url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply GrothendieckTopology.pullback_stable' | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [Coverage.toGrothendieck] | case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y :... | case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | suffices : Coverage.saturate (ExtensiveRegularCoverage' _ _ _) Xs (Z.pullback F) | case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y :... | case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | suffices : Sieve.generate (Presieve.ofArrows Xmap φ) ≤ Z.pullback F | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | intro Q q hq | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [Sieve.pullback_apply, Sieve.generate_apply] | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [Sieve.generate_apply] at hq | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | obtain ⟨E, e, r, hq⟩ := hq | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | refine' ⟨E, e, r ≫ F, ⟨_, _⟩⟩ | case this.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b ... | case this.intro.intro.intro.refine'_1
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F :... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | exact this | case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : Effecti... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | exact this | case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.a
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate_of_superset _ this | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.of | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp [ExtensiveRegularCoverage'] | case this.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | case this.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | left | case this.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y ... | case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.hS
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | refine' ⟨I, hI, Xmap, φ, ⟨rfl, _⟩⟩ | case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | suffices : Sigma.desc φ = 𝟙 _ | case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | ext | case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y := ... | case this.h.w
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xm... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app, Category.comp_id] | case this.h.w
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ Y... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h.w
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [this] | case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | exact inferInstance | case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F : (∐ fun b => Xmap b) ⟶ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.hS.h
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [h] | case this.intro.intro.intro.refine'_1
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F :... | case this.intro.intro.intro.refine'_1
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F :... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro.refine'_1
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | induction hq.1 | case this.intro.intro.intro.refine'_1
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F :... | case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro.refine'_1
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | dsimp | case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
... | case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | simp only [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app] | case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
... | case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | exact Presieve.ofArrows.mk _ | case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro.refine'_1.mk
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rw [← hq.2] | case this.intro.intro.intro.refine'_2
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F :... | case this.intro.intro.intro.refine'_2
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F :... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro.refine'_2
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | rfl | case this.intro.intro.intro.refine'_2
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)
φ : (i : I) → Xmap i ⟶ ∐ Xmap := fun i => Sigma.ι Xmap i
F :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro.intro.intro.refine'_2
X : CompHaus
S : Sieve X
Y : CompHaus
T : Presieve Y
I : Type
hI : Fintype I
Xmap : I → CompHaus
f : (a : I) → Xmap a ⟶ Y
h : T = ofArrows Xmap f
hT : Epi (Sigma.desc f)
he : EffectiveEpiFamily Xmap f ↔ Epi (Sigma.desc f)... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.top | case h.h.h.mpr.top
X : CompHaus
S : Sieve X
X✝ : CompHaus
⊢ ⊤ ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type}... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.top
X : CompHaus
S : Sieve X
X✝ : CompHaus
⊢ ⊤ ∈
GrothendieckTopology.sieves
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | apply Coverage.saturate.transitive Y T | case h.h.h.mpr.transitive
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage CompHaus) Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f S✝)
a_ih✝¹ :
T ∈
GrothendieckTopology.sieves
(Cover... | case h.h.h.mpr.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage CompHaus) Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f S✝)
a_ih✝¹ :
T ∈
GrothendieckTopology.sieves
(Cov... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.transitive
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage CompHaus) Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f S✝... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | assumption | case h.h.h.mpr.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage CompHaus) Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f S✝)
a_ih✝¹ :
T ∈
GrothendieckTopology.sieves
(Cov... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage CompHaus) Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensiveRegular_generates_coherent | [118, 1] | [227, 21] | assumption | case h.h.h.mpr.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage CompHaus) Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f S✝)
a_ih✝¹ :
T ∈
GrothendieckTopology.sieves
(Cov... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.transitive.a
X : CompHaus
S : Sieve X
Y : CompHaus
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage CompHaus) Y T
a✝ :
∀ ⦃Y_1 : CompHaus⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage CompHaus) Y_1 (Sieve.pullback f ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | cases' hS with hSIso hSSingle | B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : ... | case inl
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSIso : S ∈ ExtensiveSieve B
⊢ IsSheafFor F S
case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ... | Please generate a tactic in lean4 to solve the state.
STATE:
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | exact isSheafForExtensiveSieve hSIso hFpfp | case inl
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSIso : S ∈ ExtensiveSieve B
⊢ IsSheafFor F S | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSIso : S ∈ ExtensiveSieve B
⊢ IsSheafFor F S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [Equalizer.Presieve.sheaf_condition, Limits.Types.type_equalizer_iff_unique] | case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ∈ RegularSieve B
⊢ IsSheafFor F S | case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ∈ RegularSieve B
⊢ ∀ (y : Equalizer.FirstObj F S),
Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y → ∃! x, Equalizer.forkMap F S x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ∈ RegularSieve B
⊢ IsSheafFor F S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | intro y h | case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ∈ RegularSieve B
⊢ ∀ (y : Equalizer.FirstObj F S),
Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y → ∃! x, Equalizer.forkMap F S x = y | case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ∈ RegularSieve B
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
⊢ ∃! x, Equalizer.forkMap F S x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ∈ RegularSieve B
⊢ ∀ (y : Equalizer.FirstObj F S),
Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.second... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp [RegularSieve] at hSSingle | case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ∈ RegularSieve B
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
⊢ ∃! x, Equalizer.forkMap F S x = y | case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : ∃ X f, (S = ofArrows (fun x => X) fun x => f) ∧ Epi f
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
⊢ ∃! x, Equalizer.forkM... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : S ∈ RegularSieve B
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | obtain ⟨X, π, ⟨hS, πsurj⟩⟩ := hSSingle | case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : ∃ X f, (S = ofArrows (fun x => X) fun x => f) ∧ Epi f
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
⊢ ∃! x, Equalizer.forkM... | case inr.intro.intro.intro
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
X : CompHaus
π : X ⟶ B
hS : S = ofArrows (fun x => X) fun x => π
πsurj... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
hSSingle : ∃ X f, (S = ofArrows (fun x => X) fun x => f) ∧ Epi f
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [Presieve.ofArrows_pUnit] at hS | case inr.intro.intro.intro
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
X : CompHaus
π : X ⟶ B
hS : S = ofArrows (fun x => X) fun x => π
πsurj... | case inr.intro.intro.intro
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
X : CompHaus
π : X ⟶ B
hS : S = Presieve.singleton π
πsurj : Epi π
⊢ ∃... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
X : CompH... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | subst hS | case inr.intro.intro.intro
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
X : CompHaus
π : X ⟶ B
hS : S = Presieve.singleton π
πsurj : Epi π
⊢ ∃... | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
X : CompHaus
π : X ⟶ B
πsurj : Epi π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
S : Presieve B
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
y : Equalizer.FirstObj F S
h : Equalizer.Presieve.firstMap F S y = Equalizer.Presieve.secondMap F S y
X : CompH... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [CompHaus.epi_iff_surjective] at πsurj | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
X : CompHaus
π : X ⟶ B
πsurj : Epi π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.... | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secon... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
X : CompHaus
π : X ⟶ B
πsurj : Epi π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presie... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | specialize hFecs X B π πsurj | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secon... | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π)... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
hFecs : EqualizerCondition F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | have fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom =
F.map π.op | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π)... | case fork_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | have fmap_comp : (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π)... | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | have smap_comp : (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π)... | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | have iy_mem : F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y) | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π)... | case iy_mem
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | have uniq_F : ∃! x, F.map π.op x = (EqualizerFirstObjIso F π).hom y | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π)... | case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | obtain ⟨x,hx⟩ := uniq_F | case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π)... | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp at hx | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.sin... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [← fork_comp] at hx | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.sin... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | use x | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.sin... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.sin... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | constructor | case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.single... | case inr.intro.intro.intro.intro.left
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.s... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.sin... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp [EqualizerFirstObjIso, Equalizer.forkMap] | case fork_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case fork_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case fork_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | ext b | case fork_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case fork_comp.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :... | Please generate a tactic in lean4 to solve the state.
STATE:
case fork_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | simp only [types_comp_apply, Equalizer.firstObjEqFamily_hom, Types.pi_lift_π_apply] | case fork_comp.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fork_comp.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Eq... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp [EqualizerSecondObjIso] | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [CompHaus.fst_comp_fromExplicit, op_comp, Functor.map_comp] | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | suffices : (EqualizerFirstObjIso F π).hom ≫ F.map Limits.pullback.fst.op =
Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫
(EqualizerSecondObjIso_aux F π).hom | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp [EqualizerFirstObjIso, Equalizer.Presieve.firstMap, EqualizerSecondObjIso_aux] | case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Func... | case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Func... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | ext b | case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Func... | case this.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | simp only [types_comp_apply, Equalizer.firstObjEqFamily_hom, Types.pi_lift_π_apply] | case this.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equaliz... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | simp only [← Category.assoc] | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [this] | case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fmap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp [EqualizerSecondObjIso] | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [CompHaus.snd_comp_fromExplicit, op_comp, Functor.map_comp] | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | suffices : (EqualizerFirstObjIso F π).hom ≫ F.map Limits.pullback.snd.op =
Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫
(EqualizerSecondObjIso_aux F π).hom | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp [EqualizerFirstObjIso, Equalizer.Presieve.secondMap, EqualizerSecondObjIso_aux] | case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Func... | case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Func... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | ext b | case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Func... | case this.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | simp only [types_comp_apply, Equalizer.firstObjEqFamily_hom, Types.pi_lift_π_apply] | case this.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equaliz... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | simp only [← Category.assoc] | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [this] | case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case smap_comp
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | change ((EqualizerFirstObjIso F π).hom ≫ _) y = _ | case iy_mem
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | case iy_mem
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case iy_mem
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equaliz... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | apply Eq.symm | case iy_mem
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case iy_mem
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equaliz... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | change ((EqualizerFirstObjIso F π).hom ≫ _) y = _ | case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equal... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [fmap_comp, smap_comp] | case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equal... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp | case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equal... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [h] | case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case iy_mem.h
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equal... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [Function.bijective_iff_existsUnique] at hFecs | case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
Fu... | case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
∀ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equaliz... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | specialize hFecs ⟨(EqualizerFirstObjIso F π).hom y, iy_mem⟩ | case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
∀ ... | case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equaliz... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | obtain ⟨x, hx⟩ := hFecs | case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : ... | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equaliz... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | refine' ⟨x, _⟩ | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | dsimp [MapToEqualizer] at * | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | refine' ⟨Subtype.ext_iff.mp hx.1,_⟩ | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | intro z hz | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | apply hx.2 | case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_c... | case uniq_F.intro.a
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork... | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = E... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rwa [Subtype.ext_iff] | case uniq_F.intro.a
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case uniq_F.intro.a
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y =... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | apply_fun (EqualizerFirstObjIso F π).hom | case inr.intro.intro.intro.intro.left
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.s... | case inr.intro.intro.intro.intro.left
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.s... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.left
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presiev... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | exact hx.1 | case inr.intro.intro.intro.intro.left
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.s... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.left
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presiev... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | apply Function.Bijective.injective | case inr.intro.intro.intro.intro.left.inj
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presie... | case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Pre... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.left.inj
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Pre... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | rw [← isIso_iff_bijective] | case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Pre... | case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Pre... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | exact inferInstance | case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Pre... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | intro z hz | case inr.intro.intro.intro.intro.right
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.... | case inr.intro.intro.intro.intro.right
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.right
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presie... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | apply_fun (EqualizerFirstObjIso F π).hom at hz | case inr.intro.intro.intro.intro.right
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.... | case inr.intro.intro.intro.intro.right
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.right
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presie... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.isSheafFor_of_Dagur | [277, 1] | [352, 22] | exact hx.2 z hz | case inr.intro.intro.intro.intro.right
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.right
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presie... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final | [354, 1] | [363, 16] | rw [← extensiveRegular_generates_coherent] | A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
⊢ Presheaf.IsSheaf (coherentTopology CompHaus) F | A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
⊢ Presheaf.IsSheaf
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], E... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
⊢ Presheaf.IsSheaf (coherentTopology CompHaus) F
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final | [354, 1] | [363, 16] | refine' fun E => (Presieve.isSheaf_coverage _ _).2 _ | A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
⊢ Presheaf.IsSheaf
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], E... | A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
⊢ ∀ {X : CompHaus} (R : Presieve X),
R ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
⊢ Presheaf.IsSheaf
(Coverage.toGrothendieck CompHaus
(ExtensiveRegularCoverage' CompHaus
(_ : ∀... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final | [354, 1] | [363, 16] | intro B S hS | A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
⊢ ∀ {X : CompHaus} (R : Presieve X),
R ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (... | A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
⊢ ∀ {X : CompHaus} (R : Presieve X),
R ∈
Coverage.covering
(ExtensiveRegularCoverage'... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final | [354, 1] | [363, 16] | apply isSheafFor_of_Dagur hS | A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst :... | case hFpfp
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final | [354, 1] | [363, 16] | exact ⟨fun J inst => have := hF.1; compPreservesLimitsOfShape _ _⟩ | case hFpfp
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hFpfp
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' Co... |
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