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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | let β := finiteCoproduct.desc _ π | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | apply isIso_of_bijective _ | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso (finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π ... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pull... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso (finiteCoproduct... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | refine' ⟨extensivity_injective f HIso, fun y => _⟩ | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pull... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
⊢ ∃ a, ↑(finiteCoproduct.desc (fun a => pullback f (π a))... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Bijective ↑(f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | refine' ⟨⟨(inv β (f y)).1, ⟨⟨y, (inv β (f y)).2⟩, _⟩⟩, rfl⟩ | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
⊢ ∃ a, ↑(finiteCoproduct.desc (fun a => pullback f (π a))... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).ob... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | have inj : Function.Injective (inv β) := by intros r s hrs
convert congr_arg β hrs <;> change _ = (inv β ≫ β) _<;> simp only [IsIso.inv_hom_id]<;> rfl | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ (y, (↑... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).ob... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | apply inj | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ (y, (↑... | case a
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).ob... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | intro a | case a
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢... | case a
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
a... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompH... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | simp only [IsIso.comp_inv_eq, finiteCoproduct.ι_desc] | case a
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompH... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | intros r s hrs | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
⊢ Function.Injective ↑(CategoryTheory.inv β) | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
r s : (forget CompHaus).obj X
hrs : ↑(CategoryTheory.inv ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).ob... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | convert congr_arg β hrs <;> change _ = (inv β ≫ β) _<;> simp only [IsIso.inv_hom_id]<;> rfl | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
r s : (forget CompHaus).obj X
hrs : ↑(CategoryTheory.inv ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).ob... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | apply Eq.symm | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀... | case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
t... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).ob... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | change (_ ≫ inv β) _ = _ | case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
t... | case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
t... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompH... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | rw [this] | case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
t... | case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
t... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompH... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_explicit | [67, 1] | [85, 56] | rfl | case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
t... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
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 CompH... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | let θ := Sigma.mapIso (fun a => fromExplicitIso f (i a)) | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
H : IsIso (Sigma.desc i)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
H : IsIso (Sigma.desc i)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | let δ := FromFiniteCoproductIso (fun a => CompHaus.pullback f (i a)) | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 (i a... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | convert extensivity_explicit f HIso | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 (i a... | case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 => pul... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 : (finiteCopr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 => pul... | case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 => pul... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | simp [← Category.assoc, finiteCoproduct.ι_desc, fromExplicit] | case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 => pul... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | apply IsIso.of_isIso_comp_left θ.hom | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 (Sigm... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | apply IsIso.of_isIso_comp_left δ.hom | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 (i a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 : (finiteCopr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | let ε := ToFiniteCoproductIso Z | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 (i a... | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 (i a... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 : (finiteCopr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | convert H | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 (i a... | case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 => pul... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 : (finiteCopr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | refine' Sigma.hom_ext _ _ (fun a => _) | case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 => pul... | case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 => pul... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | simp [← Category.assoc] | case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 => pul... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity | [87, 1] | [103, 64] | apply IsIso.of_isIso_comp_left ε.hom | α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 (i a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : CompHaus
Z : α → CompHaus
i : (a : α) → Z a ⟶ X
Y : CompHaus
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 : (finiteCopr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | intro X Y Z f π hπ | ⊢ EpiPullbackOfEpi CompHaus | X Y Z : CompHaus
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 CompHaus
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | suffices : Epi (fromExplicit f π ≫ (Limits.pullback.fst : Limits.pullback f π ⟶ Y)) | X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi Limits.pullback.fst | X Y Z : CompHaus
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 : CompHaus
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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi Limits.pullback.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | rw [CompHaus.epi_iff_surjective] at hπ ⊢ | case this
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi (fromExplicit f π ≫ Limits.pullback.fst) | case this
X Y Z : CompHaus
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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Epi π
⊢ Epi (fromExplicit f π ≫ Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | intro y | case this
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
⊢ Function.Surjective ↑(fromExplicit f π ≫ Limits.pullback.fst) | case this
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).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 : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | obtain ⟨z,hz⟩ := hπ (f y) | case this
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y | case this.intro
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | have : fromExplicit f π ≫ Limits.pullback.fst = CompHaus.pullback.fst f π | case this.intro
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).obj Z
hz : ↑π z = ↑f y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y | case this
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).obj Z
hz : ↑π z = ↑f y
⊢ fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
case this.intro
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).obj Z
hz : ↑π z = ↑f y
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullback.fst) a = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | rw [this] | case this.intro
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).obj Z
hz : ↑π z = ↑f y
this : fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π
⊢ ∃ a, ↑(fromExplicit f π ≫ Limits.pullb... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | exact ⟨⟨(y, z), hz.symm⟩, rfl⟩ | case this.intro
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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
TACTIC:... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | exact @epi_of_epi _ _ _ _ _ _ _ this | X Y Z : CompHaus
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 : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.epi_pullback_of_epi | [105, 1] | [116, 33] | dsimp [fromExplicit] | case this
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).obj Z
hz : ↑π z = ↑f y
⊢ fromExplicit f π ≫ Limits.pullback.fst = pullback.fst f π | case this
X Y Z : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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 : CompHaus
f : Y ⟶ X
π : Z ⟶ X
hπ : Function.Surjective ↑π
y : (forget CompHaus).obj Y
z : (forget CompHaus).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) ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | ext X S | ⊢ Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f :... | case h.h.h
X : CompHaus
S : Sieve X
⊢ S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : F... | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → Comp... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | constructor
<;> intro h | case h.h.h
X : CompHaus
S : Sieve X
⊢ S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : F... | case h.h.h.mp
X : CompHaus
S : Sieve X
h :
S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintyp... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : CompHaus
S : Sieve X
⊢ S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullba... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [Coverage.toGrothendieck] at * | case h.h.h.mp
X : CompHaus
S : Sieve X
h :
S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintyp... | case h.h.h.mp
X : CompHaus
S : Sieve X
h :
S ∈
Coverage.saturate
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp
X : CompHaus
S : Sieve X
h :
S ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullbac... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.of | case h.h.h.mp.of
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α... | case h.h.h.mp.of.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [coherentCoverage] | case h.h.h.mp.of.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z ... | case h.h.h.mp.of.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [ExtensiveRegularCoverage'] at hT | case h.h.h.mp.of.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z ... | case h.h.h.mp.of.hS
X : CompHaus
S : Sieve X
Y : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT :
T ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Or.elim hT
<;> intro h | case h.h.h.mp.of.hS
X : CompHaus
S : Sieve X
Y : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSiev... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS
X : CompHaus
S : Sieve X
Y : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨α, x, Xmap, π, h⟩ := h | case h.h.h.mp.of.hS.left
X : CompHaus
S : Sieve X
Y : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use α | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use x | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use Xmap | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use π | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | refine' ⟨h.1,_⟩ | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Effect... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [he] | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Effect... | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Epi (S... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | letI := h.2 | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Epi (S... | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
this : I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | exact inferInstance | case h.h.h.mp.of.hS.left.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
this : I... | 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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
α : Type
x : Fintype α
Xmap : α → CompHaus
π : (a : α) → Xmap a ⟶ Y
h : T = ofArrows Xmap π ∧ IsIso (Sigma.desc π)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨Z, f, h⟩ := h | case h.h.h.mp.of.hS.right
X : CompHaus
S : Sieve X
Y : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use Unit | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use inferInstance | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ x X π, T = ofArrows X π ∧ EffectiveEpiFamily X ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use (fun _ ↦ Z) | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use (fun _ ↦ f) | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ ∃ π, T = ofArrows (fun x => Z) π ∧ EffectiveEpiFa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | refine' ⟨h.1,_⟩ | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ (T = ofArrows (fun x => Z) fun x => f) ∧ Effectiv... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 => Z... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Epi f
⊢ EffectiveEpiFamily (fun x => Z) fun x => f
TACTIC... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [he] | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 => Z... | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [CompHaus.epi_iff_surjective _] at h ⊢ | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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.Surj... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | intro x | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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.Surj... | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 Com... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨y,hy⟩ := h.2 x | case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 Com... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [← hy] | case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app] | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 CompHaus).obj Y
y : (forget Com... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [this] | case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rfl | case h.h.h.mp.of.hS.right.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
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 : (forg... | 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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ ExtensiveSieve Y ∪ RegularSieve Y
Z : CompHaus
f : Z ⟶ Y
h : (T = ofArrows (fun x => Z) fun x => f) ∧ Function.Surjective ↑f
he : (EffectiveEpiFamily (fu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.top | case h.h.h.mp.top
X : CompHaus
S : Sieve X
X✝ : CompHaus
⊢ ⊤ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.top
X : CompHaus
S : Sieve X
X✝ : CompHaus
⊢ ⊤ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X✝
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.transitive Y T | case h.h.h.mp.transitive
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → Comp... | case h.h.h.mp.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → Co... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.transitive
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | assumption | case h.h.h.mp.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → Co... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | assumption | case h.h.h.mp.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → Co... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ :
Coverage.saturate
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [coherentCoverage] at hT | case h.h.h.mpr.of
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ Coverage.covering (coherentCoverage CompHaus) Y
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ... | case h.h.h.mpr.of
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : T ∈ Coverage.covering (coherentCoverage CompHaus) Y
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' Com... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨I, hI, Xmap, f, ⟨h, hT⟩⟩ := hT | case h.h.h.mpr.of
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
hT : ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' Co... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
⊢ Sieve.generate T ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ∈
GrothendieckTopolo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
⊢ Sieve.generate T ∈
Grothendi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [he] at hT | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ∈
GrothendieckTopolo... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.siev... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : EffectiveEpiFamily Xmap f
he : EffectiveEpiFamily Xmap f ↔ E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.siev... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sig... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | let F := Sigma.desc f | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sig... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | let Z := Sieve.generate T | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sig... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sig... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sig... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 ... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sig... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.of | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (S... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [ExtensiveRegularCoverage'] | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | right | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 => S... | 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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use Xs | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 => S... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 => S... | 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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | use F | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 => S... | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 => S... | 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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 => S... | 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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | intro R g hZfg | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (S... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp at hZfg | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (S... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (S... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.intro.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (S... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveE... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveE... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 | CompHaus/ExplicitSheaves.lean | CompHaus.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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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 : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
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.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : Effecti... |
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