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 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | set g := toCompHaus.preimage <| Projective.factorThru (𝟙 _) (toCompHaus.map f) with hg | case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
⊢ IsSheafFor F (ofArrows (fun x => Y) fun x => f) | case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
⊢ IsSheafFor F (ofArrows (fun x => Y) fun x => f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | have hfg : g ≫ f = 𝟙 _ := by
refine' toCompHaus.map_injective _
rw [map_comp, hg, image_preimage, Projective.factorThru_comp, CategoryTheory.Functor.map_id] | case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (... | case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | intro y hy | case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (... | case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | refine' ⟨F.map g.op <| y f <| ofArrows.mk (), fun Z h hZ => _, fun z hz => _⟩ | case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (... | case intro.intro.intro.refine'_1
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.fa... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | rw [CompHaus.epi_iff_surjective] | X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
⊢ Epi (toCompHaus.map f) | X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
⊢ Function.Surjective ↑(toCompHaus.map f) | Please generate a tactic in lean4 to solve the state.
STATE:
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
⊢ Epi (toCompHaus.map f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | change Function.Surjective f | X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
⊢ Function.Surjective ↑(toCompHaus.map f) | X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
⊢ Function.Surjective ↑f | Please generate a tactic in lean4 to solve the state.
STATE:
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
⊢ Function.Surjective ↑(toCompHaus.map f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | rwa [← ExtrDisc.epi_iff_surjective] | X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
⊢ Function.Surjective ↑f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
⊢ Function.Surjective ↑f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | refine' toCompHaus.map_injective _ | X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) ... | X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toComp... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | rw [map_comp, hg, image_preimage, Projective.factorThru_comp, CategoryTheory.Functor.map_id] | X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toComp... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | cases' hZ with u | case intro.intro.intro.refine'_1
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.fa... | case intro.intro.intro.refine'_1.mk
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine'_1
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (t... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | have := hy (f₁ := f) (f₂ := f) (𝟙 Y) (f ≫ g) (ofArrows.mk ()) (ofArrows.mk ()) ?_ | case intro.intro.intro.refine'_1.mk
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective... | case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine'_1.mk
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X))... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | rw [op_id, F.map_id, types_id_apply] at this | case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (... | case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | rw [← types_comp_apply (F.map g.op) (F.map f.op), ← F.map_comp, ← op_comp] | case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (... | case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | exact this.symm | case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine'_1.mk.refine_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | rw [Category.id_comp, Category.assoc, hfg, Category.comp_id] | case intro.intro.intro.refine'_1.mk.refine_1
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (P... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine'_1.mk.refine_1
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHau... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | have := congr_arg (F.map g.op) <| hz f (ofArrows.mk ()) | case intro.intro.intro.refine'_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.fa... | case intro.intro.intro.refine'_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.f... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine'_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (t... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafForRegularSieve | [219, 1] | [241, 40] | rwa [← types_comp_apply (F.map f.op) (F.map g.op), ← F.map_comp, ← op_comp, hfg, op_id,
F.map_id, types_id_apply] at this | case intro.intro.intro.refine'_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (toCompHaus.map f))
hg : g = toCompHaus.preimage (Projective.f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine'_2
X : ExtrDisc
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
Y : ExtrDisc
f : Y ⟶ X
hf : Epi f
proj : Projective (toCompHaus.obj X)
this✝ : Epi (toCompHaus.map f)
g : X ⟶ Y := toCompHaus.preimage (Projective.factorThru (𝟙 (toCompHaus.obj X)) (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafFor_of_extensiveRegular | [243, 1] | [249, 44] | cases' hS with hSIso hSSingle | EverythinProj_ExtrDisc : EverythingIsProjective ExtrDisc
X : ExtrDisc
S : Presieve X
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage ExtrDisc EverythinProj_ExtrDisc
(_ :
∀ {α : Type} [inst : Fintype α] {X : ExtrDisc} {Z : α → ExtrDisc} (π : (a : α) → Z a ⟶ X) {Y : ExtrDisc}
... | case inl
EverythinProj_ExtrDisc : EverythingIsProjective ExtrDisc
X : ExtrDisc
S : Presieve X
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
hF : PreservesFiniteProducts F
hSIso : S ∈ ExtensiveSieve X
⊢ IsSheafFor F S
case inr
EverythinProj_ExtrDisc : EverythingIsProjective ExtrDisc
X : ExtrDisc
S : Presieve X
F : ExtrDiscᵒᵖ ⥤ Type (u... | Please generate a tactic in lean4 to solve the state.
STATE:
EverythinProj_ExtrDisc : EverythingIsProjective ExtrDisc
X : ExtrDisc
S : Presieve X
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage ExtrDisc EverythinProj_ExtrDisc
(_ :
∀ {α : Type} [inst : Fintype α] {X : ExtrDisc} {Z : α ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafFor_of_extensiveRegular | [243, 1] | [249, 44] | exact isSheafForExtensiveSieve hSIso hF | case inl
EverythinProj_ExtrDisc : EverythingIsProjective ExtrDisc
X : ExtrDisc
S : Presieve X
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
hF : PreservesFiniteProducts F
hSIso : S ∈ ExtensiveSieve X
⊢ IsSheafFor F S | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
EverythinProj_ExtrDisc : EverythingIsProjective ExtrDisc
X : ExtrDisc
S : Presieve X
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
hF : PreservesFiniteProducts F
hSIso : S ∈ ExtensiveSieve X
⊢ IsSheafFor F S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isSheafFor_of_extensiveRegular | [243, 1] | [249, 44] | exact isSheafForRegularSieve hSSingle F | case inr
EverythinProj_ExtrDisc : EverythingIsProjective ExtrDisc
X : ExtrDisc
S : Presieve X
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
hF : PreservesFiniteProducts F
hSSingle : S ∈ RegularSieve X
⊢ IsSheafFor F S | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
EverythinProj_ExtrDisc : EverythingIsProjective ExtrDisc
X : ExtrDisc
S : Presieve X
F : ExtrDiscᵒᵖ ⥤ Type (u + 1)
hF : PreservesFiniteProducts F
hSSingle : S ∈ RegularSieve X
⊢ IsSheafFor F S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.final | [251, 1] | [255, 65] | rw [← extensiveRegular_generates_coherent] | A : Type (u + 2)
inst✝ : Category A
F : ExtrDiscᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
⊢ Presheaf.IsSheaf (coherentTopology ExtrDisc) F | A : Type (u + 2)
inst✝ : Category A
F : ExtrDiscᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
⊢ Presheaf.IsSheaf
(Coverage.toGrothendieck ExtrDisc
(ExtensiveRegularCoverage ExtrDisc everything_proj
(_ :
∀ {α : Type} [inst : Fintype α] {X : ExtrDisc} {Z : α → ExtrDisc} (π : (a : α) → Z a ⟶ X) {Y : Ex... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝ : Category A
F : ExtrDiscᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
⊢ Presheaf.IsSheaf (coherentTopology ExtrDisc) F
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.final | [251, 1] | [255, 65] | exact fun E => (Presieve.isSheaf_coverage _ _).2 <| fun S hS => isSheafFor_of_extensiveRegular hS
⟨fun J inst => have := hF.1; compPreservesLimitsOfShape _ _⟩ | A : Type (u + 2)
inst✝ : Category A
F : ExtrDiscᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
⊢ Presheaf.IsSheaf
(Coverage.toGrothendieck ExtrDisc
(ExtensiveRegularCoverage ExtrDisc everything_proj
(_ :
∀ {α : Type} [inst : Fintype α] {X : ExtrDisc} {Z : α → ExtrDisc} (π : (a : α) → Z a ⟶ X) {Y : Ex... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝ : Category A
F : ExtrDiscᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
⊢ Presheaf.IsSheaf
(Coverage.toGrothendieck ExtrDisc
(ExtensiveRegularCoverage ExtrDisc everything_proj
(_ :
∀ {α : Type} [inst : Fintype α] {X :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ProdCoprod.lean | descOpCompCoprodToProd | [79, 1] | [84, 6] | refine' limit.hom_ext (fun a => _) | C : Type u_2
inst✝² : Category C
α : Type
inst✝¹ : Finite α
Z : α → C
inst✝ : HasFiniteCoproducts C
X✝ : C
π✝ : (a : α) → Z a ⟶ X✝
X : C
π : (a : α) → Z a ⟶ X
⊢ (Sigma.desc π).op ≫ (CoprodToProd Z).hom = Pi.lift fun a => (π a).op | C : Type u_2
inst✝² : Category C
α : Type
inst✝¹ : Finite α
Z : α → C
inst✝ : HasFiniteCoproducts C
X✝ : C
π✝ : (a : α) → Z a ⟶ X✝
X : C
π : (a : α) → Z a ⟶ X
a : Discrete α
⊢ ((Sigma.desc π).op ≫ (CoprodToProd Z).hom) ≫ limit.π (Discrete.functor fun z => (Z z).op) a =
(Pi.lift fun a => (π a).op) ≫ limit.π (Discret... | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_2
inst✝² : Category C
α : Type
inst✝¹ : Finite α
Z : α → C
inst✝ : HasFiniteCoproducts C
X✝ : C
π✝ : (a : α) → Z a ⟶ X✝
X : C
π : (a : α) → Z a ⟶ X
⊢ (Sigma.desc π).op ≫ (CoprodToProd Z).hom = Pi.lift fun a => (π a).op
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ProdCoprod.lean | descOpCompCoprodToProd | [79, 1] | [84, 6] | simp only [CoprodToProd, Category.assoc, limit.conePointUniqueUpToIso_hom_comp, oppositeFan_pt,
oppositeFan_π_app, limit.lift_π, Fan.mk_pt, Fan.mk_π_app, ← op_comp, colimit.ι_desc] | C : Type u_2
inst✝² : Category C
α : Type
inst✝¹ : Finite α
Z : α → C
inst✝ : HasFiniteCoproducts C
X✝ : C
π✝ : (a : α) → Z a ⟶ X✝
X : C
π : (a : α) → Z a ⟶ X
a : Discrete α
⊢ ((Sigma.desc π).op ≫ (CoprodToProd Z).hom) ≫ limit.π (Discrete.functor fun z => (Z z).op) a =
(Pi.lift fun a => (π a).op) ≫ limit.π (Discret... | C : Type u_2
inst✝² : Category C
α : Type
inst✝¹ : Finite α
Z : α → C
inst✝ : HasFiniteCoproducts C
X✝ : C
π✝ : (a : α) → Z a ⟶ X✝
X : C
π : (a : α) → Z a ⟶ X
a : Discrete α
⊢ ((Cofan.mk X π).ι.app { as := a.as }).op = (π a.as).op | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_2
inst✝² : Category C
α : Type
inst✝¹ : Finite α
Z : α → C
inst✝ : HasFiniteCoproducts C
X✝ : C
π✝ : (a : α) → Z a ⟶ X✝
X : C
π : (a : α) → Z a ⟶ X
a : Discrete α
⊢ ((Sigma.desc π).op ≫ (CoprodToProd Z).hom) ≫ limit.π (Discrete.functor fun z => (Z ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Sieves/ProdCoprod.lean | descOpCompCoprodToProd | [79, 1] | [84, 6] | rfl | C : Type u_2
inst✝² : Category C
α : Type
inst✝¹ : Finite α
Z : α → C
inst✝ : HasFiniteCoproducts C
X✝ : C
π✝ : (a : α) → Z a ⟶ X✝
X : C
π : (a : α) → Z a ⟶ X
a : Discrete α
⊢ ((Cofan.mk X π).ι.app { as := a.as }).op = (π a.as).op | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_2
inst✝² : Category C
α : Type
inst✝¹ : Finite α
Z : α → C
inst✝ : HasFiniteCoproducts C
X✝ : C
π✝ : (a : α) → Z a ⟶ X✝
X : C
π : (a : α) → Z a ⟶ X
a : Discrete α
⊢ ((Cofan.mk X π).ι.app { as := a.as }).op = (π a.as).op
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | constructor | ⊢ CoverDense (coherentTopology CompHaus) ExtrDisc.toCompHaus | case is_cover
⊢ ∀ (U : CompHaus), Sieve.coverByImage ExtrDisc.toCompHaus U ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) U | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ CoverDense (coherentTopology CompHaus) ExtrDisc.toCompHaus
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | intro B | case is_cover
⊢ ∀ (U : CompHaus), Sieve.coverByImage ExtrDisc.toCompHaus U ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) U | case is_cover
B : CompHaus
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B | Please generate a tactic in lean4 to solve the state.
STATE:
case is_cover
⊢ ∀ (U : CompHaus), Sieve.coverByImage ExtrDisc.toCompHaus U ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) U
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | let T := Presieve.singleton B.presentationπ | case is_cover
B : CompHaus
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B | case is_cover
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B | Please generate a tactic in lean4 to solve the state.
STATE:
case is_cover
B : CompHaus
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | let S := Sieve.generate T | case is_cover
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B | case is_cover
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B | Please generate a tactic in lean4 to solve the state.
STATE:
case is_cover
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | convert hS | case is_cover
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B | case h.e'_4
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B = S | Please generate a tactic in lean4 to solve the state.
STATE:
case is_cover
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B ∈ GrothendieckTopology.sie... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | ext Y f | case h.e'_4
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B = S | case h.e'_4.h
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
⊢ (Sieve.coverByImage ExtrDisc.toCompHaus B).arrows f ↔ S.arrows f | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
⊢ Sieve.coverByImage ExtrDisc.toCompHaus B = S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | constructor | case h.e'_4.h
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
⊢ (Sieve.coverByImage ExtrDisc.toCompHaus B).arrows f ↔ S.arrows f | case h.e'_4.h.mp
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
⊢ (Sieve.coverByImage ExtrDisc.toCompHaus B).arrows f → S.arrows f
case h.e'_4.h.mpr
B : CompHaus
T : P... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
⊢ (Sieve.coverByImage ExtrDisc.toCompHaus B).a... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | apply Coverage.saturate.of | B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B | case hS
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ T ∈ Coverage.covering (coherentCoverage CompHaus) B | Please generate a tactic in lean4 to solve the state.
STATE:
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | change ∃ _, _ | case hS
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ T ∈ Coverage.covering (coherentCoverage CompHaus) B | case hS
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ ∃ x x_1 X π, T = Presieve.ofArrows X π ∧ EffectiveEpiFamily X π | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ T ∈ Coverage.covering (coherentCoverage CompHaus) B
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | refine ⟨Unit, inferInstance,
fun _ => B.presentation.compHaus, fun _ => B.presentationπ, ?_ , ?_⟩ | case hS
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ ∃ x x_1 X π, T = Presieve.ofArrows X π ∧ EffectiveEpiFamily X π | case hS.refine_1
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ T = Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B
case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentation... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ ∃ x x_1 X π, T = Presieve.ofArrows X π ∧ EffectiveEpiFamily X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | funext X f | case hS.refine_1
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ T = Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B | case hS.refine_1.h.h
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f = Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B) f | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_1
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ T = Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | ext | case hS.refine_1.h.h
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f = Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B) f | case hS.refine_1.h.h.a
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f ↔ Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B) f | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_1.h.h
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f = Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | constructor | case hS.refine_1.h.h.a
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f ↔ Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B) f | case hS.refine_1.h.h.a.mp
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f → Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B) f
case hS.refine_1.h.h.a.mpr
B : CompHaus
T : Presiev... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_1.h.h.a
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f ↔ Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | rintro ⟨⟩ | case hS.refine_1.h.h.a.mp
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f → Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B) f | case hS.refine_1.h.h.a.mp.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
Y : CompHaus
⊢ Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B)
(CompHaus.presentationπ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_1.h.h.a.mp
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ T f → Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentati... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | refine ⟨()⟩ | case hS.refine_1.h.h.a.mp.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
Y : CompHaus
⊢ Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B)
(CompHaus.presentationπ B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_1.h.h.a.mp.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
Y : CompHaus
⊢ Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B)
(C... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | rintro ⟨⟩ | case hS.refine_1.h.h.a.mpr
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B) f → T f | case hS.refine_1.h.h.a.mpr.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
Y : CompHaus
i✝ : Unit
⊢ T (CompHaus.presentationπ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_1.h.h.a.mpr
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
X : CompHaus
f : X ⟶ B
⊢ Presieve.ofArrows (fun x => (CompHaus.presentation B).compHaus) (fun x => CompHaus.presentationπ B... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | simp | case hS.refine_1.h.h.a.mpr.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
Y : CompHaus
i✝ : Unit
⊢ T (CompHaus.presentationπ B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_1.h.h.a.mpr.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
Y : CompHaus
i✝ : Unit
⊢ T (CompHaus.presentationπ B)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | have := CompHaus.effectiveEpiFamily_tfae
(fun (_ : Unit) => B.presentation.compHaus)
(fun (_ : Unit) => B.presentationπ) | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
⊢ EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | apply (this.out 0 2).mpr | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | intro b | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | refine ⟨(), ?_⟩ | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | have hπ :
Function.Surjective B.presentationπ := by
rw [← CompHaus.epi_iff_surjective B.presentationπ]
exact CompHaus.epiPresentπ B | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | exact hπ b | case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.to... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.refine_2
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | rw [← CompHaus.epi_iff_surjective B.presentationπ] | B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.toTop), ∃ a x, ↑(Co... | B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.toTop), ∃ a x, ↑(Co... | Please generate a tactic in lean4 to solve the state.
STATE:
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | exact CompHaus.epiPresentπ B | B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun x => CompHaus.presentationπ B), ∀ (b : ↑B.toTop), ∃ a x, ↑(Co... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
this :
List.TFAE
[EffectiveEpiFamily (fun x => (CompHaus.presentation B).compHaus) fun x => CompHaus.presentationπ B,
Epi (Sigma.desc fun ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | rintro ⟨⟨obj, lift, map, fact⟩⟩ | case h.e'_4.h.mp
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
⊢ (Sieve.coverByImage ExtrDisc.toCompHaus B).arrows f → S.arrows f | case h.e'_4.h.mp.intro.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj obj ⟶ B
fac... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mp
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
⊢ (Sieve.coverByImage ExtrDisc.toCompHaus B... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | obtain ⟨obj_factors⟩ : Projective obj.compHaus := by
infer_instance | case h.e'_4.h.mp.intro.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj obj ⟶ B
fac... | case h.e'_4.h.mp.intro.mk.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj obj ⟶ B
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mp.intro.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | obtain ⟨p, p_factors⟩ := obj_factors map B.presentationπ | case h.e'_4.h.mp.intro.mk.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj obj ⟶ B
... | case h.e'_4.h.mp.intro.mk.mk.intro
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj ob... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mp.intro.mk.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrD... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | refine ⟨(CompHaus.presentation B).compHaus ,?_⟩ | case h.e'_4.h.mp.intro.mk.mk.intro
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj ob... | case h.e'_4.h.mp.intro.mk.mk.intro
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj ob... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mp.intro.mk.mk.intro
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | refine ⟨lift ≫ p, ⟨ B.presentationπ
, {
left := Presieve.singleton.mk
right := by
rw [Category.assoc, p_factors, fact]
} ⟩
⟩ | case h.e'_4.h.mp.intro.mk.mk.intro
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj ob... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mp.intro.mk.mk.intro
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | infer_instance | B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj obj ⟶ B
fact : lift ≫ map = f
⊢ Proje... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | rw [Category.assoc, p_factors, fact] | B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ExtrDisc.toCompHaus.obj obj ⟶ B
fact : lift ≫ map = f
obj_fac... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
obj : ExtrDisc
lift : Y ⟶ ExtrDisc.toCompHaus.obj obj
map : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | rintro ⟨Z, h, g, hypo1, ⟨_⟩⟩ | case h.e'_4.h.mpr
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
⊢ S.arrows f → (Sieve.coverByImage ExtrDisc.toCompHaus B).arrows f | case h.e'_4.h.mpr.intro.intro.intro.intro.refl
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y Z : CompHaus
h : Y ⟶ Z
g : Z ⟶ B
hypo1 : T g
⊢ (Sieve.coverByImage ExtrDisc.toCompHaus B).arrow... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mpr
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
f : Y ⟶ B
⊢ S.arrows f → (Sieve.coverByImage ExtrDis... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | cases hypo1 | case h.e'_4.h.mpr.intro.intro.intro.intro.refl
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y Z : CompHaus
h : Y ⟶ Z
g : Z ⟶ B
hypo1 : T g
⊢ (Sieve.coverByImage ExtrDisc.toCompHaus B).arrow... | case h.e'_4.h.mpr.intro.intro.intro.intro.refl.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
h : Y ⟶ (CompHaus.presentation B).compHaus
⊢ (Sieve.coverByImage ExtrDisc.toCompH... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mpr.intro.intro.intro.intro.refl
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y Z : CompHaus
h : Y ⟶ Z
g : Z ⟶ B
h... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | constructor | case h.e'_4.h.mpr.intro.intro.intro.intro.refl.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
h : Y ⟶ (CompHaus.presentation B).compHaus
⊢ (Sieve.coverByImage ExtrDisc.toCompH... | case h.e'_4.h.mpr.intro.intro.intro.intro.refl.mk.val
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
h : Y ⟶ (CompHaus.presentation B).compHaus
⊢ Presieve.CoverByImageStructure E... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mpr.intro.intro.intro.intro.refl.mk
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
h : Y ⟶ (CompHaus.pr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense | [41, 1] | [92, 19] | refine
{ obj := CompHaus.presentation B
lift := h
map := CompHaus.presentationπ B
fac := rfl } | case h.e'_4.h.mpr.intro.intro.intro.intro.refl.mk.val
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
h : Y ⟶ (CompHaus.presentation B).compHaus
⊢ Presieve.CoverByImageStructure E... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.mpr.intro.intro.intro.intro.refl.mk.val
B : CompHaus
T : Presieve B := Presieve.singleton (CompHaus.presentationπ B)
S : Sieve B := Sieve.generate T
hS : S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) B
Y : CompHaus
h : Y ⟶ (CompHau... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | constructor | X : ExtrDisc
S : Sieve X
⊢ (∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)) ↔
S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X | case mp
X : ExtrDisc
S : Sieve X
⊢ (∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)) →
S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X
case mpr
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X →
∃ α x Y π, EffectiveEpiF... | Please generate a tactic in lean4 to solve the state.
STATE:
X : ExtrDisc
S : Sieve X
⊢ (∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)) ↔
S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rintro ⟨α, _, Y, π, ⟨H₁, H₂⟩⟩ | case mp
X : ExtrDisc
S : Sieve X
⊢ (∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)) →
S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : ExtrDisc
S : Sieve X
⊢ (∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)) →
S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | unfold CoverDense.inducedTopology | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈
GrothendieckTopology.sieves
(LocallyCoverDense.inducedTopology (_ : LocallyCoverDense (coherentTopology CompHaus) ExtrD... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | unfold LocallyCoverDense.inducedTopology | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈
GrothendieckTopology.sieves
(LocallyCoverDense.inducedTopology (_ : LocallyCoverDense (coherentTopology CompHaus) ExtrD... | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈
GrothendieckTopology.sieves
{
sieves := fun X S =>
GrothendieckTopology.sieves (coherentTopology Comp... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈
GrothendieckTopology.sieves
(LocallyCoverDense.inducedTopo... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | simp only [ExtrDisc.toCompHaus_obj] | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈
GrothendieckTopology.sieves
{
sieves := fun X S =>
GrothendieckTopology.sieves (coherentTopology Comp... | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈ fun S =>
GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus (Sieve.functorPushforward ExtrDisc.toCompHaus S) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈
GrothendieckTopology.sieves
{
sieves := fun X S =>... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | change _ ∈ GrothendieckTopology.sieves _ _ | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈ fun S =>
GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus (Sieve.functorPushforward ExtrDisc.toCompHaus S) | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ Sieve.functorPushforward ExtrDisc.toCompHaus S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ S ∈ fun S =>
GrothendieckTopology.sieves (coherentTopology CompHaus)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | apply (coherentTopology.Sieve_iff_hasEffectiveEpiFamily (Sieve.functorPushforward _ S)).mp | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ Sieve.functorPushforward ExtrDisc.toCompHaus S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ Sieve.functorPushforward ExtrDisc.toCompHaus S ∈ GrothendieckTopology.si... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | use α, inferInstance | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a) | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | use fun i => ExtrDisc.toCompHaus.obj (Y i) | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a) | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ π,
EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) π ∧
∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHa... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward Ext... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | use fun i => ExtrDisc.toCompHaus.map (π i) | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ π,
EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) π ∧
∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHa... | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ (EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) fun i => ExtrDisc.toCompHaus.map (π i)) ∧
∀ (a : α), (Sieve.functorP... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∃ π,
EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) π ∧... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | constructor | case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ (EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) fun i => ExtrDisc.toCompHaus.map (π i)) ∧
∀ (a : α), (Sieve.functorP... | case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) fun i => ExtrDisc.toCompHaus.map (π i)
case mp.intro.intro.intro.in... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ (EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) fun i => Ex... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | simp only [ExtrDisc.toCompHaus_obj, ExtrDisc.toCompHaus_map] | case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) fun i => ExtrDisc.toCompHaus.map (π i) | case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ EffectiveEpiFamily (fun i => (Y i).compHaus) fun i => π i | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ EffectiveEpiFamily (fun i => ExtrDisc.toCompHaus.obj (Y i)) fun i =... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | simp only [(ExtrDisc.effectiveEpiFamily_tfae _ _).out 0 2] at H₁ | case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ EffectiveEpiFamily (fun i => (Y i).compHaus) fun i => π i | case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₂ : ∀ (a : α), S.arrows (π a)
H₁ : ∀ (b : CoeSort.coe X), ∃ a x, ↑(π a) x = b
⊢ EffectiveEpiFamily (fun i => (Y i).compHaus) fun i => π i | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ EffectiveEpiFamily (fun i => (Y i).compHaus) fun i => π i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | exact CompHaus.effectiveEpiFamily_of_jointly_surjective
(fun i => (Y i).compHaus) (fun i => π i) H₁ | case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₂ : ∀ (a : α), S.arrows (π a)
H₁ : ∀ (b : CoeSort.coe X), ∃ a x, ↑(π a) x = b
⊢ EffectiveEpiFamily (fun i => (Y i).compHaus) fun i => π i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₂ : ∀ (a : α), S.arrows (π a)
H₁ : ∀ (b : CoeSort.coe X), ∃ a x, ↑(π a) x = b
⊢ EffectiveEpiFamily (fun i => (Y i).compHaus) fu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | exact fun a => Sieve.image_mem_functorPushforward ExtrDisc.toCompHaus S (H₂ a) | case mp.intro.intro.intro.intro.intro.right
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows ((fun i => ExtrDisc.toCompHaus.map (π i)) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro.right
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → ExtrDisc
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | intro hS | case mpr
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X →
∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr
X : ExtrDisc
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X →
∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | unfold CoverDense.inducedTopology at hS | case mpr
X : ExtrDisc
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈
GrothendieckTopology.sieves
(LocallyCoverDense.inducedTopology (_ : LocallyCoverDense (coherentTopology CompHaus) ExtrDisc.toCompHaus)) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : ExtrDisc
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | unfold LocallyCoverDense.inducedTopology at hS | case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈
GrothendieckTopology.sieves
(LocallyCoverDense.inducedTopology (_ : LocallyCoverDense (coherentTopology CompHaus) ExtrDisc.toCompHaus)) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈
GrothendieckTopology.sieves
{
sieves := fun X S =>
GrothendieckTopology.sieves (coherentTopology CompHaus) (ExtrDisc.toCompHaus.obj X)
(Sieve.functorPushforward ExtrDisc.toCompHaus S),
top_mem' :=
(_ :
∀ (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈
GrothendieckTopology.sieves
(LocallyCoverDense.inducedTopology (_ : LocallyCoverDense (coherentTopology CompHaus) ExtrDisc.toCompHaus)) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TA... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | simp only [ExtrDisc.toCompHaus_obj] at hS | case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈
GrothendieckTopology.sieves
{
sieves := fun X S =>
GrothendieckTopology.sieves (coherentTopology CompHaus) (ExtrDisc.toCompHaus.obj X)
(Sieve.functorPushforward ExtrDisc.toCompHaus S),
top_mem' :=
(_ :
∀ (... | case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈ fun S =>
GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus (Sieve.functorPushforward ExtrDisc.toCompHaus S)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈
GrothendieckTopology.sieves
{
sieves := fun X S =>
GrothendieckTopology.sieves (coherentTopology CompHaus) (ExtrDisc.toCompHaus.obj X)
(Sieve.functorPushforward ExtrDisc.toCo... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | change _ ∈ GrothendieckTopology.sieves _ _ at hS | case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈ fun S =>
GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus (Sieve.functorPushforward ExtrDisc.toCompHaus S)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr
X : ExtrDisc
S : Sieve X
hS : Sieve.functorPushforward ExtrDisc.toCompHaus S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : ExtrDisc
S : Sieve X
hS :
S ∈ fun S =>
GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus (Sieve.functorPushforward ExtrDisc.toCompHaus S)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | replace hS := (coherentTopology.Sieve_iff_hasEffectiveEpiFamily _).mpr hS | case mpr
X : ExtrDisc
S : Sieve X
hS : Sieve.functorPushforward ExtrDisc.toCompHaus S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr
X : ExtrDisc
S : Sieve X
hS : ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : ExtrDisc
S : Sieve X
hS : Sieve.functorPushforward ExtrDisc.toCompHaus S ∈ GrothendieckTopology.sieves (coherentTopology CompHaus) X.compHaus
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rcases hS with ⟨α, _, Y, π, ⟨H₁, H₂⟩⟩ | case mpr
X : ExtrDisc
S : Sieve X
hS : ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : ExtrDisc
S : Sieve X
hS : ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | use α, inferInstance | case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | change ∀ a, ∃ (Y₀: ExtrDisc) (π₀ : Y₀⟶ X) (f₀: Y a ⟶ Y₀.compHaus), S.arrows π₀ ∧
π a = f₀ ≫ ExtrDisc.toCompHaus.map π₀ at H₂ | case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), ∃ Y₀ π₀ f₀, S.arrows π₀ ∧ π a = f₀ ≫ ExtrDisc.toCompHaus.map π₀
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), (Sieve.functorPushforward ExtrDisc.toCompHaus S).arrows (π a)
⊢ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rw [Classical.skolem] at H₂ | case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), ∃ Y₀ π₀ f₀, S.arrows π₀ ∧ π a = f₀ ≫ ExtrDisc.toCompHaus.map π₀
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∃ f, ∀ (x : α), ∃ π₀ f₀, S.arrows π₀ ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map π₀
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), ∃ Y₀ π₀ f₀, S.arrows π₀ ∧ π a = f₀ ≫ ExtrDisc.toCompHaus.map π₀
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rcases H₂ with ⟨Y₀, H₂⟩ | case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∃ f, ∀ (x : α), ∃ π₀ f₀, S.arrows π₀ ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map π₀
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
H₂ : ∀ (x : α), ∃ π₀ f₀, S.arrows π₀ ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map π₀
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a :... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∃ f, ∀ (x : α), ∃ π₀ f₀, S.arrows π₀ ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map π... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rw [Classical.skolem] at H₂ | case mpr.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
H₂ : ∀ (x : α), ∃ π₀ f₀, S.arrows π₀ ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map π₀
⊢ ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a :... | case mpr.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
H₂ : ∃ f, ∀ (x : α), ∃ f₀, S.arrows (f x) ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map (f x)
⊢ ∃ Y π, EffectiveEpiFamily Y π ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
H₂ : ∀ (x : α), ∃ π₀ f₀, S.arrows π₀ ∧ π x = f₀ ≫ ExtrDi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rcases H₂ with ⟨π₀, H₂⟩ | case mpr.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
H₂ : ∃ f, ∀ (x : α), ∃ f₀, S.arrows (f x) ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map (f x)
⊢ ∃ Y π, EffectiveEpiFamily Y π ... | case mpr.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
H₂ : ∀ (x : α), ∃ f₀, S.arrows (π₀ x) ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map (π₀ x)
⊢ ∃ Y... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
H₂ : ∃ f, ∀ (x : α), ∃ f₀, S.arrows (f x) ∧ π x = f₀ ≫ E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rw [Classical.skolem] at H₂ | case mpr.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
H₂ : ∀ (x : α), ∃ f₀, S.arrows (π₀ x) ∧ π x = f₀ ≫ ExtrDisc.toCompHaus.map (π₀ x)
⊢ ∃ Y... | case mpr.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
H₂ : ∃ f, ∀ (x : α), S.arrows (π₀ x) ∧ π x = f x ≫ ExtrDisc.toCompHaus.map (π₀ x)
⊢ ∃ Y... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
H₂ : ∀ (x : α), ∃ f₀, S.ar... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rcases H₂ with ⟨f₀, H₂⟩ | case mpr.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
H₂ : ∃ f, ∀ (x : α), S.arrows (π₀ x) ∧ π x = f x ≫ ExtrDisc.toCompHaus.map (π₀ x)
⊢ ∃ Y... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
H₂ : ∃ f, ∀ (x : α), S.arr... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | use Y₀ , π₀ | case mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | constructor | case mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x =... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | simp only [(ExtrDisc.effectiveEpiFamily_tfae _ _).out 0 2] | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x =... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x =... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | simp only [(CompHaus.effectiveEpiFamily_tfae _ _).out 0 2] at H₁ | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x =... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | intro b | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | replace H₁ := H₁ b | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | rcases H₁ with ⟨i, x, H₁⟩ | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | use i, f₀ i x | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc... | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).com... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | aesop_cat | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.left.intro.intro
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).com... |
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