url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | haveI := (hF.preserves α) | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
⊢ PreservesLimit (Discrete.functor fun a => (Z a).op) F | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimitsOfShape (Discrete α) F
⊢ PreservesLimit (Discrete.functor fun a => (Z a).op) F | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
⊢ PreservesLimit (Discrete.functor fun a => (Z a).op) F
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | infer_instance | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimitsOfShape (Discrete α) F
⊢ PreservesLimit (Discrete.functor fun a => (Z a).op) F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimitsOfShape (Discrete α) F
⊢ PreservesLimit (Discrete.functor fun a => (Z a).op) F
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | unfold fromFirst | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ Equalizer.forkMap F S ≫ fromFirst hS hF HIso = 𝟙 (F.obj X.op) | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ Equalizer.forkMap F S ≫
comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫ F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op) =
𝟙 (F.obj X.op) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ Equalizer.forkMap F S ≫ fromFirst hS hF HIso = 𝟙 (F.obj X.op)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp only [← Category.assoc] | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ Equalizer.forkMap F S ≫
comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫ F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op) =
𝟙 (F.obj X.op) | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ (((Equalizer.forkMap F S ≫ comparisoninv hS F) ≫ (PreservesProduct.iso F fun a => (Z a).op).inv) ≫
F.map (CoprodToProd Z).inv) ≫
F.map (inv (Sigma.desc π).op) =
𝟙 (F.obj X.op) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ Equalizer.forkMap F S ≫
comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫ F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op) =
𝟙 (F.obj X.op)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | rw [Functor.map_inv, IsIso.comp_inv_eq, Category.id_comp, ← Functor.mapIso_inv,
Iso.comp_inv_eq, Functor.mapIso_hom, Iso.comp_inv_eq, ← Functor.map_comp, descOpCompCoprodToProd] | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ (((Equalizer.forkMap F S ≫ comparisoninv hS F) ≫ (PreservesProduct.iso F fun a => (Z a).op).inv) ≫
F.map (CoprodToProd Z).inv) ≫
F.map (inv (Sigma.desc π).op) =
𝟙 (F.obj X.op) | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F =
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ (((Equalizer.forkMap F S ≫ comparisoninv hS F) ≫ (PreservesProduct.iso F fun a => (Z a).op).inv) ≫
F.map (CoprodToProd Z).inv) ≫
F.map (inv (Sigma.desc π).op) =
𝟙 (F.obj X.op)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | have : F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => op (Z a)).hom =
Pi.lift (fun a => F.map ((Sigma.ι Z a ≫ (Sigma.desc π)).op)) := by simp | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F =
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F =
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F =
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | erw [this] | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F =
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F = Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F =
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | refine' funext (fun s => _) | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F = Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
⊢ (Equalizer.forkMap F S ≫ comparisoninv hS F) s = Pi.lift (fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op) s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
⊢ Equalizer.forkMap F S ≫ comparisoninv hS F = Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp only [types_comp_apply, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app] | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
⊢ (Equalizer.forkMap F S ≫ comparisoninv hS F) s = Pi.lift (fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op) s | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
⊢ comparisoninv hS F (Equalizer.forkMap F S s) = Pi.lift (fun a => F.map (π a).op) s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
⊢ (Equalizer.forkMap F S ≫ comparisoninv hS F) s = Pi.lift (fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op) s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | ext a | case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
⊢ comparisoninv hS F (Equalizer.forkMap F S s) = Pi.lift (fun a => F.map (π a).op) s | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a (comparisoninv hS F (Equalizer.forkMap F S s)) =
limit.π (Discrete.functor fun a => F.obj (Z a).op) a (Pi.lift (fun a => F.map (π a).op) s) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
⊢ comparisoninv hS F (Equalizer.forkMap F S s) = Pi.lift (fun a => F.map (π a).op) s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | rw [Types.Limit.lift_π_apply] | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a (comparisoninv hS F (Equalizer.forkMap F S s)) =
limit.π (Discrete.functor fun a => F.obj (Z a).op) a (Pi.lift (fun a => F.map (π a).op) s) | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a (comparisoninv hS F (Equalizer.forkMap F S s)) =
(Fan.mk (F.obj X.op) fun a => F.map (π a).op).π.app a s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a (comparisoninv hS F (Equalizer.forkMap F S s)) =
limit.π (Discrete.functor fun a => F.obj (Z a).op) a (Pi.lift (fun a => F.map (π a).op) s)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | dsimp [comparisoninv] | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a (comparisoninv hS F (Equalizer.forkMap F S s)) =
(Fan.mk (F.obj X.op) fun a => F.map (π a).op).π.app a s | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (v hS a).fst.op))
(Equalizer.forkMap F S s)) =
F.map (π a.as).op s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a (comparisoninv hS F (Equalizer.forkMap F S s)) =
(Fan.mk (F.obj X.op) fun a => F.map (π a).op).π.app a s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp_rw [v.fst] | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (v hS a).fst.op))
(Equalizer.forkMap F S s)) =
F.map (π a.as).op s | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (Z a).op)) (Equalizer.forkMap F S s)) =
F.map (π a.as).op s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (v hS a).fst.op))
(Equalizer.forkMap F S s)) =
F.map (π a.as).op s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp only [Functor.map_id, Category.comp_id] | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (Z a).op)) (Equalizer.forkMap F S s)) =
F.map (π a.as).op s | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a)) (Equalizer.forkMap F S s)) =
F.map (π a.as).op s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (Z a).op)) (Equalizer.forkMap F S s)) =
F.map (π a.as).op s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | rw [Types.Limit.lift_π_apply] | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a)) (Equalizer.forkMap F S s)) =
F.map (π a.as).op s | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ (Fan.mk (∏ fun f => F.obj f.fst.op) fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a)).π.app a
(Equalizer.forkMap F S s) =
F.map (π a.as).op s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ limit.π (Discrete.functor fun a => F.obj (Z a).op) a
(Pi.lift (fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a)) (Equalizer.forkMap F S s)) =
F.map (π a.as).op s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp only [Fan.mk_pt, Equalizer.forkMap, Fan.mk_π_app, Types.pi_lift_π_apply, v.snd] | case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ (Fan.mk (∏ fun f => F.obj f.fst.op) fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a)).π.app a
(Equalizer.forkMap F S s) =
F.map (π a.as).op s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_1.w
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this✝ : PreservesLimit (Discrete.functor fun a => (Z a).op) F
this :
F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
s : F.obj X.op
a : Discrete α
⊢ (Fan.mk (∏ fun f => F.obj f.fst.op) fun a => Pi.π (fun f => F.obj f.fst.op) (v hS a)).π.app a
(Equalizer.forkMap F S s) =
F.map (π a.as).op s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ F.map (Pi.lift fun a => (π a).op) ≫ (PreservesProduct.iso F fun a => (Z a).op).hom =
Pi.lift fun a => F.map (Sigma.ι Z a ≫ Sigma.desc π).op
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | refine' Limits.Pi.hom_ext _ _ (fun f => _) | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ fromFirst hS hF HIso ≫ Equalizer.forkMap F S = 𝟙 (Equalizer.FirstObj F S) | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ (fromFirst hS hF HIso ≫ Equalizer.forkMap F S) ≫ Pi.π (fun f => F.obj f.fst.op) f =
𝟙 (Equalizer.FirstObj F S) ≫ Pi.π (fun f => F.obj f.fst.op) f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
⊢ fromFirst hS hF HIso ≫ Equalizer.forkMap F S = 𝟙 (Equalizer.FirstObj F S)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | dsimp [Equalizer.forkMap] | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ (fromFirst hS hF HIso ≫ Equalizer.forkMap F S) ≫ Pi.π (fun f => F.obj f.fst.op) f =
𝟙 (Equalizer.FirstObj F S) ≫ Pi.π (fun f => F.obj f.fst.op) f | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ (fromFirst hS hF HIso ≫ Pi.lift fun f => F.map (↑f.snd).op) ≫ Pi.π (fun f => F.obj f.fst.op) f =
𝟙 (Equalizer.FirstObj F S) ≫ Pi.π (fun f => F.obj f.fst.op) f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ (fromFirst hS hF HIso ≫ Equalizer.forkMap F S) ≫ Pi.π (fun f => F.obj f.fst.op) f =
𝟙 (Equalizer.FirstObj F S) ≫ Pi.π (fun f => F.obj f.fst.op) f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | rw [Category.id_comp, Category.assoc, limit.lift_π, Limits.Fan.mk_π_app] | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ (fromFirst hS hF HIso ≫ Pi.lift fun f => F.map (↑f.snd).op) ≫ Pi.π (fun f => F.obj f.fst.op) f =
𝟙 (Equalizer.FirstObj F S) ≫ Pi.π (fun f => F.obj f.fst.op) f | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ fromFirst hS hF HIso ≫ F.map (↑{ as := f }.as.snd).op = Pi.π (fun f => F.obj f.fst.op) f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ (fromFirst hS hF HIso ≫ Pi.lift fun f => F.map (↑f.snd).op) ≫ Pi.π (fun f => F.obj f.fst.op) f =
𝟙 (Equalizer.FirstObj F S) ≫ Pi.π (fun f => F.obj f.fst.op) f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp only | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ fromFirst hS hF HIso ≫ F.map (↑{ as := f }.as.snd).op = Pi.π (fun f => F.obj f.fst.op) f | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ fromFirst hS hF HIso ≫ F.map (↑f.snd).op = Pi.π (fun f => F.obj f.fst.op) f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ fromFirst hS hF HIso ≫ F.map (↑{ as := f }.as.snd).op = Pi.π (fun f => F.obj f.fst.op) f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | obtain ⟨a, ha⟩ := vsurj hS f | case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ fromFirst hS hF HIso ≫ F.map (↑f.snd).op = Pi.π (fun f => F.obj f.fst.op) f | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ fromFirst hS hF HIso ≫ F.map (↑f.snd).op = Pi.π (fun f => F.obj f.fst.op) f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
⊢ fromFirst hS hF HIso ≫ F.map (↑f.snd).op = Pi.π (fun f => F.obj f.fst.op) f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | unfold fromFirst | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ fromFirst hS hF HIso ≫ F.map (↑f.snd).op = Pi.π (fun f => F.obj f.fst.op) f | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ (comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫ F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op)) ≫
F.map (↑f.snd).op =
Pi.π (fun f => F.obj f.fst.op) f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ fromFirst hS hF HIso ≫ F.map (↑f.snd).op = Pi.π (fun f => F.obj f.fst.op) f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp only [Category.assoc] | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ (comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫ F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op)) ≫
F.map (↑f.snd).op =
Pi.π (fun f => F.obj f.fst.op) f | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫
F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op) ≫ F.map (↑f.snd).op =
Pi.π (fun f => F.obj f.fst.op) f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ (comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫ F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op)) ≫
F.map (↑f.snd).op =
Pi.π (fun f => F.obj f.fst.op) f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | rw [← Functor.map_comp, ← op_inv, ← op_comp, ← ha, v.snd hS, piCompInvdesc,
← Functor.map_comp, CoprodToProdInvComp.ι, @PreservesProduct.isoInvCompMap _ _ _ _ F _ _ _ _ (_) a] | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫
F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op) ≫ F.map (↑f.snd).op =
Pi.π (fun f => F.obj f.fst.op) f | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ comparisoninv hS F ≫ Pi.π (fun j => F.obj (Z j).op) a = Pi.π (fun f => F.obj f.fst.op) (v hS a) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ comparisoninv hS F ≫
(PreservesProduct.iso F fun a => (Z a).op).inv ≫
F.map (CoprodToProd Z).inv ≫ F.map (inv (Sigma.desc π).op) ≫ F.map (↑f.snd).op =
Pi.π (fun f => F.obj f.fst.op) f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | simp only [comparisoninv, op_id, limit.lift_π, Fan.mk_pt, Fan.mk_π_app] | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ comparisoninv hS F ≫ Pi.π (fun j => F.obj (Z j).op) a = Pi.π (fun f => F.obj f.fst.op) (v hS a) | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (v hS a).fst.op) = Pi.π (fun f => F.obj f.fst.op) (v hS a) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ comparisoninv hS F ≫ Pi.π (fun j => F.obj (Z j).op) a = Pi.π (fun f => F.obj f.fst.op) (v hS a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForDagurSieveIsIsoFork | [204, 1] | [238, 37] | erw [F.map_id, Category.comp_id] | case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (v hS a).fst.op) = Pi.π (fun f => F.obj f.fst.op) (v hS a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine'_2.intro
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
α : Type
w✝ : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
HIso : IsIso (Sigma.desc π)
this : PreservesLimit (Discrete.functor fun a => (Z a).op) F
f : (Y : C) × { f // S f }
a : α
ha : v hS a = f
⊢ Pi.π (fun f => F.obj f.fst.op) (v hS a) ≫ F.map (𝟙 (v hS a).fst.op) = Pi.π (fun f => F.obj f.fst.op) (v hS a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | refine' (Equalizer.Presieve.sheaf_condition' F <| isPullbackSieve_ExtensiveSieve hS).2 _ | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ Presieve.IsSheafFor F S | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ Nonempty
(IsLimit
(Fork.ofι (Equalizer.forkMap F fun {Y} {f} => S f)
(_ :
(Equalizer.forkMap F fun {Y} {f} => S f) ≫
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) =
(Equalizer.forkMap F fun {Y} {f} => S f) ≫
Equalizer.Presieve.secondMap' F
(_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g)))) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ Presieve.IsSheafFor F S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | rw [Limits.Types.type_equalizer_iff_unique] | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ Nonempty
(IsLimit
(Fork.ofι (Equalizer.forkMap F fun {Y} {f} => S f)
(_ :
(Equalizer.forkMap F fun {Y} {f} => S f) ≫
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) =
(Equalizer.forkMap F fun {Y} {f} => S f) ≫
Equalizer.Presieve.secondMap' F
(_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g)))) | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ ∀ (y : Equalizer.FirstObj F fun {Y} {f} => S f),
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y →
∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ Nonempty
(IsLimit
(Fork.ofι (Equalizer.forkMap F fun {Y} {f} => S f)
(_ :
(Equalizer.forkMap F fun {Y} {f} => S f) ≫
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) =
(Equalizer.forkMap F fun {Y} {f} => S f) ≫
Equalizer.Presieve.secondMap' F
(_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g))))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | dsimp [Equalizer.FirstObj] | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ ∀ (y : Equalizer.FirstObj F fun {Y} {f} => S f),
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y →
∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ ∀ (y : ∏ fun f => F.obj f.fst.op),
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y →
∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ ∀ (y : Equalizer.FirstObj F fun {Y} {f} => S f),
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y →
∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | exact isSheafForDagurSieveIsIsoFork hS hF | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ IsIso (Equalizer.forkMap F S) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
⊢ IsIso (Equalizer.forkMap F S)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | intro y _ | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
⊢ ∀ (y : ∏ fun f => F.obj f.fst.op),
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y →
∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ ∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
⊢ ∀ (y : ∏ fun f => F.obj f.fst.op),
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y →
∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | refine' ⟨inv (Equalizer.forkMap F S) y, _, fun y₁ hy₁ => _⟩ | C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ ∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y | case refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ (fun x => Equalizer.forkMap F (fun {Y} {f} => S f) x = y) (inv (Equalizer.forkMap F S) y)
case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : (fun x => Equalizer.forkMap F (fun {Y} {f} => S f) x = y) y₁
⊢ y₁ = inv (Equalizer.forkMap F S) y | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ ∃! x, Equalizer.forkMap F (fun {Y} {f} => S f) x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | change (inv (Equalizer.forkMap F S) ≫ (Equalizer.forkMap F S)) y = y | case refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ (fun x => Equalizer.forkMap F (fun {Y} {f} => S f) x = y) (inv (Equalizer.forkMap F S) y) | case refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ (inv (Equalizer.forkMap F S) ≫ Equalizer.forkMap F S) y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ (fun x => Equalizer.forkMap F (fun {Y} {f} => S f) x = y) (inv (Equalizer.forkMap F S) y)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | rw [IsIso.inv_hom_id, types_id_apply] | case refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ (inv (Equalizer.forkMap F S) ≫ Equalizer.forkMap F S) y = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
⊢ (inv (Equalizer.forkMap F S) ≫ Equalizer.forkMap F S) y = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | replace hy₁ := congr_arg (inv (Equalizer.forkMap F S)) hy₁ | case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : (fun x => Equalizer.forkMap F (fun {Y} {f} => S f) x = y) y₁
⊢ y₁ = inv (Equalizer.forkMap F S) y | case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : inv (Equalizer.forkMap F S) (Equalizer.forkMap F (fun {Y} {f} => S f) y₁) = inv (Equalizer.forkMap F S) y
⊢ y₁ = inv (Equalizer.forkMap F S) y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : (fun x => Equalizer.forkMap F (fun {Y} {f} => S f) x = y) y₁
⊢ y₁ = inv (Equalizer.forkMap F S) y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | change ((Equalizer.forkMap F S) ≫ inv (Equalizer.forkMap F S)) _ = _ at hy₁ | case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : inv (Equalizer.forkMap F S) (Equalizer.forkMap F (fun {Y} {f} => S f) y₁) = inv (Equalizer.forkMap F S) y
⊢ y₁ = inv (Equalizer.forkMap F S) y | case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : (Equalizer.forkMap F S ≫ inv (Equalizer.forkMap F S)) y₁ = inv (Equalizer.forkMap F S) y
⊢ y₁ = inv (Equalizer.forkMap F S) y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : inv (Equalizer.forkMap F S) (Equalizer.forkMap F (fun {Y} {f} => S f) y₁) = inv (Equalizer.forkMap F S) y
⊢ y₁ = inv (Equalizer.forkMap F S) y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ExtensiveRegular.lean | CategoryTheory.isSheafForExtensiveSieve | [240, 1] | [255, 44] | rwa [IsIso.hom_inv_id, types_id_apply] at hy₁ | case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : (Equalizer.forkMap F S ≫ inv (Equalizer.forkMap F S)) y₁ = inv (Equalizer.forkMap F S) y
⊢ y₁ = inv (Equalizer.forkMap F S) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
C : Type u
inst✝² : Category C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbackOfIsIsodesc C
X : C
S : Presieve X
hS : S ∈ ExtensiveSieve X
F : Cᵒᵖ ⥤ Type (max u v)
hF : PreservesFiniteProducts F
this : IsIso (Equalizer.forkMap F S)
y : ∏ fun f => F.obj f.fst.op
a✝ :
Equalizer.Presieve.firstMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y =
Equalizer.Presieve.secondMap' F (_ : ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g) y
y₁ : F.obj X.op
hy₁ : (Equalizer.forkMap F S ≫ inv (Equalizer.forkMap F S)) y₁ = inv (Equalizer.forkMap F S) y
⊢ y₁ = inv (Equalizer.forkMap F S) y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | let ζ := finiteCoproduct.desc _ (fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a ) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | let σ := finiteCoproduct.desc _ ((fun a => pullback.fst f (π a))) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | let β := finiteCoproduct.desc _ π | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have comm : ζ ≫ β = σ ≫ f := by
refine' finiteCoproduct.hom_ext _ _ _ (fun a => _)
simp [← Category.assoc, finiteCoproduct.ι_desc, pullback.condition] | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | intro R₁ R₂ hR | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have himage : (ζ ≫ β) R₁ = (ζ ≫ β) R₂ := by
rw [comm]; change f (σ R₁) = f (σ R₂); rw [hR] | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ R₁ = R₂ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | replace himage := congr_arg (inv β) himage | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂
⊢ R₁ = R₂ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₁) = ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₂)
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | change ((ζ ≫ β ≫ inv β) R₁) = ((ζ ≫ β ≫ inv β) R₂) at himage | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₁) = ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₂)
⊢ R₁ = R₂ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₁ = ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₁) = ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₂)
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [IsIso.hom_inv_id, Category.comp_id] at himage | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₁ = ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₂
⊢ R₁ = R₂ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₁ = ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | obtain ⟨a₁, r₁, h₁⟩ := finiteCoproduct.ι_jointly_surjective R₁ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
⊢ R₁ = R₂ | case intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | obtain ⟨a₂, r₂, h₂⟩ := finiteCoproduct.ι_jointly_surjective R₂ | case intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
⊢ R₁ = R₂ | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have ha₁ : a₁ = R₁.fst := (congrArg Sigma.fst h₁).symm | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
⊢ R₁ = R₂ | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have ha₂ : a₂ = R₂.fst := (congrArg Sigma.fst h₂).symm | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
⊢ R₁ = R₂ | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have ha : a₁ = a₂ := by rwa [ha₁, ha₂] | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
⊢ R₁ = R₂ | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have : R₁ ∈ Set.range (finiteCoproduct.ι _ a₂) | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ R₁ = R₂ | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ R₁ ∈ Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂)
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
this : R₁ ∈ Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂)
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | obtain ⟨xr', hr'⟩ := this | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
this : R₁ ∈ Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂)
⊢ R₁ = R₂ | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
this : R₁ ∈ Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂)
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [← hr', h₂] at hR | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ R₁ = R₂ | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have hf : ∀ (a : α), Function.Injective
((finiteCoproduct.ι _ a) ≫ (finiteCoproduct.desc _ ((fun a => pullback.fst f (π a))))) | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ R₁ = R₂ | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ ∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
hf :
∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have := hf a₂ hR | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
hf :
∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
⊢ R₁ = R₂ | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
hf :
∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
this : xr' = r₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
hf :
∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [← hr', h₂, this] | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
hf :
∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
this : xr' = r₂
⊢ R₁ = R₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
hf :
∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
this : xr' = r₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ ζ ≫ β = σ ≫ f | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
a : α
⊢ finiteCoproduct.ι (fun a => pullback f (π a)) a ≫ ζ ≫ β = finiteCoproduct.ι (fun a => pullback f (π a)) a ≫ σ ≫ f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ ζ ≫ β = σ ≫ f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | simp [← Category.assoc, finiteCoproduct.ι_desc, pullback.condition] | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
a : α
⊢ finiteCoproduct.ι (fun a => pullback f (π a)) a ≫ ζ ≫ β = finiteCoproduct.ι (fun a => pullback f (π a)) a ≫ σ ≫ f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
a : α
⊢ finiteCoproduct.ι (fun a => pullback f (π a)) a ≫ ζ ≫ β = finiteCoproduct.ι (fun a => pullback f (π a)) a ≫ σ ≫ f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [comm] | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ ↑(σ ≫ f) R₁ = ↑(σ ≫ f) R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | change f (σ R₁) = f (σ R₂) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ ↑(σ ≫ f) R₁ = ↑(σ ≫ f) R₂ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ ↑f (↑σ R₁) = ↑f (↑σ R₂) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ ↑(σ ≫ f) R₁ = ↑(σ ≫ f) R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [hR] | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ ↑f (↑σ R₁) = ↑f (↑σ R₂) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
⊢ ↑f (↑σ R₁) = ↑f (↑σ R₂)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | constructor <;> rfl | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
⊢ (↑ζ R₁).fst = R₁.fst ∧ (↑ζ R₂).fst = R₂.fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
⊢ (↑ζ R₁).fst = R₁.fst ∧ (↑ζ R₂).fst = R₂.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [← this.1, ← this.2, himage] | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
this : (↑ζ R₁).fst = R₁.fst ∧ (↑ζ R₂).fst = R₂.fst
⊢ R₁.fst = R₂.fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
this : (↑ζ R₁).fst = R₁.fst ∧ (↑ζ R₂).fst = R₂.fst
⊢ R₁.fst = R₂.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rwa [ha₁, ha₂] | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
⊢ a₁ = a₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
⊢ a₁ = a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [← finiteCoproduct.range_eq ha, h₁] | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ R₁ ∈ Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁ ∈
Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ R₁ ∈ Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | simp only [Set.mem_range, exists_apply_eq_apply] | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁ ∈
Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₁) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₁ =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
⊢ ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁ ∈
Set.range ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₁)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | intro a | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ ∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
⊢ Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
⊢ ∀ (a : α),
Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | simp only [finiteCoproduct.ι_desc] | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
⊢ Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
⊢ Function.Injective ↑(pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
⊢ Function.Injective
↑(finiteCoproduct.ι (fun a => pullback f (π a)) a ≫
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | intro x y h | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
⊢ Function.Injective ↑(pullback.fst f (π a)) | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
⊢ Function.Injective ↑(pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have h₁ := h | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
⊢ x = y | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | apply_fun f at h | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
⊢ x = y | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑f (↑(pullback.fst f (π a)) x) = ↑f (↑(pullback.fst f (π a)) y)
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | change (pullback.fst f (π a) ≫ f) x = _ at h | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑f (↑(pullback.fst f (π a)) x) = ↑f (↑(pullback.fst f (π a)) y)
⊢ x = y | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑f (↑(pullback.fst f (π a)) x) = ↑f (↑(pullback.fst f (π a)) y)
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have h' := h.symm | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
⊢ x = y | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑f (↑(pullback.fst f (π a)) y) = ↑(pullback.fst f (π a) ≫ f) x
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | change (pullback.fst f (π a) ≫ f) y = _ at h' | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑f (↑(pullback.fst f (π a)) y) = ↑(pullback.fst f (π a) ≫ f) x
⊢ x = y | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.fst f (π a) ≫ f) y = ↑(pullback.fst f (π a) ≫ f) x
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑f (↑(pullback.fst f (π a)) y) = ↑(pullback.fst f (π a) ≫ f) x
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [pullback.condition] at h' | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.fst f (π a) ≫ f) y = ↑(pullback.fst f (π a) ≫ f) x
⊢ x = y | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.fst f (π a) ≫ f) y = ↑(pullback.fst f (π a) ≫ f) x
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have : Function.Injective (π a) | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
⊢ x = y | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
⊢ Function.Injective ↑(π a)
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have h₂ := this h' | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
⊢ x = y | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | suffices : x.val = y.val | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
⊢ x = y | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this✝ : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
this : ↑x = ↑y
⊢ x = y
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
⊢ ↑x = ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | exact Prod.ext h₁ h₂.symm | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
⊢ ↑x = ↑y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
⊢ ↑x = ↑y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | intro r s hrs | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
⊢ Function.Injective ↑(π a) | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs : ↑(π a) r = ↑(π a) s
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
⊢ Function.Injective ↑(π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [← finiteCoproduct.ι_desc_apply] at hrs | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs : ↑(π a) r = ↑(π a) s
⊢ r = s | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs : ↑(π a) r = ↑(π a) s
⊢ r = s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have hrs' := hrs.symm | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
⊢ r = s | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs' : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
⊢ r = s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | rw [← finiteCoproduct.ι_desc_apply] at hrs' | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs' : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ r = s | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs' : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ r = s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | have : Function.Injective (finiteCoproduct.desc (fun a ↦ Z a) π) | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ r = s | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => Z a) π)
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
this : Function.Injective ↑(finiteCoproduct.desc (fun a => Z a) π)
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ r = s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | exact (finiteCoproduct.ι_injective a (this hrs')).symm | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
this : Function.Injective ↑(finiteCoproduct.desc (fun a => Z a) π)
⊢ r = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
this : Function.Injective ↑(finiteCoproduct.desc (fun a => Z a) π)
⊢ r = s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | apply Function.Bijective.injective | case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => Z a) π) | case this.hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => Z a) π) | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => Z a) π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | exact ConcreteCategory.bijective_of_isIso _ | case this.hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => Z a) π) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
r s : (forget Profinite).obj (Z a)
hrs✝ : ↑(π a) r = ↑(π a) s
hrs : ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r) = ↑(π a) s
hrs'✝ : ↑(π a) s = ↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
hrs' :
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) s) =
↑(finiteCoproduct.desc (fun a => Z a) π) (↑(finiteCoproduct.ι (fun a => Z a) a) r)
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => Z a) π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_injective | [10, 1] | [65, 23] | exact Subtype.ext this | case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this✝ : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
this : ↑x = ↑y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a
σ : (finiteCoproduct fun a => pullback f (π a)) ⟶ Y :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = σ ≫ f
R₁ R₂ : (forget Profinite).obj (finiteCoproduct fun a => pullback f (π a))
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
a₁ : α
r₁ : ↑((fun a => pullback f (π a)) a₁).toCompHaus.toTop
h₁✝ : R₁ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₁) r₁
a₂ : α
r₂ : ↑((fun a => pullback f (π a)) a₂).toCompHaus.toTop
h₂✝ : R₂ = ↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂
ha₁ : a₁ = R₁.fst
ha₂ : a₂ = R₂.fst
ha : a₁ = a₂
xr' : (forget Profinite).obj (pullback f (π a₂))
hR :
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr') =
↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
(↑(finiteCoproduct.ι (fun a => (fun a => pullback f (π a)) a) a₂) r₂)
hr' : ↑(finiteCoproduct.ι (fun a => pullback f (π a)) a₂) xr' = R₁
a : α
x y : (forget Profinite).obj (pullback f (π a))
h₁ : ↑(pullback.fst f (π a)) x = ↑(pullback.fst f (π a)) y
h : ↑(pullback.fst f (π a) ≫ f) x = ↑f (↑(pullback.fst f (π a)) y)
h' : ↑(pullback.snd f (π a) ≫ π a) y = ↑(pullback.snd f (π a) ≫ π a) x
this✝ : Function.Injective ↑(π a)
h₂ : ↑(pullback.snd f (π a)) y = ↑(pullback.snd f (π a)) x
this : ↑x = ↑y
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | let β := finiteCoproduct.desc _ π | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | apply isIso_of_bijective _ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | refine' ⟨extensivity_injective f HIso, fun y => _⟩ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
⊢ ∃ a, ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) a = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | refine' ⟨⟨(inv β (f y)).1, ⟨⟨y, (inv β (f y)).2⟩, _⟩⟩, rfl⟩ | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
⊢ ∃ a, ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) a = y | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
⊢ ∃ a, ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) a = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | have inj : Function.Injective (inv β) := by intros r s hrs
convert congr_arg β hrs <;> change _ = (inv β ≫ β) _<;> simp only [IsIso.inv_hom_id]<;> rfl | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd} | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd}
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | apply inj | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd} | case a
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) =
↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd}
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | intro a | case a
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a | case a
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
a : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | simp only [IsIso.comp_inv_eq, finiteCoproduct.ι_desc] | case a
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
a : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
a : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | intros r s hrs | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
⊢ Function.Injective ↑(CategoryTheory.inv β) | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
r s : (forget Profinite).obj X
hrs : ↑(CategoryTheory.inv β) r = ↑(CategoryTheory.inv β) s
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
⊢ Function.Injective ↑(CategoryTheory.inv β)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | convert congr_arg β hrs <;> change _ = (inv β ≫ β) _<;> simp only [IsIso.inv_hom_id]<;> rfl | α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
r s : (forget Profinite).obj X
hrs : ↑(CategoryTheory.inv β) r = ↑(CategoryTheory.inv β) s
⊢ r = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : (forget Profinite).obj Y
r s : (forget Profinite).obj X
hrs : ↑(CategoryTheory.inv β) r = ↑(CategoryTheory.inv β) s
⊢ r = s
TACTIC:
|
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