| mul_le_mul_of_nonneg_right hc <| abs_nonneg _)\n (Real.exp_pos _).le hB₀"
},
{
"state_after": "case intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nhBg : ∃ c, c < a ∧ ∃ B, g =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\n⊢ ∃ c, c < a ∧ ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))",
"state_before": "E : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nhBf : ∃ c, c < a ∧ ∃ B, f =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nhBg : ∃ c, c < a ∧ ∃ B, g =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\n⊢ ∃ c, c < a ∧ ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))",
"tactic": "rcases hBf with ⟨cf, hcf, Bf, hOf⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ ∃ c, c < a ∧ ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))",
"state_before": "case intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nhBg : ∃ c, c < a ∧ ∃ B, g =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\n⊢ ∃ c, c < a ∧ ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))",
"tactic": "rcases hBg with ⟨cg, hcg, Bg, hOg⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ (f - g) =O[l] fun z => expR (max 0 (max Bf Bg) * expR (max cf cg * Abs.abs (u z)))",
"state_before": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ ∃ c, c < a ∧ ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))",
"tactic": "refine' ⟨max cf cg, max_lt hcf hcg, max 0 (max Bf Bg), _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ cf ≤ max cf cg\n\ncase intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ 0 ≤ max 0 (max Bf Bg)\n\ncase intro.intro.intro.intro.intro.intro.refine'_3\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ Bf ≤ max 0 (max Bf Bg)\n\ncase intro.intro.intro.intro.intro.intro.refine'_4\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ cg ≤ max cf cg\n\ncase intro.intro.intro.intro.intro.intro.refine'_5\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ 0 ≤ max 0 (max Bf Bg)\n\ncase intro.intro.intro.intro.intro.intro.refine'_6\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ Bg ≤ max 0 (max Bf Bg)",
"state_before": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ (f - g) =O[l] fun z => expR (max 0 (max Bf Bg) * expR (max cf cg * Abs.abs (u z)))",
"tactic": "refine' (hOf.trans_le <| this _ _ _).sub (hOg.trans_le <| this _ _ _)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ cf ≤ max cf cg\n\ncase intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ 0 ≤ max 0 (max Bf Bg)\n\ncase intro.intro.intro.intro.intro.intro.refine'_3\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ Bf ≤ max 0 (max Bf Bg)\n\ncase intro.intro.intro.intro.intro.intro.refine'_4\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ cg ≤ max cf cg\n\ncase intro.intro.intro.intro.intro.intro.refine'_5\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ 0 ≤ max 0 (max Bf Bg)\n\ncase intro.intro.intro.intro.intro.intro.refine'_6\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nthis :\n ∀ {c₁ c₂ B₁ B₂ : ℝ},\n c₁ ≤ c₂ →\n 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * Abs.abs (u z)))‖ ≤ ‖expR (B₂ * expR (c₂ * Abs.abs (u z)))‖\ncf : ℝ\nhcf : cf < a\nBf : ℝ\nhOf : f =O[l] fun z => expR (Bf * expR (cf * Abs.abs (u z)))\ncg : ℝ\nhcg : cg < a\nBg : ℝ\nhOg : g =O[l] fun z => expR (Bg * expR (cg * Abs.abs (u z)))\n⊢ Bg ≤ max 0 (max Bf Bg)",
"tactic": "exacts [le_max_left _ _, le_max_left _ _, (le_max_left _ _).trans (le_max_right _ _),\n le_max_right _ _, le_max_left _ _, (le_max_right _ _).trans (le_max_right _ _)]"
},
{
"state_after": "E : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nhBf : ∃ c, c < a ∧ ∃ B, f =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nhBg : ∃ c, c < a ∧ ∃ B, g =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nc₁✝ c₂✝ B₁✝ B₂✝ : ℝ\nhc : c₁✝ ≤ c₂✝\nhB₀ : 0 ≤ B₂✝\nhB : B₁✝ ≤ B₂✝\nz : ℂ\n⊢ B₁✝ * expR (c₁✝ * Abs.abs (u z)) ≤ B₂✝ * expR (c₂✝ * Abs.abs (u z))",
"state_before": "E : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nhBf : ∃ c, c < a ∧ ∃ B, f =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nhBg : ∃ c, c < a ∧ ∃ B, g =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nc₁✝ c₂✝ B₁✝ B₂✝ : ℝ\nhc : c₁✝ ≤ c₂✝\nhB₀ : 0 ≤ B₂✝\nhB : B₁✝ ≤ B₂✝\nz : ℂ\n⊢ ‖expR (B₁✝ * expR (c₁✝ * Abs.abs (u z)))‖ ≤ ‖expR (B₂✝ * expR (c₂✝ * Abs.abs (u z)))‖",
"tactic": "rw [Real.norm_eq_abs, Real.norm_eq_abs, Real.abs_exp, Real.abs_exp, Real.exp_le_exp]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nu : ℂ → ℝ\nhBf : ∃ c, c < a ∧ ∃ B, f =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nhBg : ∃ c, c < a ∧ ∃ B, g =O[l] fun z => expR (B * expR (c * Abs.abs (u z)))\nc₁✝ c₂✝ B₁✝ B₂✝ : ℝ\nhc : c₁✝ ≤ c₂✝\nhB₀ : 0 ≤ B₂✝\nhB : B₁✝ ≤ B₂✝\nz : ℂ\n⊢ B₁✝ * expR (c₁✝ * Abs.abs (u z)) ≤ B₂✝ * expR (c₂✝ * Abs.abs (u z))",
"tactic": "exact\n mul_le_mul hB (Real.exp_le_exp.2 <| mul_le_mul_of_nonneg_right hc <| abs_nonneg _)\n (Real.exp_pos _).le hB₀"
}
] |
[
81,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
RelEmbedding.isLinearOrder
|
[] |
[
375,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
11
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
Ring.DimensionLeOne.prime_le_prime_iff_eq
|
[] |
[
1200,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1198,
1
] |
Mathlib/Computability/Language.lean
|
Language.mem_mul
|
[] |
[
117,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Dynamics/Ergodic/Ergodic.lean
|
PreErgodic.of_iterate
|
[] |
[
79,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Computability/Encoding.lean
|
Computability.encodePosNum_nonempty
|
[] |
[
133,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/RepresentationTheory/Action.lean
|
Action.ρ_one
|
[
{
"state_after": "V : Type (u + 1)\ninst✝ : LargeCategory V\nG : MonCat\nA : Action V G\n⊢ 1 = 𝟙 A.V",
"state_before": "V : Type (u + 1)\ninst✝ : LargeCategory V\nG : MonCat\nA : Action V G\n⊢ ↑A.ρ 1 = 𝟙 A.V",
"tactic": "rw [MonoidHom.map_one]"
},
{
"state_after": "no goals",
"state_before": "V : Type (u + 1)\ninst✝ : LargeCategory V\nG : MonCat\nA : Action V G\n⊢ 1 = 𝟙 A.V",
"tactic": "rfl"
}
] |
[
67,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.totalDegree_multiset_prod
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.471246\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : Multiset (MvPolynomial σ R)\nl : List (MvPolynomial σ R)\n⊢ totalDegree (Multiset.prod (Quotient.mk (List.isSetoid (MvPolynomial σ R)) l)) ≤\n Multiset.sum (Multiset.map totalDegree (Quotient.mk (List.isSetoid (MvPolynomial σ R)) l))",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.471246\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : Multiset (MvPolynomial σ R)\n⊢ totalDegree (Multiset.prod s) ≤ Multiset.sum (Multiset.map totalDegree s)",
"tactic": "refine' Quotient.inductionOn s fun l => _"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.471246\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : Multiset (MvPolynomial σ R)\nl : List (MvPolynomial σ R)\n⊢ totalDegree (List.prod l) ≤ List.sum (List.map totalDegree l)",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.471246\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : Multiset (MvPolynomial σ R)\nl : List (MvPolynomial σ R)\n⊢ totalDegree (Multiset.prod (Quotient.mk (List.isSetoid (MvPolynomial σ R)) l)) ≤\n Multiset.sum (Multiset.map totalDegree (Quotient.mk (List.isSetoid (MvPolynomial σ R)) l))",
"tactic": "rw [Multiset.quot_mk_to_coe, Multiset.coe_prod, Multiset.coe_map, Multiset.coe_sum]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.471246\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : Multiset (MvPolynomial σ R)\nl : List (MvPolynomial σ R)\n⊢ totalDegree (List.prod l) ≤ List.sum (List.map totalDegree l)",
"tactic": "exact totalDegree_list_prod l"
}
] |
[
740,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
736,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_congr
|
[
{
"state_after": "R✝ : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R✝\nm n : ℕ\ninst✝³ : Semiring R✝\np q r : R✝[X]\ninst✝² : Semiring S✝\nf✝ : R✝ →+* S✝\nx : S✝\nR : Type u_1\nS : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\ns : S\nφ : R[X]\n⊢ eval₂ f s φ = eval₂ f s φ",
"state_before": "R✝ : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R✝\nm n : ℕ\ninst✝³ : Semiring R✝\np q r : R✝[X]\ninst✝² : Semiring S✝\nf✝ : R✝ →+* S✝\nx : S✝\nR : Type u_1\nS : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf g : R →+* S\ns t : S\nφ ψ : R[X]\n⊢ f = g → s = t → φ = ψ → eval₂ f s φ = eval₂ g t ψ",
"tactic": "rintro rfl rfl rfl"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R✝\nm n : ℕ\ninst✝³ : Semiring R✝\np q r : R✝[X]\ninst✝² : Semiring S✝\nf✝ : R✝ →+* S✝\nx : S✝\nR : Type u_1\nS : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\ns : S\nφ : R[X]\n⊢ eval₂ f s φ = eval₂ f s φ",
"tactic": "rfl"
}
] |
[
58,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Data/Complex/Module.lean
|
Complex.coe_algebraMap
|
[] |
[
112,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeCommRing.isSupported_sub
|
[] |
[
198,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.eq_of_eq
|
[] |
[
444,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.constr_range
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.531644\nR : Type u_3\nR₂ : Type ?u.531650\nK : Type ?u.531653\nM : Type u_4\nM' : Type u_2\nM'' : Type ?u.531662\nV : Type u\nV' : Type ?u.531667\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx : M\nS : Type u_5\ninst✝³ : Semiring S\ninst✝² : Module S M'\ninst✝¹ : SMulCommClass R S M'\ninst✝ : Nonempty ι\nf : ι → M'\n⊢ LinearMap.range (↑(constr b S) f) = span R (range f)",
"tactic": "rw [b.constr_def S f, LinearMap.range_comp, LinearMap.range_comp, LinearEquiv.range, ←\n Finsupp.supported_univ, Finsupp.lmapDomain_supported, ← Set.image_univ, ←\n Finsupp.span_image_eq_map_total, Set.image_id]"
}
] |
[
653,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.mem_prod
|
[] |
[
1659,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1658,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.cons_vecAlt1
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\n⊢ m = n + n",
"tactic": "rwa [add_assoc n, add_comm 1, ← add_assoc, ← add_assoc, add_right_cancel_iff,\n add_right_cancel_iff] at h"
},
{
"state_after": "case h\nα : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\ni : Fin (n + 1)\n⊢ vecAlt1 h (vecCons x (vecCons y u)) i = vecCons y (vecAlt1 (_ : m = n + n) u) i",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\n⊢ vecAlt1 h (vecCons x (vecCons y u)) = vecCons y (vecAlt1 (_ : m = n + n) u)",
"tactic": "ext i"
},
{
"state_after": "case h\nα : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\ni : Fin (n + 1)\n⊢ vecCons x (vecCons y u) { val := ↑i + ↑i + 1, isLt := (_ : ↑i + ↑i + 1 < m + 1 + 1) } =\n vecCons y (fun k => u { val := ↑k + ↑k + 1, isLt := (_ : ↑k + ↑k + 1 < m) }) i",
"state_before": "case h\nα : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\ni : Fin (n + 1)\n⊢ vecAlt1 h (vecCons x (vecCons y u)) i = vecCons y (vecAlt1 (_ : m = n + n) u) i",
"tactic": "simp_rw [vecAlt1]"
},
{
"state_after": "case h.mk.zero\nα : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\nhi : Nat.zero < n + 1\n⊢ vecCons x (vecCons y u)\n { val := ↑{ val := Nat.zero, isLt := hi } + ↑{ val := Nat.zero, isLt := hi } + 1,\n isLt := (_ : ↑{ val := Nat.zero, isLt := hi } + ↑{ val := Nat.zero, isLt := hi } + 1 < m + 1 + 1) } =\n vecCons y (fun k => u { val := ↑k + ↑k + 1, isLt := (_ : ↑k + ↑k + 1 < m) }) { val := Nat.zero, isLt := hi }\n\ncase h.mk.succ\nα : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\ni : ℕ\nhi : Nat.succ i < n + 1\n⊢ vecCons x (vecCons y u)\n { val := ↑{ val := Nat.succ i, isLt := hi } + ↑{ val := Nat.succ i, isLt := hi } + 1,\n isLt := (_ : ↑{ val := Nat.succ i, isLt := hi } + ↑{ val := Nat.succ i, isLt := hi } + 1 < m + 1 + 1) } =\n vecCons y (fun k => u { val := ↑k + ↑k + 1, isLt := (_ : ↑k + ↑k + 1 < m) }) { val := Nat.succ i, isLt := hi }",
"state_before": "case h\nα : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\ni : Fin (n + 1)\n⊢ vecCons x (vecCons y u) { val := ↑i + ↑i + 1, isLt := (_ : ↑i + ↑i + 1 < m + 1 + 1) } =\n vecCons y (fun k => u { val := ↑k + ↑k + 1, isLt := (_ : ↑k + ↑k + 1 < m) }) i",
"tactic": "rcases i with ⟨⟨⟩ | i, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk.zero\nα : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\nhi : Nat.zero < n + 1\n⊢ vecCons x (vecCons y u)\n { val := ↑{ val := Nat.zero, isLt := hi } + ↑{ val := Nat.zero, isLt := hi } + 1,\n isLt := (_ : ↑{ val := Nat.zero, isLt := hi } + ↑{ val := Nat.zero, isLt := hi } + 1 < m + 1 + 1) } =\n vecCons y (fun k => u { val := ↑k + ↑k + 1, isLt := (_ : ↑k + ↑k + 1 < m) }) { val := Nat.zero, isLt := hi }",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h.mk.succ\nα : Type u\nm n o : ℕ\nm' : Type ?u.52150\nn' : Type ?u.52153\no' : Type ?u.52156\nh : m + 1 + 1 = n + 1 + (n + 1)\nx y : α\nu : Fin m → α\ni : ℕ\nhi : Nat.succ i < n + 1\n⊢ vecCons x (vecCons y u)\n { val := ↑{ val := Nat.succ i, isLt := hi } + ↑{ val := Nat.succ i, isLt := hi } + 1,\n isLt := (_ : ↑{ val := Nat.succ i, isLt := hi } + ↑{ val := Nat.succ i, isLt := hi } + 1 < m + 1 + 1) } =\n vecCons y (fun k => u { val := ↑k + ↑k + 1, isLt := (_ : ↑k + ↑k + 1 < m) }) { val := Nat.succ i, isLt := hi }",
"tactic": "simp [vecAlt1, Nat.add_succ, Nat.succ_add]"
}
] |
[
419,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.iSup_iInf_of_antitone
|
[] |
[
1507,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1504,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.induce_spanningCoe
|
[] |
[
1321,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1320,
1
] |
Mathlib/Data/Option/NAry.lean
|
Option.map₂_right_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_5\nβ : Type u_6\nγ : Type u_4\nf✝ : α → β → γ\na : Option α\nb : Option β\nc : Option γ\nδ : Type u_1\nε : Type u_2\nδ' : Type u_3\nf : δ → γ → ε\ng : α → β → δ\nf' : α → γ → δ'\ng' : δ' → β → ε\nh_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b\n⊢ map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b",
"tactic": "cases a <;> cases b <;> cases c <;> simp [h_right_comm]"
}
] |
[
143,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.gcd_gcd_self_left_left
|
[
{
"state_after": "no goals",
"state_before": "m n : Nat\n⊢ gcd (gcd m n) m = gcd m n",
"tactic": "rw [gcd_comm m n, gcd_gcd_self_left_right]"
}
] |
[
171,
45
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
170,
9
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_ennreal_ne_zero
|
[] |
[
531,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/Algebra/CharP/MixedCharZero.lean
|
split_equalCharZero_mixedCharZero
|
[
{
"state_after": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nh : ∃ p, p > 0 ∧ MixedCharZero R p\n⊢ P\n\ncase neg\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nh : ¬∃ p, p > 0 ∧ MixedCharZero R p\n⊢ P",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\n⊢ P",
"tactic": "by_cases h : ∃ p > 0, MixedCharZero R p"
},
{
"state_after": "case pos.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\np : ℕ\nH : p > 0\nhp : MixedCharZero R p\n⊢ P",
"state_before": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nh : ∃ p, p > 0 ∧ MixedCharZero R p\n⊢ P",
"tactic": "rcases h with ⟨p, ⟨H, hp⟩⟩"
},
{
"state_after": "case pos.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), p > 0 → MixedCharZero R p → P\np : ℕ\nH : p > 0\nhp : MixedCharZero R p\n⊢ P",
"state_before": "case pos.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\np : ℕ\nH : p > 0\nhp : MixedCharZero R p\n⊢ P",
"tactic": "rw [← MixedCharZero.reduce_to_p_prime] at h_mixed"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), p > 0 → MixedCharZero R p → P\np : ℕ\nH : p > 0\nhp : MixedCharZero R p\n⊢ P",
"tactic": "exact h_mixed p H hp"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nh : ¬∃ p, p > 0 ∧ MixedCharZero R p\n⊢ Algebra ℚ R",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nh : ¬∃ p, p > 0 ∧ MixedCharZero R p\n⊢ P",
"tactic": "apply h_equal"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nh : Nonempty (Algebra ℚ R)\n⊢ Algebra ℚ R",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nh : ¬∃ p, p > 0 ∧ MixedCharZero R p\n⊢ Algebra ℚ R",
"tactic": "rw [← isEmpty_algebraRat_iff_mixedCharZero, not_isEmpty_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\nP : Prop\ninst✝ : CharZero R\nh_equal : Algebra ℚ R → P\nh_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nh : Nonempty (Algebra ℚ R)\n⊢ Algebra ℚ R",
"tactic": "exact h.some"
}
] |
[
337,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.coe_pointwise_smul
|
[] |
[
373,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
|
CategoryTheory.NonPreadditiveAbelian.add_neg
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n⊢ a + -b = a - b",
"tactic": "rw [add_def, neg_neg]"
}
] |
[
400,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Data/UInt.lean
|
UInt8.toChar_aux
|
[
{
"state_after": "n : ℕ\nh : n < size\n⊢ Nat.isValidChar n\n\nn : ℕ\nh : n < size\n⊢ n < UInt32.size",
"state_before": "n : ℕ\nh : n < size\n⊢ Nat.isValidChar ↑(UInt32.ofNat n).val",
"tactic": "rw [UInt32.val_eq_of_lt]"
},
{
"state_after": "n : ℕ\nh : n < size\n⊢ n < UInt32.size",
"state_before": "n : ℕ\nh : n < size\n⊢ Nat.isValidChar n\n\nn : ℕ\nh : n < size\n⊢ n < UInt32.size",
"tactic": "exact Or.inl $ Nat.lt_trans h $ by decide"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh : n < size\n⊢ n < UInt32.size",
"tactic": "exact Nat.lt_trans h $ by decide"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh : n < size\n⊢ size < 55296",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh : n < size\n⊢ size < UInt32.size",
"tactic": "decide"
}
] |
[
124,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.image_const_sub_Ico
|
[
{
"state_after": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ico b c = Ioc (a - c) (a - b)",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => a - x) '' Ico b c = Ioc (a - c) (a - b)",
"tactic": "have := image_comp (fun x => a + x) fun x => -x"
},
{
"state_after": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ico b c = Ioc (a - c) (a - b)",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ico b c = Ioc (a - c) (a - b)",
"tactic": "dsimp [Function.comp] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ico b c = Ioc (a - c) (a - b)",
"tactic": "simp [sub_eq_add_neg, this, add_comm]"
}
] |
[
353,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PushoutCocone.mk_ι_app_left
|
[] |
[
840,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/Combinatorics/Hall/Basic.lean
|
Fintype.all_card_le_filter_rel_iff_exists_injective
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"tactic": "haveI := Classical.decEq β"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"tactic": "let r' a := univ.filter fun b => r a b"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nh : ∀ (A : Finset α), filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"tactic": "have h : ∀ A : Finset α, (univ.filter fun b : β => ∃ a ∈ A, r a b) = A.biUnion r' := by\n intro A\n ext b\n simp"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nh : ∀ (A : Finset α), filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'\nh' : ∀ (f : α → β) (x : α), r x (f x) ↔ f x ∈ r' x\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nh : ∀ (A : Finset α), filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"tactic": "have h' : ∀ (f : α → β) (x), r x (f x) ↔ f x ∈ r' x := by simp"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nh : ∀ (A : Finset α), filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'\nh' : ∀ (f : α → β) (x : α), r x (f x) ↔ f x ∈ r' x\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (Finset.biUnion A fun a => filter (fun b => r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), f x ∈ filter (fun b => r x b) univ",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nh : ∀ (A : Finset α), filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'\nh' : ∀ (f : α → β) (x : α), r x (f x) ↔ f x ∈ r' x\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (filter (fun b => ∃ a, a ∈ A ∧ r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)",
"tactic": "simp_rw [h, h']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nh : ∀ (A : Finset α), filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'\nh' : ∀ (f : α → β) (x : α), r x (f x) ↔ f x ∈ r' x\n⊢ (∀ (A : Finset α), Finset.card A ≤ Finset.card (Finset.biUnion A fun a => filter (fun b => r a b) univ)) ↔\n ∃ f, Function.Injective f ∧ ∀ (x : α), f x ∈ filter (fun b => r x b) univ",
"tactic": "apply Finset.all_card_le_biUnion_card_iff_exists_injective"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nA : Finset α\n⊢ filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\n⊢ ∀ (A : Finset α), filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'",
"tactic": "intro A"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nA : Finset α\nb : β\n⊢ b ∈ filter (fun b => ∃ a, a ∈ A ∧ r a b) univ ↔ b ∈ Finset.biUnion A r'",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nA : Finset α\n⊢ filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'",
"tactic": "ext b"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nA : Finset α\nb : β\n⊢ b ∈ filter (fun b => ∃ a, a ∈ A ∧ r a b) univ ↔ b ∈ Finset.biUnion A r'",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nthis : DecidableEq β\nr' : α → Finset β := fun a => filter (fun b => r a b) univ\nh : ∀ (A : Finset α), filter (fun b => ∃ a, a ∈ A ∧ r a b) univ = Finset.biUnion A r'\n⊢ ∀ (f : α → β) (x : α), r x (f x) ↔ f x ∈ r' x",
"tactic": "simp"
}
] |
[
228,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
NNReal.tsum_eq_add_tsum_ite
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.330764\nγ : Type ?u.330767\nf : α → ℝ≥0\nhf : Summable f\ni i' : α\n⊢ update (fun x => f x) i 0 i' ≤ f i'",
"state_before": "α : Type u_1\nβ : Type ?u.330764\nγ : Type ?u.330767\nf : α → ℝ≥0\nhf : Summable f\ni : α\n⊢ (∑' (x : α), f x) = f i + ∑' (x : α), if x = i then 0 else f x",
"tactic": "refine' tsum_eq_add_tsum_ite' i (NNReal.summable_of_le (fun i' => _) hf)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.330764\nγ : Type ?u.330767\nf : α → ℝ≥0\nhf : Summable f\ni i' : α\n⊢ (if i' = i then 0 else f i') ≤ f i'",
"state_before": "α : Type u_1\nβ : Type ?u.330764\nγ : Type ?u.330767\nf : α → ℝ≥0\nhf : Summable f\ni i' : α\n⊢ update (fun x => f x) i 0 i' ≤ f i'",
"tactic": "rw [Function.update_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.330764\nγ : Type ?u.330767\nf : α → ℝ≥0\nhf : Summable f\ni i' : α\n⊢ (if i' = i then 0 else f i') ≤ f i'",
"tactic": "split_ifs <;> simp only [zero_le', le_rfl]"
}
] |
[
1234,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1230,
1
] |
Mathlib/Order/MinMax.lean
|
le_max_iff
|
[] |
[
42,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/Topology/Algebra/Order/Compact.lean
|
Continuous.exists_forall_ge
|
[] |
[
333,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Filter.EventuallyEq.trans_isLittleO
|
[] |
[
383,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.opow_mul
|
[
{
"state_after": "case pos\na b c : Ordinal\nb0 : b = 0\n⊢ a ^ (b * c) = (a ^ b) ^ c\n\ncase neg\na b c : Ordinal\nb0 : ¬b = 0\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"state_before": "a b c : Ordinal\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"tactic": "by_cases b0 : b = 0"
},
{
"state_after": "case pos\na b c : Ordinal\nb0 : ¬b = 0\na0 : a = 0\n⊢ a ^ (b * c) = (a ^ b) ^ c\n\ncase neg\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"state_before": "case neg\na b c : Ordinal\nb0 : ¬b = 0\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"tactic": "by_cases a0 : a = 0"
},
{
"state_after": "case neg.inl\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 = a\n⊢ a ^ (b * c) = (a ^ b) ^ c\n\ncase neg.inr\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"state_before": "case neg\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"tactic": "cases' eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with a1 a1"
},
{
"state_after": "case neg.inr.H₁\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\n⊢ a ^ (b * 0) = (a ^ b) ^ 0\n\ncase neg.inr.H₂\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\n⊢ ∀ (o : Ordinal), a ^ (b * o) = (a ^ b) ^ o → a ^ (b * succ o) = (a ^ b) ^ succ o\n\ncase neg.inr.H₃\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\n⊢ ∀ (o : Ordinal), IsLimit o → (∀ (o' : Ordinal), o' < o → a ^ (b * o') = (a ^ b) ^ o') → a ^ (b * o) = (a ^ b) ^ o",
"state_before": "case neg.inr\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"tactic": "apply limitRecOn c"
},
{
"state_after": "no goals",
"state_before": "case pos\na b c : Ordinal\nb0 : b = 0\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"tactic": "simp only [b0, zero_mul, opow_zero, one_opow]"
},
{
"state_after": "case pos\nb c : Ordinal\nb0 : ¬b = 0\n⊢ 0 ^ (b * c) = (0 ^ b) ^ c",
"state_before": "case pos\na b c : Ordinal\nb0 : ¬b = 0\na0 : a = 0\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"tactic": "subst a"
},
{
"state_after": "case pos\nb c : Ordinal\nb0 : ¬b = 0\nc0 : c = 0\n⊢ 0 ^ (b * c) = (0 ^ b) ^ c\n\ncase neg\nb c : Ordinal\nb0 : ¬b = 0\nc0 : ¬c = 0\n⊢ 0 ^ (b * c) = (0 ^ b) ^ c",
"state_before": "case pos\nb c : Ordinal\nb0 : ¬b = 0\n⊢ 0 ^ (b * c) = (0 ^ b) ^ c",
"tactic": "by_cases c0 : c = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nb c : Ordinal\nb0 : ¬b = 0\nc0 : ¬c = 0\n⊢ 0 ^ (b * c) = (0 ^ b) ^ c",
"tactic": "simp only [zero_opow b0, zero_opow c0, zero_opow (mul_ne_zero b0 c0)]"
},
{
"state_after": "no goals",
"state_before": "case pos\nb c : Ordinal\nb0 : ¬b = 0\nc0 : c = 0\n⊢ 0 ^ (b * c) = (0 ^ b) ^ c",
"tactic": "simp only [c0, mul_zero, opow_zero]"
},
{
"state_after": "case neg.inl\nb c : Ordinal\nb0 : ¬b = 0\na0 : ¬1 = 0\n⊢ 1 ^ (b * c) = (1 ^ b) ^ c",
"state_before": "case neg.inl\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 = a\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"tactic": "subst a1"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nb c : Ordinal\nb0 : ¬b = 0\na0 : ¬1 = 0\n⊢ 1 ^ (b * c) = (1 ^ b) ^ c",
"tactic": "simp only [one_opow]"
},
{
"state_after": "no goals",
"state_before": "case neg.inr.H₁\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\n⊢ a ^ (b * 0) = (a ^ b) ^ 0",
"tactic": "simp only [mul_zero, opow_zero]"
},
{
"state_after": "case neg.inr.H₂\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nIH : a ^ (b * c) = (a ^ b) ^ c\n⊢ a ^ (b * succ c) = (a ^ b) ^ succ c",
"state_before": "case neg.inr.H₂\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\n⊢ ∀ (o : Ordinal), a ^ (b * o) = (a ^ b) ^ o → a ^ (b * succ o) = (a ^ b) ^ succ o",
"tactic": "intro c IH"
},
{
"state_after": "no goals",
"state_before": "case neg.inr.H₂\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nIH : a ^ (b * c) = (a ^ b) ^ c\n⊢ a ^ (b * succ c) = (a ^ b) ^ succ c",
"tactic": "rw [mul_succ, opow_add, IH, opow_succ]"
},
{
"state_after": "case neg.inr.H₃\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nl : IsLimit c\nIH : ∀ (o' : Ordinal), o' < c → a ^ (b * o') = (a ^ b) ^ o'\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"state_before": "case neg.inr.H₃\na b c : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\n⊢ ∀ (o : Ordinal), IsLimit o → (∀ (o' : Ordinal), o' < o → a ^ (b * o') = (a ^ b) ^ o') → a ^ (b * o) = (a ^ b) ^ o",
"tactic": "intro c l IH"
},
{
"state_after": "case neg.inr.H₃\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nl : IsLimit c\nIH : ∀ (o' : Ordinal), o' < c → a ^ (b * o') = (a ^ b) ^ o'\nd : Ordinal\n⊢ (∀ (b_1 : Ordinal), b_1 < c → ((fun x x_1 => x ^ x_1) a ∘ (fun x x_1 => x * x_1) b) b_1 ≤ d) ↔ (a ^ b) ^ c ≤ d",
"state_before": "case neg.inr.H₃\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nl : IsLimit c\nIH : ∀ (o' : Ordinal), o' < c → a ^ (b * o') = (a ^ b) ^ o'\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"tactic": "refine'\n eq_of_forall_ge_iff fun d =>\n (((opow_isNormal a1).trans (mul_isNormal (Ordinal.pos_iff_ne_zero.2 b0))).limit_le\n l).trans\n _"
},
{
"state_after": "case neg.inr.H₃\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nl : IsLimit c\nIH : ∀ (o' : Ordinal), o' < c → a ^ (b * o') = (a ^ b) ^ o'\nd : Ordinal\n⊢ (∀ (b_1 : Ordinal), b_1 < c → a ^ (b * b_1) ≤ d) ↔ (a ^ b) ^ c ≤ d",
"state_before": "case neg.inr.H₃\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nl : IsLimit c\nIH : ∀ (o' : Ordinal), o' < c → a ^ (b * o') = (a ^ b) ^ o'\nd : Ordinal\n⊢ (∀ (b_1 : Ordinal), b_1 < c → ((fun x x_1 => x ^ x_1) a ∘ (fun x x_1 => x * x_1) b) b_1 ≤ d) ↔ (a ^ b) ^ c ≤ d",
"tactic": "dsimp only [Function.comp]"
},
{
"state_after": "case neg.inr.H₃\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nl : IsLimit c\nIH : ∀ (o' : Ordinal), o' < c → a ^ (b * o') = (a ^ b) ^ o'\nd : Ordinal\n⊢ (∀ (b_1 : Ordinal), b_1 < c → (a ^ b) ^ b_1 ≤ d) ↔ (a ^ b) ^ c ≤ d",
"state_before": "case neg.inr.H₃\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nl : IsLimit c\nIH : ∀ (o' : Ordinal), o' < c → a ^ (b * o') = (a ^ b) ^ o'\nd : Ordinal\n⊢ (∀ (b_1 : Ordinal), b_1 < c → a ^ (b * b_1) ≤ d) ↔ (a ^ b) ^ c ≤ d",
"tactic": "simp (config := { contextual := true }) only [IH]"
},
{
"state_after": "no goals",
"state_before": "case neg.inr.H₃\na b c✝ : Ordinal\nb0 : ¬b = 0\na0 : ¬a = 0\na1 : 1 < a\nc : Ordinal\nl : IsLimit c\nIH : ∀ (o' : Ordinal), o' < c → a ^ (b * o') = (a ^ b) ^ o'\nd : Ordinal\n⊢ (∀ (b_1 : Ordinal), b_1 < c → (a ^ b) ^ b_1 ≤ d) ↔ (a ^ b) ^ c ≤ d",
"tactic": "exact (opow_le_of_limit (opow_ne_zero _ a0) l).symm"
}
] |
[
248,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.insert'_eq_insertWith
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nsize✝ : ℕ\nl : Ordnode α\ny : α\nr : Ordnode α\n⊢ (match cmpLE x y with\n | Ordering.lt => balanceL (insert' x l) y r\n | Ordering.eq => node size✝ l y r\n | Ordering.gt => balanceR l y (insert' x r)) =\n match cmpLE x y with\n | Ordering.lt => balanceL (insertWith id x l) y r\n | Ordering.eq => node size✝ l (id y) r\n | Ordering.gt => balanceR l y (insertWith id x r)",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nsize✝ : ℕ\nl : Ordnode α\ny : α\nr : Ordnode α\n⊢ insert' x (node size✝ l y r) = insertWith id x (node size✝ l y r)",
"tactic": "unfold insert' insertWith"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nsize✝ : ℕ\nl : Ordnode α\ny : α\nr : Ordnode α\n⊢ (match cmpLE x y with\n | Ordering.lt => balanceL (insert' x l) y r\n | Ordering.eq => node size✝ l y r\n | Ordering.gt => balanceR l y (insert' x r)) =\n match cmpLE x y with\n | Ordering.lt => balanceL (insertWith id x l) y r\n | Ordering.eq => node size✝ l (id y) r\n | Ordering.gt => balanceR l y (insertWith id x r)",
"tactic": "cases cmpLE x y <;> simp [insert'_eq_insertWith]"
}
] |
[
1556,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1552,
1
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseInducing.tendsto_comap_nhds_nhds
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map g (comap g (𝓝 d)) ≤ 𝓝 d\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"tactic": "have lim1 : map g (comap g (𝓝 d)) ≤ 𝓝 d := map_comap_le"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map h (map g (comap g (𝓝 d))) ≤ map h (𝓝 d)\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map g (comap g (𝓝 d)) ≤ 𝓝 d\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"tactic": "replace lim1 : map h (map g (comap g (𝓝 d))) ≤ map h (𝓝 d) := map_mono lim1"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map f (comap g (𝓝 d)) ≤ comap i (map h (𝓝 d))\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map h (map g (comap g (𝓝 d))) ≤ map h (𝓝 d)\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"tactic": "rw [Filter.map_map, comm, ← Filter.map_map, map_le_iff_le_comap] at lim1"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map f (comap g (𝓝 d)) ≤ comap i (map h (𝓝 d))\nlim2 : comap i (map h (𝓝 d)) ≤ comap i (𝓝 (i a))\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map f (comap g (𝓝 d)) ≤ comap i (map h (𝓝 d))\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"tactic": "have lim2 : comap i (map h (𝓝 d)) ≤ comap i (𝓝 (i a)) := comap_mono H"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map f (comap g (𝓝 d)) ≤ comap i (map h (𝓝 d))\nlim2 : comap i (map h (𝓝 d)) ≤ 𝓝 a\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map f (comap g (𝓝 d)) ≤ comap i (map h (𝓝 d))\nlim2 : comap i (map h (𝓝 d)) ≤ comap i (𝓝 (i a))\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"tactic": "rw [← di.nhds_eq_comap] at lim2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_4\nδ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : DenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map f (comap g (𝓝 d)) ≤ comap i (map h (𝓝 d))\nlim2 : comap i (map h (𝓝 d)) ≤ 𝓝 a\n⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)",
"tactic": "exact le_trans lim1 lim2"
}
] |
[
128,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Data/Semiquot.lean
|
Semiquot.eq_mk_of_mem
|
[] |
[
66,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Order/Lattice.lean
|
inf_left_comm
|
[] |
[
524,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_unit
|
[] |
[
415,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean
|
ModuleCat.ihom_map_apply
|
[] |
[
61,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.nndist_le_two_nnnorm
|
[] |
[
1011,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1010,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Finset.infsep_pos_iff_nontrivial
|
[] |
[
570,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.NullMeasurableSet.congr
|
[] |
[
146,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
11
] |
Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean
|
spectrum.map_polynomial_aeval_of_degree_pos
|
[
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\n⊢ σ (↑(aeval a) p) = (fun x => eval x p) '' σ a",
"tactic": "refine' Set.eq_of_subset_of_subset (fun k hk => _) (subset_polynomial_aeval a p)"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "have hprod := eq_prod_roots_of_splits_id (IsAlgClosed.splits (C k - p))"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "have h_ne : C k - p ≠ 0 := ne_zero_of_degree_gt <| by\n rwa [degree_sub_eq_right_of_degree_lt (lt_of_le_of_lt degree_C_le hdeg)]"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "have lead_ne := leadingCoeff_ne_zero.mpr h_ne"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "have lead_unit := (Units.map ↑ₐ.toMonoidHom (Units.mk0 _ lead_ne)).isUnit"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "have p_a_eq : aeval a (C k - p) = ↑ₐ k - aeval a p := by\n simp only [aeval_C, AlgHom.map_sub, sub_left_inj]"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk :\n ¬(IsUnit (↑↑ₐ (leadingCoeff (↑C k - p))) ∧\n IsUnit (↑(aeval a) (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p))))))\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "rw [mem_iff, ← p_a_eq, hprod, aeval_mul,\n ((Commute.all _ _).map (aeval a : 𝕜[X] →ₐ[𝕜] A)).isUnit_mul_iff, aeval_C] at hk"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\nhk : ∃ k_1, k_1 ∈ σ a ∧ eval k_1 (↑C k - p) = 0\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk :\n ¬(IsUnit (↑↑ₐ (leadingCoeff (↑C k - p))) ∧\n IsUnit (↑(aeval a) (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p))))))\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "replace hk := exists_mem_of_not_isUnit_aeval_prod (not_and.mp hk lead_unit)"
},
{
"state_after": "case intro.intro\n𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\nr : 𝕜\nr_mem : r ∈ σ a\nr_ev : eval r (↑C k - p) = 0\n⊢ k ∈ (fun x => eval x p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\nhk : ∃ k_1, k_1 ∈ σ a ∧ eval k_1 (↑C k - p) = 0\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "rcases hk with ⟨r, r_mem, r_ev⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\nr : 𝕜\nr_mem : r ∈ σ a\nr_ev : eval r (↑C k - p) = 0\n⊢ k ∈ (fun x => eval x p) '' σ a",
"tactic": "exact ⟨r, r_mem, symm (by simpa [eval_sub, eval_C, sub_eq_zero] using r_ev)⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\n⊢ ?m.271688 < degree (↑C k - p)",
"tactic": "rwa [degree_sub_eq_right_of_degree_lt (lt_of_le_of_lt degree_C_le hdeg)]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhk : k ∈ σ (↑(aeval a) p)\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\n⊢ ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p",
"tactic": "simp only [aeval_C, AlgHom.map_sub, sub_left_inj]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < degree p\nk : 𝕜\nhprod : ↑C k - p = ↑C (leadingCoeff (↑C k - p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (↑C k - p)))\nh_ne : ↑C k - p ≠ 0\nlead_ne : leadingCoeff (↑C k - p) ≠ 0\nlead_unit : IsUnit ↑(↑(Units.map ↑↑ₐ) (Units.mk0 (leadingCoeff (↑C k - p)) lead_ne))\np_a_eq : ↑(aeval a) (↑C k - p) = ↑↑ₐ k - ↑(aeval a) p\nr : 𝕜\nr_mem : r ∈ σ a\nr_ev : eval r (↑C k - p) = 0\n⊢ k = (fun x => eval x p) r",
"tactic": "simpa [eval_sub, eval_C, sub_eq_zero] using r_ev"
}
] |
[
122,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.choose_symm
|
[
{
"state_after": "no goals",
"state_before": "n k : ℕ\nhk : k ≤ n\n⊢ choose n (n - k) = choose n k",
"tactic": "rw [choose_eq_factorial_div_factorial hk, choose_eq_factorial_div_factorial (Nat.sub_le _ _),\n tsub_tsub_cancel_of_le hk, mul_comm]"
}
] |
[
197,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.aeval_X_left
|
[] |
[
1459,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1458,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.abs_integral_mono_interval
|
[] |
[
1418,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1410,
1
] |
Mathlib/Data/Nat/Interval.lean
|
Nat.Ioc_succ_singleton
|
[
{
"state_after": "no goals",
"state_before": "a b c : ℕ\n⊢ Ioc b (b + 1) = {b + 1}",
"tactic": "rw [← Nat.Icc_succ_left, Icc_self]"
}
] |
[
195,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.inf_singleton
|
[] |
[
354,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
Set.iUnion_Ici
|
[] |
[
73,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
IsLocalizedModule.iso_symm_comp
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\n⊢ ↑(LinearMap.comp (↑(LinearEquiv.symm (iso S f))) f) m = ↑(LocalizedModule.mkLinearMap S M) m",
"state_before": "R : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\n⊢ LinearMap.comp (↑(LinearEquiv.symm (iso S f))) f = LocalizedModule.mkLinearMap S M",
"tactic": "ext m"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\n⊢ ↑↑(LinearEquiv.symm (iso S f)) (↑f m) = LocalizedModule.mk m 1",
"state_before": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\n⊢ ↑(LinearMap.comp (↑(LinearEquiv.symm (iso S f))) f) m = ↑(LocalizedModule.mkLinearMap S M) m",
"tactic": "rw [LinearMap.comp_apply, LocalizedModule.mkLinearMap_apply]"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\n⊢ ↑(LinearEquiv.symm (iso S f)) (↑f m) = LocalizedModule.mk m 1",
"state_before": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\n⊢ ↑↑(LinearEquiv.symm (iso S f)) (↑f m) = LocalizedModule.mk m 1",
"tactic": "change (iso S f).symm _ = _"
},
{
"state_after": "case h.eq1\nR : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\n⊢ 1 • ↑f m = ↑f m",
"state_before": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\n⊢ ↑(LinearEquiv.symm (iso S f)) (↑f m) = LocalizedModule.mk m 1",
"tactic": "rw [iso_symm_apply']"
},
{
"state_after": "no goals",
"state_before": "case h.eq1\nR : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.700210\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\n⊢ 1 • ↑f m = ↑f m",
"tactic": "exact one_smul _ _"
}
] |
[
841,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/Init/Classical.lean
|
Classical.cases
|
[] |
[
31,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.map_coe_nnreal_restrict
|
[] |
[
780,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
778,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.xor_not_not
|
[
{
"state_after": "no goals",
"state_before": "⊢ ∀ (a b : Bool), (xor (!a) !b) = xor a b",
"tactic": "decide"
}
] |
[
282,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.coe_add_get
|
[
{
"state_after": "no goals",
"state_before": "x : ℕ\ny : PartENat\nh : (↑x + y).Dom\n⊢ Part.get (↑x + y) h = x + Part.get y (_ : y.Dom)",
"tactic": "rfl"
}
] |
[
186,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
aestronglyMeasurable_add_measure_iff
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.400090\nι : Type ?u.400093\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ninst✝ : PseudoMetrizableSpace β\nν : Measure α\n⊢ AEStronglyMeasurable f (bif true then μ else ν) ∧ AEStronglyMeasurable f (bif false then μ else ν) ↔\n AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.400090\nι : Type ?u.400093\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ninst✝ : PseudoMetrizableSpace β\nν : Measure α\n⊢ AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν",
"tactic": "rw [← sum_cond, aestronglyMeasurable_sum_measure_iff, Bool.forall_bool, and_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.400090\nι : Type ?u.400093\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ninst✝ : PseudoMetrizableSpace β\nν : Measure α\n⊢ AEStronglyMeasurable f (bif true then μ else ν) ∧ AEStronglyMeasurable f (bif false then μ else ν) ↔\n AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν",
"tactic": "rfl"
}
] |
[
1694,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1691,
1
] |
Mathlib/Logic/Basic.lean
|
Decidable.or_not_of_imp
|
[] |
[
366,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
11
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
IsUnit.tendsto_const_smul_iff
|
[] |
[
404,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
8
] |
Mathlib/RepresentationTheory/Action.lean
|
Action.associator_inv_hom
|
[
{
"state_after": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nX Y Z : Action V G\n⊢ ((𝟙 X.V ⊗ 𝟙 (Y.V ⊗ Z.V)) ≫ (α_ X.V Y.V Z.V).inv) ≫ (𝟙 (X.V ⊗ Y.V) ⊗ 𝟙 Z.V) = (α_ X.V Y.V Z.V).inv",
"state_before": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nX Y Z : Action V G\n⊢ (α_ X Y Z).inv.hom = (α_ X.V Y.V Z.V).inv",
"tactic": "dsimp [Monoidal.transport_associator]"
},
{
"state_after": "no goals",
"state_before": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nX Y Z : Action V G\n⊢ ((𝟙 X.V ⊗ 𝟙 (Y.V ⊗ Z.V)) ≫ (α_ X.V Y.V Z.V).inv) ≫ (𝟙 (X.V ⊗ Y.V) ⊗ 𝟙 Z.V) = (α_ X.V Y.V Z.V).inv",
"tactic": "simp"
}
] |
[
522,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
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