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 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | apply Eq.symm | 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)
this... | case h
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 ... | 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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | change (_ ≫ inv β) _ = _ | case h
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 ... | case h
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
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 Pr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | rw [this] | case h
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 ... | case h
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
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 Pr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity_explicit | [67, 1] | [85, 56] | rfl | case h
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
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 Pr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | let θ := Sigma.mapIso (fun a => fromExplicitIso f (i a)) | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | let δ := FromFiniteCoproductIso (fun a => Profinite.pullback f (i a)) | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
⊢ IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
⊢ IsIso (θ.h... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | convert extensivity_explicit f HIso | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (... | case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteC... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => ... | case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | simp [← Category.assoc, finiteCoproduct.ι_desc, fromExplicit] | case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | apply IsIso.of_isIso_comp_left θ.hom | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
this : IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
⊢ IsIso (S... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
this : IsIso... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | apply IsIso.of_isIso_comp_left δ.hom | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteC... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | let ε := ToFiniteCoproductIso Z | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (... | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteC... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | convert H | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (... | case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteC... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | refine' Sigma.hom_ext _ _ (fun a => _) | case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => ... | case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | simp [← Category.assoc] | case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensivity | [87, 1] | [103, 64] | apply IsIso.of_isIso_comp_left ε.hom | α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : Profinite
Z : α → Profinite
i : (a : α) → Z a ⟶ X
Y : Profinite
f : Y ⟶ X
H : IsIso (Sigma.desc i)
θ : (∐ fun a => pullback f (i a)) ≅ ∐ fun a => Limits.pullback f (i a) := Sigma.mapIso fun a => fromExplicitIso f (i a)
δ : (finiteC... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | intro X Y Z f π hπ | ⊢ EpiPullbackOfEpi Profinite | X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi Limits.pullback.fst | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ EpiPullbackOfEpi Profinite
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | suffices : Epi (fromExplicit f π ≫ (Limits.pullback.fst : Limits.pullback f π ⟶ Y)) | X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi Limits.pullback.fst | X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
this : Epi (fromExplicit f π ≫ Limits.pullback.fst)
⊢ Epi Limits.pullback.fst
case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi (fromExplicit f π ≫ Limits.pullback.fst) | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi Limits.pullback.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | rw [Profinite.epi_iff_surjective] at hπ ⊢ | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi (fromExplicit f π ≫ Limits.pullback.fst) | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
⊢ Function.Surjective ↑(fromExplicit f π ≫ Limits.pullback.fst) | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi (fromExplicit f π ≫ Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | intro y | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
⊢ Function.Surjective ↑(fromExplicit f π ≫ Limits.pullback.fst) | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
⊢ Function.Surjective ↑(fromExplicit f π ≫ Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | obtain ⟨z,hz⟩ := hπ (f y) | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y | case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | have : fromExplicit f π ≫ Limits.pullback.fst = Profinite.pullback.fst f π | case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | rw [this] | case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
this : fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y | case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
this : fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
⊢ ∃ a, ↑(pullback.fst f π) a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
this : fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | exact ⟨⟨(y, z), hz.symm⟩, rfl⟩ | case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
this : fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
⊢ ∃ a, ↑(pullback.fst f π) a = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
this : fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
⊢ ∃ a, ↑(pullback.fst f π) a = y
TACT... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | exact @epi_of_epi _ _ _ _ _ _ _ this | X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
this : Epi (fromExplicit f π ≫ Limits.pullback.fst)
⊢ Epi Limits.pullback.fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
this : Epi (fromExplicit f π ≫ Limits.pullback.fst)
⊢ Epi Limits.pullback.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | dsimp [fromExplicit] | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ Limits.pullback.lift (pullback.fst f π) (pullback.snd f π) (_ : pullback.fst f π ≫ f = pullback.snd f π ≫ π) ≫
Limits.pullback.fst =
pullback.fst f π | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.epi_pullback_of_epi | [105, 1] | [116, 33] | simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app] | case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ Limits.pullback.lift (pullback.fst f π) (pullback.snd f π) (_ : pullback.fst f π ≫ f = pullback.snd f π ≫ π) ≫
Limits.pullback.fst =
pullback.fst f π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z : Profinite
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget Profinite).obj Y
z : (forget Profinite).obj Z
hz : ↑π z = ↑f y
⊢ Limits.pullback.lift (pullback.fst f π) (pullback.snd f π) (_ : pullback.fst f π ≫ f = pullback.snd f π ≫ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | ext X S | ⊢ Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite} {Z : α → Profinite} (π : (a : α) → Z a ⟶ X) {Y : Profinite}
... | case h.h.h
X : Profinite
S : Sieve X
⊢ S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst... | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite} {Z : α → ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | constructor
<;> intro h | case h.h.h
X : Profinite
S : Sieve X
⊢ S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst... | case h.h.h.mp
X : Profinite
S : Sieve X
h :
S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : Profinite
S : Sieve X
⊢ S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [Coverage.toGrothendieck] at * | case h.h.h.mp
X : Profinite
S : Sieve X
h :
S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fi... | case h.h.h.mp
X : Profinite
S : Sieve X
h :
S ∈
Coverage.saturate
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite} {Z : α → Profinite} (π : (a : α) →... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp
X : Profinite
S : Sieve X
h :
S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pul... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.of | case h.h.h.mp.of
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite} {... | case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [coherentCoverage] | case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite... | case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [ExtensiveRegularCoverage'] at hT | case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite... | case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Or.elim hT
<;> intro h | case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.left
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
h : T ∈ ExtensiveSieve Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
case h.h.h.mp.of.hS.right
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ Regular... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨α, x, Xmap, π, h⟩ := h | case h.h.h.mp.of.hS.left
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
h : T ∈ ExtensiveSieve Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
h : T ∈ ExtensiveSieve Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use α | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ ∃ x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use x | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ ∃ x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ ∃ X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use Xmap | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ ∃ X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ ∃ π, T = ofArrows Xmap π ∧ EffectiveEpiFamily Xmap π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ ∃ π, T = ofArrows Xmap π ∧ EffectiveEpiFamily Xmap π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ T = ofArrows Xmap π ∧ EffectiveEpiFamily Xmap π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | refine' ⟨h.1,_⟩ | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ T = ofArrows Xmap π ∧ EffectiveEpiFamily Xmap π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ EffectiveEpiFamily Xmap π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | have he := (effectiveEpiFamily_tfae Xmap π).out 0 1 | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
⊢ EffectiveEpiFamily Xmap π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Eff... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [he] | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Eff... | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Epi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | letI := h.2 | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Epi... | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
this ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | exact inferInstance | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
this ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → Profinite
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨Z, f, h⟩ := h | case h.h.h.mp.of.hS.right
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
h : T ∈ RegularSieve Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
h : T ∈ RegularSieve Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use Unit | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFami... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use inferInstance | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ x X π, T = ofArrows X π ∧ EffectiveEpiFamily... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use (fun _ ↦ Z) | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ π, T = ofArrows (fun x => Z) π ∧ EffectiveEpiFamily (fun x => Z) π | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ X π, T = ofArrows X π ∧ EffectiveEpiFamily X... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use (fun _ ↦ f) | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ π, T = ofArrows (fun x => Z) π ∧ EffectiveEpiFamily (fun x => Z) π | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ (T = ofArrows (fun x => Z) fun x => f) ∧ EffectiveEpiFamily (fun x => Z) fun x => f | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ π, T = ofArrows (fun x => Z) π ∧ EffectiveEp... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | refine' ⟨h.1,_⟩ | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ (T = ofArrows (fun x => Z) fun x => f) ∧ EffectiveEpiFamily (fun x => Z) fun x => f | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ EffectiveEpiFamily (fun x => Z) fun x => f | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ (T = ofArrows (fun x => Z) fun x => f) ∧ Effec... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | have he := (effectiveEpiFamily_tfae (fun (_ : Unit) ↦ Z) (fun _ ↦ f)).out 0 1 | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ EffectiveEpiFamily (fun x => Z) fun x => f | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
⊢ EffectiveEpiFamily (fun x =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ EffectiveEpiFamily (fun x => Z) fun x => f
TAC... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [he] | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
⊢ EffectiveEpiFamily (fun x =... | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
⊢ Epi (Sigma.desc fun x => f) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
he : (EffectiveEpiFamily (fun x => Z) fun x => f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [Profinite.epi_iff_surjective _] at h ⊢ | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
⊢ Epi (Sigma.desc fun x => f) | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
⊢ Function.S... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
he : (EffectiveEpiFamily (fun x => Z) fun x => f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | intro x | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
⊢ Function.S... | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (forget ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨y,hy⟩ := h.2 x | case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (forget ... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use Sigma.ι (fun (_ : Unit) ↦ Z) Unit.unit y | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [← hy] | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | suffices : (f : Z → Y) = Sigma.ι (fun (_ : Unit) ↦ Z) Unit.unit ≫ Sigma.desc (fun _ ↦ f) | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app] | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (forget Profinite).obj Y
y : (forget... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Si... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [this] | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rfl | case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : Profinite
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.top | case h.h.h.mp.top
X : Profinite
S : Sieve X
X✝ : Profinite
⊢ ⊤ ∈ GrothendieckTopology.sieves (coherentTopology Profinite) X✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.top
X : Profinite
S : Sieve X
X✝ : Profinite
⊢ ⊤ ∈ GrothendieckTopology.sieves (coherentTopology Profinite) X✝
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.transitive Y T | case h.h.h.mp.transitive
X : Profinite
S : Sieve X
Y : Profinite
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite} {Z : α →... | case h.h.h.mp.transitive.a
X : Profinite
S : Sieve X
Y : Profinite
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite} {Z : α... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.transitive
X : Profinite
S : Sieve X
Y : Profinite
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | assumption | case h.h.h.mp.transitive.a
X : Profinite
S : Sieve X
Y : Profinite
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite} {Z : α... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.transitive.a
X : Profinite
S : Sieve X
Y : Profinite
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | assumption | case h.h.h.mp.transitive.a
X : Profinite
S : Sieve X
Y : Profinite
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : Profinite} {Z : α... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.transitive.a
X : Profinite
S : Sieve X
Y : Profinite
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [coherentCoverage] at hT | case h.h.h.mpr.of
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ Coverage.covering (coherentCoverage Profinite) Y
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (... | case h.h.h.mpr.of
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : T ∈ Coverage.covering (coherentCoverage Profinite) Y
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage'... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨I, hI, Xmap, f, ⟨h, hT⟩⟩ := hT | case h.h.h.mpr.of
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage' Profinite
(_ : ∀ {X Y Z : Profinite} (f : Y ⟶ X) (... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinit... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
hT : ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinite
(ExtensiveRegularCoverage'... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | have he := (effectiveEpiFamily_tfae Xmap f).out 0 1 | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck Profinit... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
⊢ Sieve.generate T ∈
GrothendieckTop... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
⊢ Sieve.generate T ∈
Grothe... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [he] at hT | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
⊢ Sieve.generate T ∈
GrothendieckTop... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
⊢ Sieve.generate T ∈
GrothendieckTopology.s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
he : EffectiveEpiFamily Xmap f ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | let φ := fun (i : I) ↦ Sigma.ι Xmap i | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
⊢ Sieve.generate T ∈
GrothendieckTopology.s... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | let F := Sigma.desc f | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | let Z := Sieve.generate T | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | let Xs := (∐ fun (i : I) => Xmap i) | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | let Zf : Sieve Y := Sieve.generate
(Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F)) | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.transitive Y Zf | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.of | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [ExtensiveRegularCoverage'] | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [Set.mem_union, Set.mem_setOf_eq] | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | right | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use Xs | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i =... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use F | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i =... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | refine' ⟨rfl, inferInstance⟩ | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | intro R g hZfg | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp at hZfg | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [Presieve.ofArrows_pUnit] at hZfg | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sig... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [Presieve.singleton] at hW | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : Effecti... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | induction hW | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : Effecti... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [← hW', Sieve.pullback_comp Z] | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : Effe... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | suffices : Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
(ExtensiveRegularCoverage' _ _ _).toGrothendieck R | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : Effe... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply GrothendieckTopology.pullback_stable' | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | case this.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [Coverage.toGrothendieck] | case this.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | case this.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | suffices : Coverage.saturate (ExtensiveRegularCoverage' _ _ _) Xs (Z.pullback F) | case this.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | case this.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.a
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | suffices : Sieve.generate (Presieve.ofArrows Xmap φ) ≤ Z.pullback F | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | intro Q q hq | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [Sieve.pullback_apply, Sieve.generate_apply] | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [Sieve.generate_apply] at hq | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨E, e, r, hq⟩ := hq | case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | case this.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | refine' ⟨E, e, r ≫ F, ⟨_, _⟩⟩ | case this.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun... | case this.intro.intro.intro.refine'_1
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitSheaves.lean | Profinite.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | exact this | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : Profinite
S : Sieve X
Y : Profinite
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → Profinite
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : Effe... |
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