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/Group/Commute.lean
|
Commute.units_inv_right
|
[] |
[
223,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.set_integral_congr_set_ae
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3053\nE : Type u_2\nF : Type ?u.3059\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nhst : s =ᵐ[μ] t\n⊢ (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in t, f x ∂μ",
"tactic": "rw [Measure.restrict_congr_set hst]"
}
] |
[
96,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_iInf_eq
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.279682\nι : Sort x\nf : ι → Filter α\nm : α → β\nhf : Directed (fun x x_1 => x ≥ x_1) f\ninst✝ : Nonempty ι\ns : Set β\nhs : m ⁻¹' s ∈ iInf f\ni : ι\nhi : m ⁻¹' s ∈ f i\n⊢ m ⁻¹' s ∈ f i",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.279682\nι : Sort x\nf : ι → Filter α\nm : α → β\nhf : Directed (fun x x_1 => x ≥ x_1) f\ninst✝ : Nonempty ι\ns : Set β\nhs : m ⁻¹' s ∈ iInf f\ni : ι\nhi : m ⁻¹' s ∈ f i\n⊢ map m (f i) ≤ 𝓟 s",
"tactic": "simp only [le_principal_iff, mem_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.279682\nι : Sort x\nf : ι → Filter α\nm : α → β\nhf : Directed (fun x x_1 => x ≥ x_1) f\ninst✝ : Nonempty ι\ns : Set β\nhs : m ⁻¹' s ∈ iInf f\ni : ι\nhi : m ⁻¹' s ∈ f i\n⊢ m ⁻¹' s ∈ f i",
"tactic": "assumption"
}
] |
[
2496,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2488,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
Real.tendsto_abs_tan_atTop
|
[] |
[
50,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.le_arcsin_iff_sin_le'
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx : x ∈ Ioc (-(π / 2)) (π / 2)\n⊢ x ≤ arcsin y ↔ sin x ≤ y",
"tactic": "rw [← neg_le_neg_iff, ← arcsin_neg, arcsin_le_iff_le_sin' ⟨neg_le_neg hx.2, neg_lt.2 hx.1⟩,\n sin_neg, neg_le_neg_iff]"
}
] |
[
172,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.blocks_pos'
|
[] |
[
194,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tendsto_ofReal_atTop
|
[] |
[
193,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.fromBlocks_neg
|
[
{
"state_after": "case a.h\nl : Type u_3\nm : Type u_4\nn : Type u_2\no : Type u_5\np : Type ?u.30648\nq : Type ?u.30651\nm' : o → Type ?u.30656\nn' : o → Type ?u.30661\np' : o → Type ?u.30666\nR : Type u_1\nS : Type ?u.30672\nα : Type ?u.30675\nβ : Type ?u.30678\ninst✝ : Neg R\nA : Matrix n l R\nB : Matrix n m R\nC : Matrix o l R\nD : Matrix o m R\ni : n ⊕ o\nj : l ⊕ m\n⊢ (-fromBlocks A B C D) i j = fromBlocks (-A) (-B) (-C) (-D) i j",
"state_before": "l : Type u_3\nm : Type u_4\nn : Type u_2\no : Type u_5\np : Type ?u.30648\nq : Type ?u.30651\nm' : o → Type ?u.30656\nn' : o → Type ?u.30661\np' : o → Type ?u.30666\nR : Type u_1\nS : Type ?u.30672\nα : Type ?u.30675\nβ : Type ?u.30678\ninst✝ : Neg R\nA : Matrix n l R\nB : Matrix n m R\nC : Matrix o l R\nD : Matrix o m R\n⊢ -fromBlocks A B C D = fromBlocks (-A) (-B) (-C) (-D)",
"tactic": "ext i j"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type u_3\nm : Type u_4\nn : Type u_2\no : Type u_5\np : Type ?u.30648\nq : Type ?u.30651\nm' : o → Type ?u.30656\nn' : o → Type ?u.30661\np' : o → Type ?u.30666\nR : Type u_1\nS : Type ?u.30672\nα : Type ?u.30675\nβ : Type ?u.30678\ninst✝ : Neg R\nA : Matrix n l R\nB : Matrix n m R\nC : Matrix o l R\nD : Matrix o m R\ni : n ⊕ o\nj : l ⊕ m\n⊢ (-fromBlocks A B C D) i j = fromBlocks (-A) (-B) (-C) (-D) i j",
"tactic": "cases i <;> cases j <;> simp [fromBlocks]"
}
] |
[
234,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BoundedOrderHom.coe_comp
|
[] |
[
647,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
646,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.constantCoeff_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nr : R\n⊢ ↑constantCoeff (↑C r) = r",
"tactic": "classical simp [constantCoeff_eq]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nr : R\n⊢ ↑constantCoeff (↑C r) = r",
"tactic": "simp [constantCoeff_eq]"
}
] |
[
871,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
870,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
LinearEquiv.uniformEmbedding
|
[] |
[
2151,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2143,
11
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.reindex_updateColumn
|
[] |
[
2924,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2921,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.Relabelling.mk_leftMovesEquiv
|
[] |
[
1043,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1042,
1
] |
Mathlib/Data/Multiset/Lattice.lean
|
Multiset.inf_coe
|
[] |
[
122,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
AntitoneOn.mul_strictAnti'
|
[] |
[
1510,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1506,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.cast_mul'
|
[] |
[
384,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Control/Traversable/Instances.lean
|
Sum.naturality
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα : Type u_1\nβ : Type u\nf : α → F β\nx : σ ⊕ α\n⊢ (fun {α} => ApplicativeTransformation.app η α) (Sum.traverse f x) =\n Sum.traverse ((fun {α} => ApplicativeTransformation.app η α) ∘ f) x",
"tactic": "cases x <;> simp! [Sum.traverse, functor_norm, ApplicativeTransformation.preserves_map,\n ApplicativeTransformation.preserves_seq, ApplicativeTransformation.preserves_pure]"
}
] |
[
193,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
11
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.eq_zero_of_fst_eq_zero
|
[
{
"state_after": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.370358\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nz : S\nx : R\ny : { x // x ∈ M }\nh : z * ↑(algebraMap R S) ↑y = 0\nhx : x = 0\n⊢ z = 0",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.370358\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nz : S\nx : R\ny : { x // x ∈ M }\nh : z * ↑(algebraMap R S) ↑y = ↑(algebraMap R S) x\nhx : x = 0\n⊢ z = 0",
"tactic": "rw [hx, (algebraMap R S).map_zero] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.370358\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nz : S\nx : R\ny : { x // x ∈ M }\nh : z * ↑(algebraMap R S) ↑y = 0\nhx : x = 0\n⊢ z = 0",
"tactic": "exact (IsUnit.mul_left_eq_zero (IsLocalization.map_units S y)).1 h"
}
] |
[
224,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Order/PFilter.lean
|
Order.PFilter.sInf_gc
|
[
{
"state_after": "P : Type u_1\ninst✝ : CompleteSemilatticeInf P\nF✝ : PFilter P\nx : P\nF : (PFilter P)ᵒᵈ\n⊢ ↑toDual (principal x) ≤ F ↔ ∀ (b : P), b ∈ ↑ofDual F → x ≤ b",
"state_before": "P : Type u_1\ninst✝ : CompleteSemilatticeInf P\nF✝ : PFilter P\nx : P\nF : (PFilter P)ᵒᵈ\n⊢ (fun x => ↑toDual (principal x)) x ≤ F ↔ x ≤ (fun F => sInf ↑(↑ofDual F)) F",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "P : Type u_1\ninst✝ : CompleteSemilatticeInf P\nF✝ : PFilter P\nx : P\nF : (PFilter P)ᵒᵈ\n⊢ ↑toDual (principal x) ≤ F ↔ ∀ (b : P), b ∈ ↑ofDual F → x ≤ b",
"tactic": "rfl"
}
] |
[
181,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_arrow
|
[] |
[
498,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/CategoryTheory/Elements.lean
|
CategoryTheory.CategoryOfElements.costructuredArrow_yoneda_equivalence_naturality
|
[
{
"state_after": "case h_obj\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\n⊢ ∀ (X : (Functor.Elements F₁)ᵒᵖ),\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X = (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X\n\ncase h_map\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\n⊢ autoParam\n (∀ (X Y : (Functor.Elements F₁)ᵒᵖ) (f : X ⟶ Y),\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).map f =\n eqToHom (_ : ?F.obj X = ?G.obj X) ≫\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).map f ≫\n eqToHom\n (_ :\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj Y =\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj Y))\n _auto✝",
"state_before": "C : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\n⊢ Functor.op (map α) ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α",
"tactic": "fapply Functor.ext"
},
{
"state_after": "case h_obj\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX : (Functor.Elements F₁)ᵒᵖ\n⊢ (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X = (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X",
"state_before": "case h_obj\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\n⊢ ∀ (X : (Functor.Elements F₁)ᵒᵖ),\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X = (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X",
"tactic": "intro X"
},
{
"state_after": "case h_obj\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX : (Functor.Elements F₁)ᵒᵖ\n⊢ CostructuredArrow.mk\n ((yonedaSections ((map α).obj X.unop).op.unop.fst.unop F₂).inv { down := ((map α).obj X.unop).op.unop.snd }) =\n CostructuredArrow.mk ((yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α)",
"state_before": "case h_obj\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX : (Functor.Elements F₁)ᵒᵖ\n⊢ (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X = (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X",
"tactic": "simp only [CostructuredArrow.map_mk, toCostructuredArrow_obj, Functor.op_obj,\n Functor.comp_obj]"
},
{
"state_after": "case h_obj.e_f\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX : (Functor.Elements F₁)ᵒᵖ\n⊢ (yonedaSections ((map α).obj X.unop).op.unop.fst.unop F₂).inv { down := ((map α).obj X.unop).op.unop.snd } =\n (yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α",
"state_before": "case h_obj\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX : (Functor.Elements F₁)ᵒᵖ\n⊢ CostructuredArrow.mk\n ((yonedaSections ((map α).obj X.unop).op.unop.fst.unop F₂).inv { down := ((map α).obj X.unop).op.unop.snd }) =\n CostructuredArrow.mk ((yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α)",
"tactic": "congr"
},
{
"state_after": "case h_obj.e_f.w.h.h\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX : (Functor.Elements F₁)ᵒᵖ\nx✝ : Cᵒᵖ\nf : (yoneda.obj ((map α).obj X.unop).op.unop.fst.unop).obj x✝\n⊢ ((yonedaSections ((map α).obj X.unop).op.unop.fst.unop F₂).inv { down := ((map α).obj X.unop).op.unop.snd }).app x✝\n f =\n ((yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α).app x✝ f",
"state_before": "case h_obj.e_f\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX : (Functor.Elements F₁)ᵒᵖ\n⊢ (yonedaSections ((map α).obj X.unop).op.unop.fst.unop F₂).inv { down := ((map α).obj X.unop).op.unop.snd } =\n (yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α",
"tactic": "ext _ f"
},
{
"state_after": "no goals",
"state_before": "case h_obj.e_f.w.h.h\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX : (Functor.Elements F₁)ᵒᵖ\nx✝ : Cᵒᵖ\nf : (yoneda.obj ((map α).obj X.unop).op.unop.fst.unop).obj x✝\n⊢ ((yonedaSections ((map α).obj X.unop).op.unop.fst.unop F₂).inv { down := ((map α).obj X.unop).op.unop.snd }).app x✝\n f =\n ((yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α).app x✝ f",
"tactic": "simpa using congr_fun (α.naturality f.op).symm (unop X).snd"
},
{
"state_after": "case h_map\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX Y : (Functor.Elements F₁)ᵒᵖ\nf : X ⟶ Y\n⊢ (Functor.op (map α) ⋙ toCostructuredArrow F₂).map f =\n eqToHom\n (_ :\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X =\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X) ≫\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).map f ≫\n eqToHom\n (_ :\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj Y =\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj Y)",
"state_before": "case h_map\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\n⊢ autoParam\n (∀ (X Y : (Functor.Elements F₁)ᵒᵖ) (f : X ⟶ Y),\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).map f =\n eqToHom\n (_ :\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X =\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X) ≫\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).map f ≫\n eqToHom\n (_ :\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj Y =\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj Y))\n _auto✝",
"tactic": "intro X Y f"
},
{
"state_after": "case h_map.h\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX Y : (Functor.Elements F₁)ᵒᵖ\nf : X ⟶ Y\n⊢ ((Functor.op (map α) ⋙ toCostructuredArrow F₂).map f).left =\n (eqToHom\n (_ :\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X =\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X) ≫\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).map f ≫\n eqToHom\n (_ :\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj Y =\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj Y)).left",
"state_before": "case h_map\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX Y : (Functor.Elements F₁)ᵒᵖ\nf : X ⟶ Y\n⊢ (Functor.op (map α) ⋙ toCostructuredArrow F₂).map f =\n eqToHom\n (_ :\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X =\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X) ≫\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).map f ≫\n eqToHom\n (_ :\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj Y =\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj Y)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h_map.h\nC : Type u\ninst✝ : Category C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\nX Y : (Functor.Elements F₁)ᵒᵖ\nf : X ⟶ Y\n⊢ ((Functor.op (map α) ⋙ toCostructuredArrow F₂).map f).left =\n (eqToHom\n (_ :\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj X =\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X) ≫\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).map f ≫\n eqToHom\n (_ :\n (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj Y =\n (Functor.op (map α) ⋙ toCostructuredArrow F₂).obj Y)).left",
"tactic": "simp [CostructuredArrow.eqToHom_left]"
}
] |
[
283,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.left_mem_Ioc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4365\ninst✝ : Preorder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ a ∈ Ioc a b ↔ False",
"tactic": "simp [lt_irrefl]"
}
] |
[
200,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Std/Data/List/Lemmas.lean
|
List.range_eq_nil
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ range n = [] ↔ n = 0",
"tactic": "rw [← length_eq_zero, length_range]"
}
] |
[
1904,
38
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1903,
9
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_mul_eq_right
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.68899\nι : Type ?u.68902\nM : Type u_2\nN : Type ?u.68908\ninst✝ : MulOneClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nf g : α → M\nh : Disjoint (mulSupport f) (mulSupport g)\nx : α\nhx : x ∈ mulSupport g\n⊢ (f * g) x = g x",
"state_before": "α : Type u_1\nβ : Type ?u.68899\nι : Type ?u.68902\nM : Type u_2\nN : Type ?u.68908\ninst✝ : MulOneClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nf g : α → M\nh : Disjoint (mulSupport f) (mulSupport g)\n⊢ mulIndicator (mulSupport g) (f * g) = g",
"tactic": "refine' (mulIndicator_congr fun x hx => _).trans mulIndicator_mulSupport"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.68899\nι : Type ?u.68902\nM : Type u_2\nN : Type ?u.68908\ninst✝ : MulOneClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nf g : α → M\nh : Disjoint (mulSupport f) (mulSupport g)\nx : α\nhx : x ∈ mulSupport g\nthis : f x = 1\n⊢ (f * g) x = g x",
"state_before": "α : Type u_1\nβ : Type ?u.68899\nι : Type ?u.68902\nM : Type u_2\nN : Type ?u.68908\ninst✝ : MulOneClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nf g : α → M\nh : Disjoint (mulSupport f) (mulSupport g)\nx : α\nhx : x ∈ mulSupport g\n⊢ (f * g) x = g x",
"tactic": "have : f x = 1 := nmem_mulSupport.1 (disjoint_right.1 h hx)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.68899\nι : Type ?u.68902\nM : Type u_2\nN : Type ?u.68908\ninst✝ : MulOneClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nf g : α → M\nh : Disjoint (mulSupport f) (mulSupport g)\nx : α\nhx : x ∈ mulSupport g\nthis : f x = 1\n⊢ (f * g) x = g x",
"tactic": "rw [Pi.mul_apply, this, one_mul]"
}
] |
[
458,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
deriv_const
|
[] |
[
695,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Mathlib/Order/SymmDiff.lean
|
Pi.symmDiff_def
|
[] |
[
854,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.zpow_pos
|
[
{
"state_after": "case ofNat\nα : Type ?u.409403\nβ : Type ?u.409406\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nh'a : a ≠ ⊤\na✝ : ℕ\n⊢ 0 < a ^ Int.ofNat a✝\n\ncase negSucc\nα : Type ?u.409403\nβ : Type ?u.409406\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nh'a : a ≠ ⊤\na✝ : ℕ\n⊢ 0 < a ^ Int.negSucc a✝",
"state_before": "α : Type ?u.409403\nβ : Type ?u.409406\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nh'a : a ≠ ⊤\nn : ℤ\n⊢ 0 < a ^ n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case ofNat\nα : Type ?u.409403\nβ : Type ?u.409406\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nh'a : a ≠ ⊤\na✝ : ℕ\n⊢ 0 < a ^ Int.ofNat a✝",
"tactic": "exact ENNReal.pow_pos ha.bot_lt _"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nα : Type ?u.409403\nβ : Type ?u.409406\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nh'a : a ≠ ⊤\na✝ : ℕ\n⊢ 0 < a ^ Int.negSucc a✝",
"tactic": "simp only [h'a, pow_eq_top_iff, zpow_negSucc, Ne.def, not_false, ENNReal.inv_pos, false_and]"
}
] |
[
1865,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1862,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
Orthonormal.equiv_symm
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3085209\ninst✝⁸ : IsROrC 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\nι' : Type u_4\nι'' : Type ?u.3085264\nE' : Type u_5\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\nE'' : Type ?u.3085285\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : InnerProductSpace 𝕜 E''\nv : Basis ι 𝕜 E\nhv : Orthonormal 𝕜 ↑v\nv' : Basis ι' 𝕜 E'\nhv' : Orthonormal 𝕜 ↑v'\ne : ι ≃ ι'\ni : ι'\n⊢ ↑(equiv hv hv' e) (↑(LinearIsometryEquiv.symm (equiv hv hv' e)) (↑v' i)) =\n ↑(equiv hv hv' e) (↑(equiv hv' hv e.symm) (↑v' i))",
"tactic": "simp only [LinearIsometryEquiv.apply_symm_apply, Orthonormal.equiv_apply, e.apply_symm_apply]"
}
] |
[
1425,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1421,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhds_eq_iInf_abs_sub
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na : α\n⊢ ((⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)) =\n (⨅ (x : α) (_ : x > 0), 𝓟 {a_1 | -x < a - a_1}) ⊓ ⨅ (x : α) (_ : x > 0), 𝓟 {a_1 | a - a_1 < x}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na : α\n⊢ 𝓝 a = ⨅ (r : α) (_ : r > 0), 𝓟 {b | abs (a - b) < r}",
"tactic": "simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]"
},
{
"state_after": "case refine_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na : α\n⊢ (⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) = ⨅ (x : α) (_ : x > 0), 𝓟 {a_1 | a - a_1 < x}\n\ncase refine_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na : α\n⊢ (⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)) = ⨅ (x : α) (_ : x > 0), 𝓟 {a_1 | -x < a - a_1}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na : α\n⊢ ((⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)) =\n (⨅ (x : α) (_ : x > 0), 𝓟 {a_1 | -x < a - a_1}) ⊓ ⨅ (x : α) (_ : x > 0), 𝓟 {a_1 | a - a_1 < x}",
"tactic": "refine (congr_arg₂ _ ?_ ?_).trans inf_comm"
},
{
"state_after": "case refine_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na x : α\n⊢ (⨅ (_ : ↑(Equiv.subLeft a) x > 0), 𝓟 {a_1 | a - a_1 < ↑(Equiv.subLeft a) x}) = ⨅ (_ : x ∈ Iio a), 𝓟 (Ioi x)",
"state_before": "case refine_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na : α\n⊢ (⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) = ⨅ (x : α) (_ : x > 0), 𝓟 {a_1 | a - a_1 < x}",
"tactic": "refine (Equiv.subLeft a).iInf_congr fun x => ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na x : α\n⊢ (⨅ (_ : ↑(Equiv.subLeft a) x > 0), 𝓟 {a_1 | a - a_1 < ↑(Equiv.subLeft a) x}) = ⨅ (_ : x ∈ Iio a), 𝓟 (Ioi x)",
"tactic": "simp [Ioi]"
},
{
"state_after": "case refine_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na x : α\n⊢ (⨅ (_ : ↑(Equiv.subRight a) x > 0), 𝓟 {a_1 | -↑(Equiv.subRight a) x < a - a_1}) = ⨅ (_ : x ∈ Ioi a), 𝓟 (Iio x)",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na : α\n⊢ (⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)) = ⨅ (x : α) (_ : x > 0), 𝓟 {a_1 | -x < a - a_1}",
"tactic": "refine (Equiv.subRight a).iInf_congr fun x => ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\na x : α\n⊢ (⨅ (_ : ↑(Equiv.subRight a) x > 0), 𝓟 {a_1 | -↑(Equiv.subRight a) x < a - a_1}) = ⨅ (_ : x ∈ Ioi a), 𝓟 (Iio x)",
"tactic": "simp [Iio]"
}
] |
[
1856,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1852,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval₂_id
|
[] |
[
1176,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1175,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.snorm_indicator_eq_snorm_restrict
|
[
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : p = 0\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)\n\ncase neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)",
"state_before": "α : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)",
"tactic": "by_cases hp_zero : p = 0"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : p = ⊤\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)\n\ncase neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)",
"tactic": "by_cases hp_top : p = ∞"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) =\n (∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)",
"tactic": "simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top]"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) = ∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) =\n (∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"tactic": "suffices (∫⁻ x, (‖s.indicator f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) =\n ∫⁻ x in s, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ by rw [this]"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) =\n ∫⁻ (a : α), Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a ∂μ",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) = ∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ",
"tactic": "rw [← lintegral_indicator _ hs]"
},
{
"state_after": "case neg.e_f\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (fun x => ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p) = fun a =>\n Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) =\n ∫⁻ (a : α), Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a ∂μ",
"tactic": "congr"
},
{
"state_after": "case neg.e_f\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (fun x => Set.indicator s (fun x => ↑‖f x‖₊) x ^ ENNReal.toReal p) = fun a =>\n Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a",
"state_before": "case neg.e_f\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (fun x => ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p) = fun a =>\n Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a",
"tactic": "simp_rw [nnnorm_indicator_eq_indicator_nnnorm, ENNReal.coe_indicator]"
},
{
"state_after": "case neg.e_f\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\nh_zero : (fun x => x ^ ENNReal.toReal p) 0 = 0\n⊢ (fun x => Set.indicator s (fun x => ↑‖f x‖₊) x ^ ENNReal.toReal p) = fun a =>\n Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a",
"state_before": "case neg.e_f\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (fun x => Set.indicator s (fun x => ↑‖f x‖₊) x ^ ENNReal.toReal p) = fun a =>\n Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a",
"tactic": "have h_zero : (fun x => x ^ p.toReal) (0 : ℝ≥0∞) = 0 := by\n simp [ENNReal.toReal_pos hp_zero hp_top]"
},
{
"state_after": "no goals",
"state_before": "case neg.e_f\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\nh_zero : (fun x => x ^ ENNReal.toReal p) 0 = 0\n⊢ (fun x => Set.indicator s (fun x => ↑‖f x‖₊) x ^ ENNReal.toReal p) = fun a =>\n Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a",
"tactic": "exact (Set.indicator_comp_of_zero (g := fun x : ℝ≥0∞ => x ^ p.toReal) h_zero).symm"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : p = 0\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)",
"tactic": "simp only [hp_zero, snorm_exponent_zero]"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : p = ⊤\n⊢ snormEssSup (Set.indicator s f) μ = snormEssSup f (Measure.restrict μ s)",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : p = ⊤\n⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s)",
"tactic": "simp_rw [hp_top, snorm_exponent_top]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : p = ⊤\n⊢ snormEssSup (Set.indicator s f) μ = snormEssSup f (Measure.restrict μ s)",
"tactic": "exact snormEssSup_indicator_eq_snormEssSup_restrict hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\nthis : (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) = ∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ\n⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) =\n (∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.658895\nF : Type u_2\nG : Type ?u.658901\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc : E\nf✝ : α → E\nhf : AEStronglyMeasurable f✝ μ\nf : α → F\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ (fun x => x ^ ENNReal.toReal p) 0 = 0",
"tactic": "simp [ENNReal.toReal_pos hp_zero hp_top]"
}
] |
[
668,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
StrictAnti.mul_const'
|
[] |
[
1422,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1421,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.zero_rpow_of_pos
|
[
{
"state_after": "y : ℝ\nh : 0 < y\n⊢ Option.some 0 ^ y = Option.some 0",
"state_before": "y : ℝ\nh : 0 < y\n⊢ 0 ^ y = 0",
"tactic": "rw [← ENNReal.coe_zero, ← ENNReal.some_eq_coe]"
},
{
"state_after": "no goals",
"state_before": "y : ℝ\nh : 0 < y\n⊢ (if 0 = 0 ∧ y < 0 then ⊤ else ↑(NNReal.rpow 0 y)) = Option.some 0",
"tactic": "simp [h, asymm h, ne_of_gt h]"
}
] |
[
327,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Data/MvPolynomial/Equiv.lean
|
MvPolynomial.finSuccEquiv_X_succ
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.994751\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nj : Fin n\n⊢ ↑(finSuccEquiv R n) (X (Fin.succ j)) = ↑Polynomial.C (X j)",
"tactic": "simp"
}
] |
[
362,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.uniformContinuous
|
[] |
[
269,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
11
] |
Mathlib/Data/Set/Finite.lean
|
Set.infinite_coe_iff
|
[] |
[
1282,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1281,
1
] |
Mathlib/Topology/Order/Basic.lean
|
tendsto_order
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\nf : β → α\na : α\nx : Filter β\n⊢ ((∀ (i : α), i ∈ Iio a → ∀ᶠ (a : β) in x, f a ∈ Ioi i) ∧ ∀ (i : α), i ∈ Ioi a → ∀ᶠ (a : β) in x, f a ∈ Iio i) ↔\n (∀ (a' : α), a' < a → ∀ᶠ (b : β) in x, a' < f b) ∧ ∀ (a' : α), a' > a → ∀ᶠ (b : β) in x, f b < a'",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\nf : β → α\na : α\nx : Filter β\n⊢ Tendsto f x (𝓝 a) ↔ (∀ (a' : α), a' < a → ∀ᶠ (b : β) in x, a' < f b) ∧ ∀ (a' : α), a' > a → ∀ᶠ (b : β) in x, f b < a'",
"tactic": "simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\nf : β → α\na : α\nx : Filter β\n⊢ ((∀ (i : α), i ∈ Iio a → ∀ᶠ (a : β) in x, f a ∈ Ioi i) ∧ ∀ (i : α), i ∈ Ioi a → ∀ᶠ (a : β) in x, f a ∈ Iio i) ↔\n (∀ (a' : α), a' < a → ∀ᶠ (b : β) in x, a' < f b) ∧ ∀ (a' : α), a' > a → ∀ᶠ (b : β) in x, f b < a'",
"tactic": "rfl"
}
] |
[
935,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
933,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.image_fst_graph
|
[] |
[
101,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Order/PrimeIdeal.lean
|
Order.Ideal.PrimePair.disjoint
|
[] |
[
78,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
11
] |
Mathlib/AlgebraicGeometry/LocallyRingedSpace.lean
|
AlgebraicGeometry.LocallyRingedSpace.comp_val
|
[] |
[
173,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Computability/Language.lean
|
Language.mul_self_kstar_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.121636\nγ : Type ?u.121639\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ l∗ * l = l * l∗",
"tactic": "simp only [kstar_eq_iSup_pow, mul_iSup, iSup_mul, ← pow_succ, ← pow_succ']"
}
] |
[
281,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/CategoryTheory/Action.lean
|
CategoryTheory.ActionCategory.stabilizerIsoEnd_symm_apply
|
[] |
[
131,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Analysis/NormedSpace/LpEquiv.lean
|
coe_lpBcfₗᵢ
|
[] |
[
181,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Topology/MetricSpace/MetricSeparated.lean
|
IsMetricSeparated.mono_left
|
[] |
[
74,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/GroupTheory/DoubleCoset.lean
|
Doset.union_quotToDoset
|
[
{
"state_after": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx : G\n⊢ (x ∈ ⋃ (q : Quotient ↑H ↑K), quotToDoset H K q) ↔ x ∈ Set.univ",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\n⊢ (⋃ (q : Quotient ↑H ↑K), quotToDoset H K q) = Set.univ",
"tactic": "ext x"
},
{
"state_after": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx : G\n⊢ ∃ i x_1, x_1 ∈ H ∧ ∃ y, y ∈ K ∧ x = x_1 * Quotient.out' i * y",
"state_before": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx : G\n⊢ (x ∈ ⋃ (q : Quotient ↑H ↑K), quotToDoset H K q) ↔ x ∈ Set.univ",
"tactic": "simp only [Set.mem_iUnion, quotToDoset, mem_doset, SetLike.mem_coe, exists_prop, Set.mem_univ,\n iff_true_iff]"
},
{
"state_after": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx : G\n⊢ ∃ x_1, x_1 ∈ H ∧ ∃ y, y ∈ K ∧ x = x_1 * Quotient.out' (mk H K x) * y",
"state_before": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx : G\n⊢ ∃ i x_1, x_1 ∈ H ∧ ∃ y, y ∈ K ∧ x = x_1 * Quotient.out' i * y",
"tactic": "use mk H K x"
},
{
"state_after": "case h.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx h k : G\nh3 : h ∈ H\nh4 : k ∈ K\nh5 : Quotient.out' (mk H K x) = h * x * k\n⊢ ∃ x_1, x_1 ∈ H ∧ ∃ y, y ∈ K ∧ x = x_1 * Quotient.out' (mk H K x) * y",
"state_before": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx : G\n⊢ ∃ x_1, x_1 ∈ H ∧ ∃ y, y ∈ K ∧ x = x_1 * Quotient.out' (mk H K x) * y",
"tactic": "obtain ⟨h, k, h3, h4, h5⟩ := mk_out'_eq_mul H K x"
},
{
"state_after": "case h.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx h k : G\nh3 : h ∈ H\nh4 : k ∈ K\nh5 : Quotient.out' (mk H K x) = h * x * k\n⊢ x = h⁻¹ * Quotient.out' (mk H K x) * k⁻¹",
"state_before": "case h.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx h k : G\nh3 : h ∈ H\nh4 : k ∈ K\nh5 : Quotient.out' (mk H K x) = h * x * k\n⊢ ∃ x_1, x_1 ∈ H ∧ ∃ y, y ∈ K ∧ x = x_1 * Quotient.out' (mk H K x) * y",
"tactic": "refine' ⟨h⁻¹, H.inv_mem h3, k⁻¹, K.inv_mem h4, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.56988\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\nx h k : G\nh3 : h ∈ H\nh4 : k ∈ K\nh5 : Quotient.out' (mk H K x) = h * x * k\n⊢ x = h⁻¹ * Quotient.out' (mk H K x) * k⁻¹",
"tactic": "simp only [h5, Subgroup.coe_mk, ← mul_assoc, one_mul, mul_left_inv, mul_inv_cancel_right]"
}
] |
[
170,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
Commute.list_prod_right
|
[
{
"state_after": "case nil\nι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝ l₁ l₂ : List M\na : M\nl : List M\ny : M\nh✝ : ∀ (x : M), x ∈ l → Commute y x\nh : ∀ (x : M), x ∈ [] → Commute y x\n⊢ Commute y (prod [])\n\ncase cons\nι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝¹ l₁ l₂ : List M\na : M\nl✝ : List M\ny : M\nh✝ : ∀ (x : M), x ∈ l✝ → Commute y x\nz : M\nl : List M\nIH : (∀ (x : M), x ∈ l → Commute y x) → Commute y (prod l)\nh : ∀ (x : M), x ∈ z :: l → Commute y x\n⊢ Commute y (prod (z :: l))",
"state_before": "ι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝ l₁ l₂ : List M\na : M\nl : List M\ny : M\nh : ∀ (x : M), x ∈ l → Commute y x\n⊢ Commute y (prod l)",
"tactic": "induction' l with z l IH"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝ l₁ l₂ : List M\na : M\nl : List M\ny : M\nh✝ : ∀ (x : M), x ∈ l → Commute y x\nh : ∀ (x : M), x ∈ [] → Commute y x\n⊢ Commute y (prod [])",
"tactic": "simp"
},
{
"state_after": "case cons\nι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝¹ l₁ l₂ : List M\na : M\nl✝ : List M\ny : M\nh✝ : ∀ (x : M), x ∈ l✝ → Commute y x\nz : M\nl : List M\nIH : (∀ (x : M), x ∈ l → Commute y x) → Commute y (prod l)\nh : Commute y z ∧ ∀ (x : M), x ∈ l → Commute y x\n⊢ Commute y (prod (z :: l))",
"state_before": "case cons\nι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝¹ l₁ l₂ : List M\na : M\nl✝ : List M\ny : M\nh✝ : ∀ (x : M), x ∈ l✝ → Commute y x\nz : M\nl : List M\nIH : (∀ (x : M), x ∈ l → Commute y x) → Commute y (prod l)\nh : ∀ (x : M), x ∈ z :: l → Commute y x\n⊢ Commute y (prod (z :: l))",
"tactic": "rw [List.forall_mem_cons] at h"
},
{
"state_after": "case cons\nι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝¹ l₁ l₂ : List M\na : M\nl✝ : List M\ny : M\nh✝ : ∀ (x : M), x ∈ l✝ → Commute y x\nz : M\nl : List M\nIH : (∀ (x : M), x ∈ l → Commute y x) → Commute y (prod l)\nh : Commute y z ∧ ∀ (x : M), x ∈ l → Commute y x\n⊢ Commute y (z * prod l)",
"state_before": "case cons\nι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝¹ l₁ l₂ : List M\na : M\nl✝ : List M\ny : M\nh✝ : ∀ (x : M), x ∈ l✝ → Commute y x\nz : M\nl : List M\nIH : (∀ (x : M), x ∈ l → Commute y x) → Commute y (prod l)\nh : Commute y z ∧ ∀ (x : M), x ∈ l → Commute y x\n⊢ Commute y (prod (z :: l))",
"tactic": "rw [List.prod_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.71204\nα : Type ?u.71207\nM : Type u_1\nN : Type ?u.71213\nP : Type ?u.71216\nM₀ : Type ?u.71219\nG : Type ?u.71222\nR : Type ?u.71225\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝¹ l₁ l₂ : List M\na : M\nl✝ : List M\ny : M\nh✝ : ∀ (x : M), x ∈ l✝ → Commute y x\nz : M\nl : List M\nIH : (∀ (x : M), x ∈ l → Commute y x) → Commute y (prod l)\nh : Commute y z ∧ ∀ (x : M), x ∈ l → Commute y x\n⊢ Commute y (z * prod l)",
"tactic": "exact Commute.mul_right h.1 (IH h.2)"
}
] |
[
252,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cofork.IsColimit.hom_ext
|
[] |
[
427,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Data/List/Range.lean
|
List.nthLe_range'_1
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nn m i : ℕ\nH : i < length (range' n m)\n⊢ nthLe (range' n m) i H = n + i",
"tactic": "simp"
}
] |
[
65,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
SubsemiringClass.coe_pow
|
[
{
"state_after": "case zero\nR✝ : Type u\nS : Type v\nT : Type w\ninst✝⁴ : NonAssocSemiring R✝\nM : Submonoid R✝\ninst✝³ : SetLike S R✝\nhSR : SubsemiringClass S R✝\ns : S\nR : Type u_1\ninst✝² : Semiring R\ninst✝¹ : SetLike S R\ninst✝ : SubsemiringClass S R\nx : { x // x ∈ s }\n⊢ ↑(x ^ Nat.zero) = ↑x ^ Nat.zero\n\ncase succ\nR✝ : Type u\nS : Type v\nT : Type w\ninst✝⁴ : NonAssocSemiring R✝\nM : Submonoid R✝\ninst✝³ : SetLike S R✝\nhSR : SubsemiringClass S R✝\ns : S\nR : Type u_1\ninst✝² : Semiring R\ninst✝¹ : SetLike S R\ninst✝ : SubsemiringClass S R\nx : { x // x ∈ s }\nn : ℕ\nih : ↑(x ^ n) = ↑x ^ n\n⊢ ↑(x ^ Nat.succ n) = ↑x ^ Nat.succ n",
"state_before": "R✝ : Type u\nS : Type v\nT : Type w\ninst✝⁴ : NonAssocSemiring R✝\nM : Submonoid R✝\ninst✝³ : SetLike S R✝\nhSR : SubsemiringClass S R✝\ns : S\nR : Type u_1\ninst✝² : Semiring R\ninst✝¹ : SetLike S R\ninst✝ : SubsemiringClass S R\nx : { x // x ∈ s }\nn : ℕ\n⊢ ↑(x ^ n) = ↑x ^ n",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR✝ : Type u\nS : Type v\nT : Type w\ninst✝⁴ : NonAssocSemiring R✝\nM : Submonoid R✝\ninst✝³ : SetLike S R✝\nhSR : SubsemiringClass S R✝\ns : S\nR : Type u_1\ninst✝² : Semiring R\ninst✝¹ : SetLike S R\ninst✝ : SubsemiringClass S R\nx : { x // x ∈ s }\n⊢ ↑(x ^ Nat.zero) = ↑x ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR✝ : Type u\nS : Type v\nT : Type w\ninst✝⁴ : NonAssocSemiring R✝\nM : Submonoid R✝\ninst✝³ : SetLike S R✝\nhSR : SubsemiringClass S R✝\ns : S\nR : Type u_1\ninst✝² : Semiring R\ninst✝¹ : SetLike S R\ninst✝ : SubsemiringClass S R\nx : { x // x ∈ s }\nn : ℕ\nih : ↑(x ^ n) = ↑x ^ n\n⊢ ↑(x ^ Nat.succ n) = ↑x ^ Nat.succ n",
"tactic": "simp [pow_succ, ih]"
}
] |
[
122,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Std/Data/String/Lemmas.lean
|
String.validFor_toSubstring
|
[
{
"state_after": "no goals",
"state_before": "s : String\n⊢ (toSubstring s).str.data = [] ++ s.data ++ []",
"tactic": "simp [toSubstring]"
},
{
"state_after": "no goals",
"state_before": "s : String\n⊢ (toSubstring s).stopPos.byteIdx = utf8Len [] + utf8Len s.data",
"tactic": "simp [toSubstring, endPos, utf8ByteSize]"
}
] |
[
801,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
800,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sdiff_sdiff_comm
|
[] |
[
632,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
631,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformOnFun.topologicalSpace_eq
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.57789\nι : Type ?u.57792\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\n⊢ (⨅ (i : Set α) (_ : i ∈ 𝔖), TopologicalSpace.induced (restrict i) UniformSpace.toTopologicalSpace) =\n ⨅ (s : Set α) (_ : s ∈ 𝔖),\n TopologicalSpace.induced (restrict s ∘ ↑UniformFun.toFun) (UniformFun.topologicalSpace (↑s) β)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.57789\nι : Type ?u.57792\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\n⊢ topologicalSpace α β 𝔖 =\n ⨅ (s : Set α) (_ : s ∈ 𝔖),\n TopologicalSpace.induced (restrict s ∘ ↑UniformFun.toFun) (UniformFun.topologicalSpace (↑s) β)",
"tactic": "simp only [UniformOnFun.topologicalSpace, toTopologicalSpace_iInf, toTopologicalSpace_iInf,\n toTopologicalSpace_comap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.57789\nι : Type ?u.57792\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\n⊢ (⨅ (i : Set α) (_ : i ∈ 𝔖), TopologicalSpace.induced (restrict i) UniformSpace.toTopologicalSpace) =\n ⨅ (s : Set α) (_ : s ∈ 𝔖),\n TopologicalSpace.induced (restrict s ∘ ↑UniformFun.toFun) (UniformFun.topologicalSpace (↑s) β)",
"tactic": "rfl"
}
] |
[
633,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
627,
11
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.interpolate_eq_iff_values_eq_on
|
[] |
[
371,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_subset_iUnion_const
|
[] |
[
1298,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1297,
1
] |
Mathlib/GroupTheory/Subsemigroup/Center.lean
|
Subsemigroup.center_eq_top
|
[] |
[
188,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
MulHom.comp_coprod
|
[
{
"state_after": "no goals",
"state_before": "A : Type ?u.44652\nB : Type ?u.44655\nG : Type ?u.44658\nH : Type ?u.44661\nM : Type u_3\nN : Type u_4\nP : Type u_2\ninst✝³ : Mul M\ninst✝² : Mul N\ninst✝¹ : CommSemigroup P\nf✝ : M →ₙ* P\ng✝ : N →ₙ* P\nQ : Type u_1\ninst✝ : CommSemigroup Q\nh : P →ₙ* Q\nf : M →ₙ* P\ng : N →ₙ* P\nx : M × N\n⊢ ↑(comp h (coprod f g)) x = ↑(coprod (comp h f) (comp h g)) x",
"tactic": "simp"
}
] |
[
434,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.map.unique
|
[
{
"state_after": "case mk\nα : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\nf : α → β\nx y : FreeGroup α\ng : FreeGroup α →* FreeGroup β\nhg : ∀ (x : α), ↑g (FreeGroup.of x) = FreeGroup.of (f x)\nx✝ : FreeGroup α\nL : List (α × Bool)\n⊢ ↑g (Quot.mk Red.Step L) = ↑(map f) (Quot.mk Red.Step L)",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\nf : α → β\nx y : FreeGroup α\ng : FreeGroup α →* FreeGroup β\nhg : ∀ (x : α), ↑g (FreeGroup.of x) = FreeGroup.of (f x)\n⊢ ∀ {x : FreeGroup α}, ↑g x = ↑(map f) x",
"tactic": "rintro ⟨L⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\nf : α → β\nx y : FreeGroup α\ng : FreeGroup α →* FreeGroup β\nhg : ∀ (x : α), ↑g (FreeGroup.of x) = FreeGroup.of (f x)\nx✝ : FreeGroup α\nL : List (α × Bool)\n⊢ ↑g (Quot.mk Red.Step L) = ↑(map f) (Quot.mk Red.Step L)",
"tactic": "exact List.recOn L g.map_one fun ⟨x, b⟩ t (ih : g (FreeGroup.mk t) = map f (FreeGroup.mk t)) =>\n Bool.recOn b\n (show g ((FreeGroup.of x)⁻¹ * FreeGroup.mk t) =\n FreeGroup.map f ((FreeGroup.of x)⁻¹ * FreeGroup.mk t) by\n simp [g.map_mul, g.map_inv, hg, ih])\n (show g (FreeGroup.of x * FreeGroup.mk t) =\n FreeGroup.map f (FreeGroup.of x * FreeGroup.mk t) by simp [g.map_mul, hg, ih])"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\nf : α → β\nx✝² y : FreeGroup α\ng : FreeGroup α →* FreeGroup β\nhg : ∀ (x : α), ↑g (FreeGroup.of x) = FreeGroup.of (f x)\nx✝¹ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\nt : List (α × Bool)\nih : ↑g (FreeGroup.mk t) = ↑(map f) (FreeGroup.mk t)\nx : α\nb : Bool\n⊢ ↑g ((FreeGroup.of x)⁻¹ * FreeGroup.mk t) = ↑(map f) ((FreeGroup.of x)⁻¹ * FreeGroup.mk t)",
"tactic": "simp [g.map_mul, g.map_inv, hg, ih]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\nf : α → β\nx✝² y : FreeGroup α\ng : FreeGroup α →* FreeGroup β\nhg : ∀ (x : α), ↑g (FreeGroup.of x) = FreeGroup.of (f x)\nx✝¹ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\nt : List (α × Bool)\nih : ↑g (FreeGroup.mk t) = ↑(map f) (FreeGroup.mk t)\nx : α\nb : Bool\n⊢ ↑g (FreeGroup.of x * FreeGroup.mk t) = ↑(map f) (FreeGroup.of x * FreeGroup.mk t)",
"tactic": "simp [g.map_mul, hg, ih]"
}
] |
[
842,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.cmpLT.trans_r
|
[] |
[
37,
84
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
36,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.disjoint_comm
|
[] |
[
2901,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2900,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.last_pos
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ 0 < last (n + 1)",
"tactic": "simp [lt_iff_val_lt_val]"
}
] |
[
545,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/CategoryTheory/Bicategory/Functor.lean
|
CategoryTheory.Pseudofunctor.to_oplax_mapComp
|
[] |
[
485,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Order/SemiconjSup.lean
|
Function.sSup_div_semiconj
|
[] |
[
120,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.insertNth_zero_right
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\ninst✝ : (j : Fin (n + 1)) → Zero (α j)\ni : Fin (n + 1)\nx : α i\n⊢ Pi.single i x i = x ∧ 0 = fun j => Pi.single i x (↑(succAbove i) j)",
"tactic": "simp [succAbove_ne, Pi.zero_def]"
}
] |
[
756,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castPred_last
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ ↑(castPred (last (n + 1))) = ↑(last n)",
"tactic": "simp [castPred, predAbove, castSucc_lt_last]"
}
] |
[
2319,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2318,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toNNReal_eq_toNNReal_iff
|
[] |
[
380,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/CategoryTheory/Quotient.lean
|
CategoryTheory.Quotient.sound
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝ : Category C\nr : HomRel C\na b : C\nf₁ f₂ : a ⟶ b\nh : r f₁ f₂\n⊢ (functor r).map f₁ = (functor r).map f₂",
"tactic": "simpa using Quot.sound (CompClosure.intro (𝟙 a) f₁ f₂ (𝟙 b) h)"
}
] |
[
141,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
11
] |
Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean
|
Matrix.compl_apply
|
[
{
"state_after": "V : Type u_2\nα : Type u_1\nβ : Type ?u.12780\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq V\nA : Matrix V V α\ninst✝¹ : Zero α\ninst✝ : One α\ni j : V\n⊢ (if i = j then 0 else if A i j = 0 then 1 else 0) = 0 ∨ (if i = j then 0 else if A i j = 0 then 1 else 0) = 1",
"state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.12780\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq V\nA : Matrix V V α\ninst✝¹ : Zero α\ninst✝ : One α\ni j : V\n⊢ compl A i j = 0 ∨ compl A i j = 1",
"tactic": "unfold compl"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.12780\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq V\nA : Matrix V V α\ninst✝¹ : Zero α\ninst✝ : One α\ni j : V\n⊢ (if i = j then 0 else if A i j = 0 then 1 else 0) = 0 ∨ (if i = j then 0 else if A i j = 0 then 1 else 0) = 1",
"tactic": "split_ifs <;> simp"
}
] |
[
114,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.prod_self_subset_prod_self
|
[] |
[
90,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
interior_closedBall'
|
[
{
"state_after": "case inl\nα : Type ?u.306784\nβ : Type ?u.306787\nγ : Type ?u.306790\nι : Type ?u.306793\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.306886\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\n⊢ interior (closedBall x 0) = ball x 0\n\ncase inr\nα : Type ?u.306784\nβ : Type ?u.306787\nγ : Type ?u.306790\nι : Type ?u.306793\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.306886\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ interior (closedBall x r) = ball x r",
"state_before": "α : Type ?u.306784\nβ : Type ?u.306787\nγ : Type ?u.306790\nι : Type ?u.306793\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.306886\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\n⊢ interior (closedBall x r) = ball x r",
"tactic": "rcases eq_or_ne r 0 with (rfl | hr)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.306784\nβ : Type ?u.306787\nγ : Type ?u.306790\nι : Type ?u.306793\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.306886\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\n⊢ interior (closedBall x 0) = ball x 0",
"tactic": "rw [closedBall_zero, ball_zero, interior_singleton]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type ?u.306784\nβ : Type ?u.306787\nγ : Type ?u.306790\nι : Type ?u.306793\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.306886\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ interior (closedBall x r) = ball x r",
"tactic": "exact interior_closedBall x hr"
}
] |
[
387,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Analysis/NormedSpace/DualNumber.lean
|
DualNumber.exp_eps
|
[] |
[
38,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.add_im
|
[] |
[
127,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
subset_interior_mul'
|
[] |
[
1311,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1310,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_inv_distrib
|
[] |
[
1807,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1806,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.apply_coe_mem_map
|
[] |
[
1426,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1425,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_iUnion₂
|
[] |
[
247,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.span_singleton_mul_left_mono
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.238401\ninst✝¹ : CommSemiring R\nI J K L : Ideal R\ninst✝ : IsDomain R\nx : R\nhx : x ≠ 0\n⊢ I * span {x} ≤ J * span {x} ↔ I ≤ J",
"tactic": "simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx"
}
] |
[
576,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.image_symm
|
[] |
[
456,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\n⊢ ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p) ↔\n ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\n⊢ ↑(reflection s₁) p = ↑(reflection s₂) p ↔ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"tactic": "rw [reflection_apply, reflection_apply]"
},
{
"state_after": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\n⊢ ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p) →\n ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)\n\ncase mpr\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\n⊢ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p) →\n ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p)",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\n⊢ ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p) ↔\n ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"tactic": "constructor"
},
{
"state_after": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\nh :\n ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p)\n⊢ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"state_before": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\n⊢ ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p) →\n ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"tactic": "intro h"
},
{
"state_after": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\nh : 2 = 0 ∨ ↑(↑(orthogonalProjection s₁) p) -ᵥ ↑(↑(orthogonalProjection s₂) p) = 0\n⊢ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"state_before": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\nh :\n ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p)\n⊢ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"tactic": "rw [← @vsub_eq_zero_iff_eq V, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_comm, add_sub_assoc,\n vsub_sub_vsub_cancel_right, ←\n two_smul ℝ ((orthogonalProjection s₁ p : P) -ᵥ orthogonalProjection s₂ p), smul_eq_zero] at h"
},
{
"state_after": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\nh : ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)\n⊢ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"state_before": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\nh : 2 = 0 ∨ ↑(↑(orthogonalProjection s₁) p) -ᵥ ↑(↑(orthogonalProjection s₂) p) = 0\n⊢ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"tactic": "norm_num at h"
},
{
"state_after": "no goals",
"state_before": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\nh : ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)\n⊢ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)",
"tactic": "exact h"
},
{
"state_after": "case mpr\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\nh : ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)\n⊢ ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p)",
"state_before": "case mpr\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\n⊢ ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p) →\n ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p)",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nV : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s₁ }\ninst✝² : Nonempty { x // x ∈ s₂ }\ninst✝¹ : CompleteSpace { x // x ∈ direction s₁ }\ninst✝ : CompleteSpace { x // x ∈ direction s₂ }\np : P\nh : ↑(↑(orthogonalProjection s₁) p) = ↑(↑(orthogonalProjection s₂) p)\n⊢ ↑(↑(orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₁) p) =\n ↑(↑(orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s₂) p)",
"tactic": "rw [h]"
}
] |
[
613,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
600,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.insert_val_of_not_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.103482\nγ : Type ?u.103485\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝ b a : α\ns : Finset α\nh : ¬a ∈ s\n⊢ (insert a s).val = a ::ₘ s.val",
"tactic": "rw [insert_val, ndinsert_of_not_mem h]"
}
] |
[
1056,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/Combinatorics/Quiver/Path.lean
|
Quiver.Path.comp_cons
|
[] |
[
100,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
Setoid.IsPartition.pairwiseDisjoint
|
[] |
[
225,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
TopCat.pullbackIsoProdSubtype_inv_fst
|
[
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullbackIsoProdSubtype f g).inv ≫ pullback.fst = pullbackFst f g",
"tactic": "simp [pullbackCone, pullbackIsoProdSubtype]"
}
] |
[
103,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.inv_tendsto_zero
|
[] |
[
154,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.get_equiv
|
[] |
[
1018,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1016,
1
] |
Mathlib/Data/Rat/Lemmas.lean
|
Rat.pnatDen_one
|
[] |
[
342,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/CategoryTheory/Monoidal/Mon_.lean
|
Mon_.Mon_tensor_one_mul
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ ((M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X))) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =\n (λ_ (M.X ⊗ N.X)).hom",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =\n (λ_ (M.X ⊗ N.X)).hom",
"tactic": "rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n (tensor_μ C (tensorUnit', 𝟙_ C) (M.X, N.X) ≫ ((M.one ⊗ 𝟙 M.X) ⊗ N.one ⊗ 𝟙 N.X)) ≫ (M.mul ⊗ N.mul) =\n (λ_ (M.X ⊗ N.X)).hom",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ ((M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X))) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =\n (λ_ (M.X ⊗ N.X)).hom",
"tactic": "slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (tensorUnit', 𝟙_ C) (M.X, N.X) ≫ ((λ_ M.X).hom ⊗ (λ_ N.X).hom) =\n (λ_ (M.X ⊗ N.X)).hom",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n (tensor_μ C (tensorUnit', 𝟙_ C) (M.X, N.X) ≫ ((M.one ⊗ 𝟙 M.X) ⊗ N.one ⊗ 𝟙 N.X)) ≫ (M.mul ⊗ N.mul) =\n (λ_ (M.X ⊗ N.X)).hom",
"tactic": "slice_lhs 3 4 => rw [← tensor_comp, one_mul M, one_mul N]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (λ_ (M.X ⊗ N.X)).hom =\n ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (tensorUnit', 𝟙_ C) (M.X, N.X) ≫ ((λ_ M.X).hom ⊗ (λ_ N.X).hom)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (tensorUnit', 𝟙_ C) (M.X, N.X) ≫ ((λ_ M.X).hom ⊗ (λ_ N.X).hom) =\n (λ_ (M.X ⊗ N.X)).hom",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (λ_ (M.X ⊗ N.X)).hom =\n ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (tensorUnit', 𝟙_ C) (M.X, N.X) ≫ ((λ_ M.X).hom ⊗ (λ_ N.X).hom)",
"tactic": "exact tensor_left_unitality C M.X N.X"
}
] |
[
418,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
MeasurePreservingQuotientGroup.mk'
|
[
{
"state_after": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : MeasureTheory.Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nc : ℝ≥0\nh : ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) = ↑c\n⊢ ↑c • haarMeasure K = c • haarMeasure K",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : MeasureTheory.Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nc : ℝ≥0\nh : ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) = ↑c\n⊢ map (↑(QuotientGroup.mk' Γ)) (Measure.restrict μ 𝓕) = c • haarMeasure K",
"tactic": "rw [h𝓕.map_restrict_quotient K h𝓕_finite, h]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : TopologicalGroup G\ninst✝⁸ : BorelSpace G\nμ : MeasureTheory.Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain { x // x ∈ ↑Subgroup.opposite Γ } 𝓕\ninst✝⁷ : Countable { x // x ∈ Γ }\ninst✝⁶ : MeasurableSpace (G ⧸ Γ)\ninst✝⁵ : BorelSpace (G ⧸ Γ)\ninst✝⁴ : T2Space (G ⧸ Γ)\ninst✝³ : SecondCountableTopology (G ⧸ Γ)\nK : PositiveCompacts (G ⧸ Γ)\ninst✝² : Subgroup.Normal Γ\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsMulRightInvariant μ\nh𝓕_finite : ↑↑μ 𝓕 < ⊤\nc : ℝ≥0\nh : ↑↑μ (𝓕 ∩ ↑(QuotientGroup.mk' Γ) ⁻¹' ↑K) = ↑c\n⊢ ↑c • haarMeasure K = c • haarMeasure K",
"tactic": "rfl"
}
] |
[
159,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Order/Basic.lean
|
eq_of_ge_of_not_gt
|
[] |
[
425,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Topology/Order/Basic.lean
|
frontier_Iic
|
[] |
[
2341,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2340,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.coe_pureOneHom
|
[] |
[
161,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Topology/Algebra/ContinuousMonoidHom.lean
|
ContinuousMonoidHom.inducing_toContinuousMap
|
[] |
[
281,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.C_surjective
|
[
{
"state_after": "R✝ : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ✝ : Type ?u.315261\na✝ a' a₁ a₂ : R✝\ne : ℕ\nn m : σ✝\ns : σ✝ →₀ ℕ\ninst✝³ : CommSemiring R✝\ninst✝² : CommSemiring S₁\np✝ q : MvPolynomial σ✝ R✝\nR : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : IsEmpty σ\np : MvPolynomial σ R\na : σ →₀ ℕ\n⊢ ↑(↑C (toFun p 0)) a = ↑p a",
"state_before": "R✝ : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ✝ : Type ?u.315261\na a' a₁ a₂ : R✝\ne : ℕ\nn m : σ✝\ns : σ✝ →₀ ℕ\ninst✝³ : CommSemiring R✝\ninst✝² : CommSemiring S₁\np q : MvPolynomial σ✝ R✝\nR : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : IsEmpty σ\n⊢ Surjective ↑C",
"tactic": "refine' fun p => ⟨p.toFun 0, Finsupp.ext fun a => _⟩"
},
{
"state_after": "R✝ : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ✝ : Type ?u.315261\na✝ a' a₁ a₂ : R✝\ne : ℕ\nn m : σ✝\ns : σ✝ →₀ ℕ\ninst✝³ : CommSemiring R✝\ninst✝² : CommSemiring S₁\np✝ q : MvPolynomial σ✝ R✝\nR : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : IsEmpty σ\np : MvPolynomial σ R\na : σ →₀ ℕ\n⊢ toFun p 0 = ↑p 0",
"state_before": "R✝ : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ✝ : Type ?u.315261\na✝ a' a₁ a₂ : R✝\ne : ℕ\nn m : σ✝\ns : σ✝ →₀ ℕ\ninst✝³ : CommSemiring R✝\ninst✝² : CommSemiring S₁\np✝ q : MvPolynomial σ✝ R✝\nR : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : IsEmpty σ\np : MvPolynomial σ R\na : σ →₀ ℕ\n⊢ ↑(↑C (toFun p 0)) a = ↑p a",
"tactic": "simp only [C_apply, ←single_eq_monomial, (Finsupp.ext isEmptyElim (α := σ) : a = 0),\n single_eq_same]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ✝ : Type ?u.315261\na✝ a' a₁ a₂ : R✝\ne : ℕ\nn m : σ✝\ns : σ✝ →₀ ℕ\ninst✝³ : CommSemiring R✝\ninst✝² : CommSemiring S₁\np✝ q : MvPolynomial σ✝ R✝\nR : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : IsEmpty σ\np : MvPolynomial σ R\na : σ →₀ ℕ\n⊢ toFun p 0 = ↑p 0",
"tactic": "rfl"
}
] |
[
248,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.model_distinctConstantsTheory
|
[
{
"state_after": "L : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\n⊢ (∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ →\n M ⊨ φ) ↔\n Set.InjOn (fun i => ↑(Language.con L i)) s",
"state_before": "L : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\n⊢ M ⊨ distinctConstantsTheory L s ↔ Set.InjOn (fun i => ↑(Language.con L i)) s",
"tactic": "simp only [distinctConstantsTheory, Theory.model_iff, Set.mem_image, Set.mem_inter,\n Set.mem_prod, Set.mem_compl, Prod.exists, forall_exists_index, and_imp]"
},
{
"state_after": "case refine'_1\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh :\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : (fun i => ↑(Language.con L i)) a = (fun i => ↑(Language.con L i)) b\n⊢ a = b\n\ncase refine'_2\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\n⊢ Set.InjOn (fun i => ↑(Language.con L i)) s →\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ",
"state_before": "L : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\n⊢ (∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ →\n M ⊨ φ) ↔\n Set.InjOn (fun i => ↑(Language.con L i)) s",
"tactic": "refine' ⟨fun h a as b bs ab => _, _⟩"
},
{
"state_after": "case refine'_1\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh :\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : a ≠ b\n⊢ ↑(Language.con L a) ≠ ↑(Language.con L b)",
"state_before": "case refine'_1\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh :\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : (fun i => ↑(Language.con L i)) a = (fun i => ↑(Language.con L i)) b\n⊢ a = b",
"tactic": "contrapose! ab"
},
{
"state_after": "case refine'_1\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh :\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : a ≠ b\nh' : M ⊨ Formula.not (Term.equal (Constants.term (Language.con L a)) (Constants.term (Language.con L b)))\n⊢ ↑(Language.con L a) ≠ ↑(Language.con L b)",
"state_before": "case refine'_1\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh :\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : a ≠ b\n⊢ ↑(Language.con L a) ≠ ↑(Language.con L b)",
"tactic": "have h' := h _ a b ⟨⟨as, bs⟩, ab⟩ rfl"
},
{
"state_after": "case refine'_1\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh :\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : a ≠ b\nh' : ¬↑(Language.con L a) = ↑(Language.con L b)\n⊢ ↑(Language.con L a) ≠ ↑(Language.con L b)",
"state_before": "case refine'_1\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh :\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : a ≠ b\nh' : M ⊨ Formula.not (Term.equal (Constants.term (Language.con L a)) (Constants.term (Language.con L b)))\n⊢ ↑(Language.con L a) ≠ ↑(Language.con L b)",
"tactic": "simp only [Sentence.Realize, Formula.realize_not, Formula.realize_equal,\n Term.realize_constants] at h'"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh :\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : a ≠ b\nh' : ¬↑(Language.con L a) = ↑(Language.con L b)\n⊢ ↑(Language.con L a) ≠ ↑(Language.con L b)",
"tactic": "exact h'"
},
{
"state_after": "case refine'_2.intro.intro\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh : Set.InjOn (fun i => ↑(Language.con L i)) s\na b : α\nab : (a, b) ∈ Set.diagonal αᶜ\nas : (a, b).fst ∈ s\nbs : (a, b).snd ∈ s\n⊢ M ⊨ Formula.not (Term.equal (Constants.term (Language.con L a)) (Constants.term (Language.con L b)))",
"state_before": "case refine'_2\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\n⊢ Set.InjOn (fun i => ↑(Language.con L i)) s →\n ∀ (φ : Sentence (L[[α]])) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ Set.diagonal αᶜ →\n Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = φ → M ⊨ φ",
"tactic": "rintro h φ a b ⟨⟨as, bs⟩, ab⟩ rfl"
},
{
"state_after": "case refine'_2.intro.intro\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh : Set.InjOn (fun i => ↑(Language.con L i)) s\na b : α\nab : (a, b) ∈ Set.diagonal αᶜ\nas : (a, b).fst ∈ s\nbs : (a, b).snd ∈ s\n⊢ ¬↑(Language.con L a) = ↑(Language.con L b)",
"state_before": "case refine'_2.intro.intro\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh : Set.InjOn (fun i => ↑(Language.con L i)) s\na b : α\nab : (a, b) ∈ Set.diagonal αᶜ\nas : (a, b).fst ∈ s\nbs : (a, b).snd ∈ s\n⊢ M ⊨ Formula.not (Term.equal (Constants.term (Language.con L a)) (Constants.term (Language.con L b)))",
"tactic": "simp only [Sentence.Realize, Formula.realize_not, Formula.realize_equal, Term.realize_constants]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nL : Language\nL' : Language\nM✝ : Type w\nN : Type ?u.854555\nP : Type ?u.854558\ninst✝³ : Structure L M✝\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nM : Type w\ninst✝ : Structure (L[[α]]) M\ns : Set α\nh : Set.InjOn (fun i => ↑(Language.con L i)) s\na b : α\nab : (a, b) ∈ Set.diagonal αᶜ\nas : (a, b).fst ∈ s\nbs : (a, b).snd ∈ s\n⊢ ¬↑(Language.con L a) = ↑(Language.con L b)",
"tactic": "exact fun contra => ab (h as bs contra)"
}
] |
[
1088,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1076,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.toMeasure_zero
|
[] |
[
218,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.prime_C_iff
|
[
{
"state_after": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\n⊢ Prime (↑C r)",
"state_before": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\n⊢ Prime r → Prime (↑C r)",
"tactic": "intro hr"
},
{
"state_after": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nthis : r ≠ 0\n⊢ Prime (↑C r)",
"state_before": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\n⊢ Prime (↑C r)",
"tactic": "have := hr.1"
},
{
"state_after": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Ideal.IsPrime (Ideal.span {r})\nthis : r ≠ 0\n⊢ Ideal.IsPrime (Ideal.span {↑C r})\n\nR : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Ideal.IsPrime (Ideal.span {r})\nthis : r ≠ 0\n⊢ ↑C r ≠ 0\n\nR : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nthis : r ≠ 0\n⊢ r ≠ 0",
"state_before": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nthis : r ≠ 0\n⊢ Prime (↑C r)",
"tactic": "rw [← Ideal.span_singleton_prime] at hr⊢"
},
{
"state_after": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Ideal.IsPrime (Ideal.span {r})\nthis : r ≠ 0\n⊢ Ideal.IsPrime (Ideal.map C (Ideal.span {r}))",
"state_before": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Ideal.IsPrime (Ideal.span {r})\nthis : r ≠ 0\n⊢ Ideal.IsPrime (Ideal.span {↑C r})",
"tactic": "rw [← Set.image_singleton, ← Ideal.map_span]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Ideal.IsPrime (Ideal.span {r})\nthis : r ≠ 0\n⊢ Ideal.IsPrime (Ideal.map C (Ideal.span {r}))",
"tactic": "apply Ideal.isPrime_map_C_of_isPrime hr"
},
{
"state_after": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Ideal.IsPrime (Ideal.span {r})\nthis : r ≠ 0\nh : ↑C r = 0\n⊢ False",
"state_before": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Ideal.IsPrime (Ideal.span {r})\nthis : r ≠ 0\n⊢ ↑C r ≠ 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Ideal.IsPrime (Ideal.span {r})\nthis : r ≠ 0\nh : ↑C r = 0\n⊢ False",
"tactic": "apply (this (C_eq_zero.mp h))"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.257711\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nthis : r ≠ 0\n⊢ r ≠ 0",
"tactic": "assumption"
}
] |
[
762,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
1
] |
Mathlib/Algebra/Order/Kleene.lean
|
add_idem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.17357\nι : Type ?u.17360\nπ : ι → Type ?u.17365\ninst✝ : IdemSemiring α\na✝ b c a : α\n⊢ a + a = a",
"tactic": "simp"
}
] |
[
144,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.smul_def
|
[] |
[
476,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.aecover_Ico_of_Ioo
|
[] |
[
254,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
|
Complex.log_inv_eq_ite
|
[
{
"state_after": "case pos\nx : ℂ\nhx : x = 0\n⊢ log x⁻¹ = if arg x = π then -↑(starRingEnd ℂ) (log x) else -log x\n\ncase neg\nx : ℂ\nhx : ¬x = 0\n⊢ log x⁻¹ = if arg x = π then -↑(starRingEnd ℂ) (log x) else -log x",
"state_before": "x : ℂ\n⊢ log x⁻¹ = if arg x = π then -↑(starRingEnd ℂ) (log x) else -log x",
"tactic": "by_cases hx : x = 0"
},
{
"state_after": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ (if arg x = π then log x else ↑(starRingEnd ℂ) (log x)) - ↑(Real.log (↑normSq x)) =\n if arg x = π then -↑(starRingEnd ℂ) (log x) else -log x\n\ncase neg.hr\nx : ℂ\nhx : ¬x = 0\n⊢ 0 < (↑normSq x)⁻¹\n\ncase neg.hx\nx : ℂ\nhx : ¬x = 0\n⊢ ↑(starRingEnd ℂ) x ≠ 0",
"state_before": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ log x⁻¹ = if arg x = π then -↑(starRingEnd ℂ) (log x) else -log x",
"tactic": "rw [inv_def, log_mul_ofReal, Real.log_inv, ofReal_neg, ← sub_eq_neg_add, log_conj_eq_ite]"
},
{
"state_after": "no goals",
"state_before": "case pos\nx : ℂ\nhx : x = 0\n⊢ log x⁻¹ = if arg x = π then -↑(starRingEnd ℂ) (log x) else -log x",
"tactic": "simp [hx]"
},
{
"state_after": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ (if arg x = π then ↑(Real.log (↑abs x)) + ↑(arg x) * I else ↑(Real.log (↑abs x)) + -(↑(arg x) * I)) -\n ↑2 * ↑(Real.log (↑abs x)) =\n if arg x = π then -↑(Real.log (↑abs x)) + ↑(arg x) * I else -↑(Real.log (↑abs x)) + -(↑(arg x) * I)",
"state_before": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ (if arg x = π then log x else ↑(starRingEnd ℂ) (log x)) - ↑(Real.log (↑normSq x)) =\n if arg x = π then -↑(starRingEnd ℂ) (log x) else -log x",
"tactic": "simp_rw [log, map_add, map_mul, conj_ofReal, conj_I, normSq_eq_abs, Real.log_pow,\n Nat.cast_two, ofReal_mul, ofReal_bit0, ofReal_one, neg_add, mul_neg, two_mul, neg_neg]"
},
{
"state_after": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ (if arg x = π then ↑(Real.log (↑abs x)) + ↑(arg x) * I else ↑(Real.log (↑abs x)) + -(↑(arg x) * I)) -\n 2 * ↑(Real.log (↑abs x)) =\n if arg x = π then -↑(Real.log (↑abs x)) + ↑(arg x) * I else -↑(Real.log (↑abs x)) + -(↑(arg x) * I)",
"state_before": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ (if arg x = π then ↑(Real.log (↑abs x)) + ↑(arg x) * I else ↑(Real.log (↑abs x)) + -(↑(arg x) * I)) -\n ↑2 * ↑(Real.log (↑abs x)) =\n if arg x = π then -↑(Real.log (↑abs x)) + ↑(arg x) * I else -↑(Real.log (↑abs x)) + -(↑(arg x) * I)",
"tactic": "norm_num"
},
{
"state_after": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ (if arg x = π then ↑(Real.log (↑abs x)) + ↑(arg x) * I else ↑(Real.log (↑abs x)) + -(↑(arg x) * I)) -\n (↑(Real.log (↑abs x)) + ↑(Real.log (↑abs x))) =\n if arg x = π then -↑(Real.log (↑abs x)) + ↑(arg x) * I else -↑(Real.log (↑abs x)) + -(↑(arg x) * I)",
"state_before": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ (if arg x = π then ↑(Real.log (↑abs x)) + ↑(arg x) * I else ↑(Real.log (↑abs x)) + -(↑(arg x) * I)) -\n 2 * ↑(Real.log (↑abs x)) =\n if arg x = π then -↑(Real.log (↑abs x)) + ↑(arg x) * I else -↑(Real.log (↑abs x)) + -(↑(arg x) * I)",
"tactic": "rw [two_mul]"
},
{
"state_after": "case neg.inl\nx : ℂ\nhx : ¬x = 0\nh✝ : arg x = π\n⊢ ↑(Real.log (↑abs x)) + ↑(arg x) * I - (↑(Real.log (↑abs x)) + ↑(Real.log (↑abs x))) =\n -↑(Real.log (↑abs x)) + ↑(arg x) * I\n\ncase neg.inr\nx : ℂ\nhx : ¬x = 0\nh✝ : ¬arg x = π\n⊢ ↑(Real.log (↑abs x)) + -(↑(arg x) * I) - (↑(Real.log (↑abs x)) + ↑(Real.log (↑abs x))) =\n -↑(Real.log (↑abs x)) + -(↑(arg x) * I)",
"state_before": "case neg\nx : ℂ\nhx : ¬x = 0\n⊢ (if arg x = π then ↑(Real.log (↑abs x)) + ↑(arg x) * I else ↑(Real.log (↑abs x)) + -(↑(arg x) * I)) -\n (↑(Real.log (↑abs x)) + ↑(Real.log (↑abs x))) =\n if arg x = π then -↑(Real.log (↑abs x)) + ↑(arg x) * I else -↑(Real.log (↑abs x)) + -(↑(arg x) * I)",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nx : ℂ\nhx : ¬x = 0\nh✝ : arg x = π\n⊢ ↑(Real.log (↑abs x)) + ↑(arg x) * I - (↑(Real.log (↑abs x)) + ↑(Real.log (↑abs x))) =\n -↑(Real.log (↑abs x)) + ↑(arg x) * I",
"tactic": "rw [add_sub_right_comm, sub_add_cancel']"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nx : ℂ\nhx : ¬x = 0\nh✝ : ¬arg x = π\n⊢ ↑(Real.log (↑abs x)) + -(↑(arg x) * I) - (↑(Real.log (↑abs x)) + ↑(Real.log (↑abs x))) =\n -↑(Real.log (↑abs x)) + -(↑(arg x) * I)",
"tactic": "rw [add_sub_right_comm, sub_add_cancel']"
},
{
"state_after": "no goals",
"state_before": "case neg.hr\nx : ℂ\nhx : ¬x = 0\n⊢ 0 < (↑normSq x)⁻¹",
"tactic": "rwa [inv_pos, Complex.normSq_pos]"
},
{
"state_after": "no goals",
"state_before": "case neg.hx\nx : ℂ\nhx : ¬x = 0\n⊢ ↑(starRingEnd ℂ) x ≠ 0",
"tactic": "rwa [map_ne_zero]"
}
] |
[
133,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Data/Nat/Pow.lean
|
Nat.pow_right_strictMono
|
[] |
[
105,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
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