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 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)
⊢ 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 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
⊢ Function.Bijective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : 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 a))
TACTIC:
|
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 => 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
y : (forget CompHaus).obj Y
⊢ ∃ a, ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) a = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : 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 => pullback.fst f (π a))
TACTIC:
|
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)) fun a => pullback.fst f (π a)) a = 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
y : (forget CompHaus).obj Y
⊢ (y, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : 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)) fun a => pullback.fst f (π a)) a = y
TACTIC:
|
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.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd} | 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, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : 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.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd}
TACTIC:
|
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, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd} | case a
α : Type
inst✝ : Fintype α
X : 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)
⊢ ↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) =
↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : 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, (↑(CategoryTheory.inv β) (↑f y)).snd) ∈ {xy | ↑f xy.fst = ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) xy.snd}
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | 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)
⊢ ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z 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)
a : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : 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 : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
TACTIC:
|
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 : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : 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 : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
TACTIC:
|
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 β) r = ↑(CategoryTheory.inv β) s
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : 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)
TACTIC:
|
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 β) r = ↑(CategoryTheory.inv β) s
⊢ r = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : 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 β) r = ↑(CategoryTheory.inv β) s
⊢ r = s
TACTIC:
|
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 : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) =
↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd) | 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)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd) =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) | 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).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) =
↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd)
TACTIC:
|
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)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd) =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) | 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)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst ≫ CategoryTheory.inv β) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) | 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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd) =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst)
TACTIC:
|
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)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst ≫ CategoryTheory.inv β) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) | 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)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(finiteCoproduct.ι Z (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) | 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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst ≫ CategoryTheory.inv β) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst)
TACTIC:
|
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)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(finiteCoproduct.ι Z (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst) | 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 CompHaus).obj Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(finiteCoproduct.ι Z (↑(CategoryTheory.inv β) (↑f y)).fst) (y, (↑(CategoryTheory.inv β) (↑f y)).snd).snd =
↑(CategoryTheory.inv β) (↑f (y, (↑(CategoryTheory.inv β) (↑f y)).snd).fst)
TACTIC:
|
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) :=
FromFiniteCoproductIso fun a => pullback f (i a)
⊢ IsIso (θ.hom ≫ 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)
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)
TACTIC:
|
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) :=
FromFiniteCoproductIso fun a => pullback f (i a)
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ IsIso (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) | 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 => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.desc (fun a => pullback f (i a)) fun a => pullback.fst 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ IsIso (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
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 => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.desc (fun a => pullback f (i a)) fun a => pullback.fst 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 => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
a : α
⊢ (finiteCoproduct.ι (fun a => pullback f (i a)) a ≫ δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.ι (fun a => pullback f (i a)) a ≫
finiteCoproduct.desc (fun a => pullback f (i a)) fun a => pullback.fst f (i a) | 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.desc (fun a => pullback f (i a)) fun a => pullback.fst f (i a)
TACTIC:
|
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 => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
a : α
⊢ (finiteCoproduct.ι (fun a => pullback f (i a)) a ≫ δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.ι (fun a => pullback f (i a)) a ≫
finiteCoproduct.desc (fun a => pullback f (i a)) fun a => pullback.fst f (i a) | 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
a : α
⊢ (finiteCoproduct.ι (fun a => pullback f (i a)) a ≫ δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.ι (fun a => pullback f (i a)) a ≫
finiteCoproduct.desc (fun a => pullback f (i a)) fun a => pullback.fst f (i a)
TACTIC:
|
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 (Sigma.desc fun x => Limits.pullback.fst) | 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.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
⊢ 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] | 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) :=
FromFiniteCoproductIso fun a => pullback f (i a)
this : IsIso (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
⊢ IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) | 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
this : IsIso (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
⊢ IsIso (θ.hom ≫ 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 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) :=
FromFiniteCoproductIso fun a => pullback f (i a)
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i) | 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) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i) | 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i)
TACTIC:
|
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) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ IsIso (ε.hom ≫ finiteCoproduct.desc Z i) | 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 => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.desc i | 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ IsIso (ε.hom ≫ finiteCoproduct.desc Z i)
TACTIC:
|
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 => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.desc i | 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 => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
a : α
⊢ Sigma.ι Z a ≫ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.ι Z a ≫ Sigma.desc i | 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.desc i
TACTIC:
|
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 => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
a : α
⊢ Sigma.ι Z a ≫ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.ι Z a ≫ Sigma.desc i | 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
a : α
⊢ Sigma.ι Z a ≫ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.ι Z a ≫ Sigma.desc i
TACTIC:
|
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) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
this : IsIso (ε.hom ≫ finiteCoproduct.desc Z i)
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i) | 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 : (finiteCoproduct fun a => pullback f (i a)) ≅ ∐ fun a => pullback f (i a) :=
FromFiniteCoproductIso fun a => pullback f (i a)
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
this : IsIso (ε.hom ≫ finiteCoproduct.desc Z i)
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i)
TACTIC:
|
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 : (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 | 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.pullback.fst) a = y
TACTIC:
|
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) ≫
Limits.pullback.fst =
pullback.fst f π
TACTIC:
|
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 : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))) =
coherentTopology CompHaus | 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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
X ↔
S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X | 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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))) =
coherentTopology CompHaus
TACTIC:
|
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
X ↔
S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X | 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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X
case h.h.h.mpr
X : CompHaus
S : Sieve X
h : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) 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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
X | 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.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
X ↔
S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X
TACTIC:
|
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X | 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X | 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.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
X
⊢ S ∈ 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.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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y
⊢ Sieve.generate T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y | 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 : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y
⊢ T ∈ Coverage.covering (coherentCoverage CompHaus) Y | 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)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y
⊢ Sieve.generate T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
TACTIC:
|
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 : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y
⊢ T ∈ Coverage.covering (coherentCoverage CompHaus) Y | 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 : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
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)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y
⊢ T ∈ Coverage.covering (coherentCoverage CompHaus) Y
TACTIC:
|
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 : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y
⊢ ∃ α x X π, T = ofArrows X π ∧ EffectiveEpiFamily X π | 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)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
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] | 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 ∪ RegularSieve Y
h : T ∈ 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 ∈ 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)
⊢ ∃ 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 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)
⊢ ∃ 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 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)
⊢ ∃ 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 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)
⊢ ∃ π, T = ofArrows Xmap π ∧ EffectiveEpiFamily Xmap π
TACTIC:
|
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)
⊢ T = ofArrows Xmap π ∧ EffectiveEpiFamily Xmap π
TACTIC:
|
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)
⊢ 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)
⊢ EffectiveEpiFamily Xmap π
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.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)
⊢ 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)
⊢ Epi (Sigma.desc 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)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ EffectiveEpiFamily Xmap π
TACTIC:
|
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 (Sigma.desc 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)
this : IsIso (Sigma.desc π) := h.right
⊢ Epi (Sigma.desc 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)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
⊢ Epi (Sigma.desc π)
TACTIC:
|
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 : IsIso (Sigma.desc π) := h.right
⊢ Epi (Sigma.desc π) | 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)
he : EffectiveEpiFamily Xmap π ↔ Epi (Sigma.desc π)
this : IsIso (Sigma.desc π) := h.right
⊢ Epi (Sigma.desc π)
TACTIC:
|
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 X π
TACTIC:
|
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 π
TACTIC:
|
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
TACTIC:
|
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 ∧ EffectiveEpiFamily (fun x => Z) π
TACTIC:
|
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) ∧ 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] | 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) 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
⊢ 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) 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)
⊢ 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) ↔ Epi (Sigma.desc fun x => 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 [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.Surjective ↑(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) ↔ Epi (Sigma.desc fun x => f)
⊢ Epi (Sigma.desc fun x => f)
TACTIC:
|
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.Surjective ↑(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)
x : (forget CompHaus).obj Y
⊢ ∃ a, ↑(Sigma.desc fun x => f) a = x | 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 => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
⊢ Function.Surjective ↑(Sigma.desc fun x => f)
TACTIC:
|
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 CompHaus).obj Y
⊢ ∃ a, ↑(Sigma.desc fun x => f) a = x | 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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ∃ a, ↑(Sigma.desc fun x => f) a = x | 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 => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (forget CompHaus).obj Y
⊢ ∃ a, ↑(Sigma.desc fun x => f) a = x
TACTIC:
|
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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ∃ a, ↑(Sigma.desc fun x => f) a = x | 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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = x | 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 (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ∃ a, ↑(Sigma.desc fun x => f) a = x
TACTIC:
|
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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = x | 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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑f y | 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 (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = x
TACTIC:
|
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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑f 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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
this : ↑f = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f)
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑f y
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 CompHaus).obj Z
hy : ↑f y = x
⊢ ↑f = ↑(Sigma.ι (fun x => Z) () ≫ 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.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 CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑f y
TACTIC:
|
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 CompHaus).obj Z
hy : ↑f y = x
⊢ ↑f = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f) | 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.desc fun x => f)
x : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
⊢ ↑f = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f)
TACTIC:
|
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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
this : ↑f = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f)
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑f 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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
this : ↑f = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f)
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f) y | 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 (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
this : ↑f = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f)
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑f y
TACTIC:
|
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 : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
this : ↑f = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f)
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f) y | 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 (fun x => Z) fun x => f) ↔ Epi (Sigma.desc fun x => f)
x : (forget CompHaus).obj Y
y : (forget CompHaus).obj Z
hy : ↑f y = x
this : ↑f = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f)
⊢ ↑(Sigma.desc fun x => f) (↑(Sigma.ι (fun x => Z) ()) y) = ↑(Sigma.ι (fun x => Z) () ≫ Sigma.desc fun x => f) y
TACTIC:
|
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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
a_ih✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Sieve.pullback f S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y_1
⊢ S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y | 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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
a_ih✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Sieve.pullback f S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y_1
⊢ Coverage.saturate (coherentCoverage CompHaus) Y T
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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
a_ih✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Sieve.pullback f S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y_1
⊢ ∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f S✝) | 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)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
a_ih✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Sieve.pullback f S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y_1
⊢ S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
TACTIC:
|
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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
a_ih✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Sieve.pullback f S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y_1
⊢ Coverage.saturate (coherentCoverage CompHaus) Y T | 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)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
a_ih✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Sieve.pullback f S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y_1
⊢ Coverage.saturate (coherentCoverage CompHaus) Y T
TACTIC:
|
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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
a_ih✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Sieve.pullback f S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y_1
⊢ ∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f S✝) | 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)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y
a_ih✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Sieve.pullback f S✝ ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) Y_1
⊢ ∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f S✝)
TACTIC:
|
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 ⟶ 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 : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 ⟶ 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 : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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' 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 : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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 ⟶ 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 : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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
(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 : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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' 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 : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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
(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 : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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 ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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.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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 ↔ Epi (Sigma.desc f)
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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.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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 Xmap i
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 (Sigma.desc f)
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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 Xmap i
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y | 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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y Zf
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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
Zf.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f (Sieve.generate T)) | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ Sieve.generate T ∈
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 : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst))))
Y
TACTIC:
|
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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y Zf | 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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => Xs) fun x => F) ∈
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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y Zf
TACTIC:
|
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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => Xs) fun x => F) ∈
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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y | 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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ ExtensiveSieve Y ∪ RegularSieve Y | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => Xs) fun x => F) ∈
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 → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y
TACTIC:
|
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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ ExtensiveSieve Y ∪ RegularSieve Y | 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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ ExtensiveSieve Y ∨
(ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ RegularSieve Y | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ ExtensiveSieve Y ∪ RegularSieve Y
TACTIC:
|
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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ ExtensiveSieve Y ∨
(ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ RegularSieve Y | 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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ RegularSieve Y | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ ExtensiveSieve Y ∨
(ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ RegularSieve Y
TACTIC:
|
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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ RegularSieve Y | 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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ∃ f_1, ((ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) = ofArrows (fun x => Xs) fun x => f_1) ∧ Epi f_1 | 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 ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ (ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) ∈ RegularSieve Y
TACTIC:
|
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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ∃ f_1, ((ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) = ofArrows (fun x => Xs) fun x => f_1) ∧ Epi f_1 | 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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ((ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) = ofArrows (fun x => Xs) fun x => F) ∧ Epi F | 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 ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ∃ f_1, ((ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) = ofArrows (fun x => Xs) fun x => f_1) ∧ Epi f_1
TACTIC:
|
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 => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ((ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) = ofArrows (fun x => Xs) fun x => F) ∧ Epi F | 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 ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ((ofArrows (fun x => ∐ fun i => Xmap i) fun x => Sigma.desc f) = ofArrows (fun x => Xs) fun x => F) ∧ Epi F
TACTIC:
|
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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
Zf.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f (Sieve.generate T)) | 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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : Zf.arrows g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
⊢ ∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄,
Zf.arrows f →
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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
Y_1 (Sieve.pullback f (Sieve.generate T))
TACTIC:
|
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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : Zf.arrows g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : ∃ Y_1 h g_1, ofArrows (fun x => ∐ fun i => Xmap i) (fun x => Sigma.desc f) g_1 ∧ h ≫ g_1 = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : Zf.arrows g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T))
TACTIC:
|
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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : ∃ Y_1 h g_1, ofArrows (fun x => ∐ fun i => Xmap i) (fun x => Sigma.desc f) g_1 ∧ h ≫ g_1 = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : ∃ Y_1 h g_1, Presieve.singleton (Sigma.desc f) g_1 ∧ h ≫ g_1 = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : ∃ Y_1 h g_1, ofArrows (fun x => ∐ fun i => Xmap i) (fun x => Sigma.desc f) g_1 ∧ h ≫ g_1 = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T))
TACTIC:
|
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.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : ∃ Y_1 h g_1, Presieve.singleton (Sigma.desc f) g_1 ∧ h ≫ g_1 = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
ψ : R ⟶ W
σ : W ⟶ Y
hW : Presieve.singleton (Sigma.desc f) σ
hW' : ψ ≫ σ = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
hZfg : ∃ Y_1 h g_1, Presieve.singleton (Sigma.desc f) g_1 ∧ h ≫ g_1 = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T))
TACTIC:
|
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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
ψ : R ⟶ W
σ : W ⟶ Y
hW : Presieve.singleton (Sigma.desc f) σ
hW' : ψ ≫ σ = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
ψ : R ⟶ W
σ : W ⟶ Y
hW : singleton' (Sigma.desc f) σ
hW' : ψ ≫ σ = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
ψ : R ⟶ W
σ : W ⟶ Y
hW : Presieve.singleton (Sigma.desc f) σ
hW' : ψ ≫ σ = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T))
TACTIC:
|
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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
ψ : R ⟶ W
σ : W ⟶ Y
hW : singleton' (Sigma.desc f) σ
hW' : ψ ≫ σ = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => Xmap i
hW' : ψ ≫ Sigma.desc f = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
ψ : R ⟶ W
σ : W ⟶ Y
hW : singleton' (Sigma.desc f) σ
hW' : ψ ≫ σ = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T))
TACTIC:
|
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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => Xmap i
hW' : ψ ≫ Sigma.desc f = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T)) | 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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => Xmap i
hW' : ψ ≫ Sigma.desc f = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback ψ (Sieve.pullback (Sigma.desc f) Z)) | 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 : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => Xmap i
hW' : ψ ≫ Sigma.desc f = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback g (Sieve.generate T))
TACTIC:
|
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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => Xmap i
hW' : ψ ≫ Sigma.desc f = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback ψ (Sieve.pullback (Sigma.desc f) 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 ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => Xmap i
hW' : ψ ≫ Sigma.desc f = g
this :
Sieve.pullback ψ (Sieve.pullback F Z) ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus (ExtensiveRegularCoverage' CompHaus ?m.588102 ?m.588103)) R
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback ψ (Sieve.pullback (Sigma.desc f) Z))
case this
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 Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => Xmap i
hW' : ψ ≫ Sigma.desc f = g
⊢ Sieve.pullback ψ (Sieve.pullback F Z) ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus (ExtensiveRegularCoverage' CompHaus ?m.588102 ?m.588103)) R | 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 : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := Sigma.desc f
Z : Sieve Y := Sieve.generate T
Xs : CompHaus := ∐ fun i => Xmap i
Zf : Sieve Y := Sieve.generate (ofArrows (fun x => Xs) fun x => F)
R : CompHaus
g : R ⟶ Y
W : CompHaus
σ : W ⟶ Y
ψ : R ⟶ ∐ fun i => Xmap i
hW' : ψ ≫ Sigma.desc f = g
⊢ 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 ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
R (Sieve.pullback ψ (Sieve.pullback (Sigma.desc f) Z))
TACTIC:
|
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