url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | rintro ⟨α, ⟨h, ⟨Y, ⟨π, hπ⟩⟩⟩⟩ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ (∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)) →
S ∈ GrothendieckTopology.sieves (coherentTopology C) X | case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology C) X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ (∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)) →
S ∈ GrothendieckTopology.sieves (coherentTopology C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | change Coverage.saturate _ _ _ | case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology C) X | case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
⊢ Coverage.saturate (coherentCoverage C) X S | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | let T := Sieve.generate (Presieve.ofArrows _ π) | case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
⊢ Coverage.saturate (coherentCoverage C) X S | case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ Coverage.saturate (coherentCoverage C) X S | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
⊢ Coverage.saturate (coherentCoverage C) X S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | apply Coverage.saturate_of_superset (coherentCoverage C) h_le (_) | case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ Coverage.saturate (coherentCoverage C) X S | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ Coverage.saturate (coherentCoverage C) X T | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ Coverage.saturate (coherentCoverage C) X S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | refine Coverage.saturate.of X _ ?_ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ Coverage.saturate (coherentCoverage C) X T | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π ∈ Coverage.covering (coherentCoverage C) X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ Coverage.saturate (coherentCoverage C) X T
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | rw [Sieve.sets_iff_generate (Presieve.ofArrows _ π) S] | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ T ≤ S | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ ofArrows (fun i => Y i) π ≤ S.arrows | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ T ≤ S
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | refine Presieve.le_of_factorsThru_sieve (Presieve.ofArrows (fun i => Y i) π) S ?h | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ ofArrows (fun i => Y i) π ≤ S.arrows | case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ FactorsThru (ofArrows (fun i => Y i) π) S.arrows | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ ofArrows (fun i => Y i) π ≤ S.arrows
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | intro Y g f | case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ FactorsThru (ofArrows (fun i => Y i) π) S.arrows | case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
g : Y ⟶ X
f : ofArrows (fun i => Y✝ i) π g
⊢ ∃ W i e, S.arrows e ∧ i ≫ e = g | Please generate a tactic in lean4 to solve the state.
STATE:
case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
⊢ FactorsThru (ofArrows (fun i => Y i) π) S.arrows
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | use Y, 𝟙 Y | case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
g : Y ⟶ X
f : ofArrows (fun i => Y✝ i) π g
⊢ ∃ W i e, S.arrows e ∧ i ≫ e = g | case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
g : Y ⟶ X
f : ofArrows (fun i => Y✝ i) π g
⊢ ∃ e, S.arrows e ∧ 𝟙 Y ≫ e = g | Please generate a tactic in lean4 to solve the state.
STATE:
case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
g : Y ⟶ X
f : ofArrows (fun i => Y✝ i) π g
⊢ ∃ W i e, S.arrows e ∧ i ≫ e = g
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | rcases f with ⟨i⟩ | case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
g : Y ⟶ X
f : ofArrows (fun i => Y✝ i) π g
⊢ ∃ e, S.arrows e ∧ 𝟙 Y ≫ e = g | case h.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ ∃ e, S.arrows e ∧ 𝟙 (Y✝ i) ≫ e = π i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
g : Y ⟶ X
f : ofArrows (fun i => Y✝ i) π g
⊢ ∃ e, S.arrows e ∧ 𝟙 Y ≫ e = g
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | use (π i) | case h.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ ∃ e, S.arrows e ∧ 𝟙 (Y✝ i) ≫ e = π i | case h.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ S.arrows (π i) ∧ 𝟙 (Y✝ i) ≫ π i = π i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ ∃ e, S.arrows e ∧ 𝟙 (Y✝ i) ≫ e = π i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | constructor | case h.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ S.arrows (π i) ∧ 𝟙 (Y✝ i) ≫ π i = π i | case h.mk.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ S.arrows (π i)
case h.mk.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ 𝟙 (Y✝ i) ≫ π i = π i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ S.arrows (π i) ∧ 𝟙 (Y✝ i) ≫ π i = π i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | exact hπ.2 i | case h.mk.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ S.arrows (π i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ S.arrows (π i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | exact Category.id_comp (π i) | case h.mk.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ 𝟙 (Y✝ i) ≫ π i = π i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
hπ : EffectiveEpiFamily Y✝ π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y✝ i) π)
Y : C
i : α
⊢ 𝟙 (Y✝ i) ≫ π i = π i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | unfold coherentCoverage | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π ∈ Coverage.covering (coherentCoverage C) X | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π ∈
Coverage.covering
{ covering := fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π},
pullback :=
(_ :
∀ ⦃B₁ B₂ : C⦄ (f : B₂ ⟶ B₁) (S : Presieve B₁),
S ∈ (fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π}) B₁ →
∃ T,
T ∈ (fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π}) B₂ ∧
FactorsThruAlong T S f) }
X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π ∈ Coverage.covering (coherentCoverage C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | simp | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π ∈
Coverage.covering
{ covering := fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π},
pullback :=
(_ :
∀ ⦃B₁ B₂ : C⦄ (f : B₂ ⟶ B₁) (S : Presieve B₁),
S ∈ (fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π}) B₁ →
∃ T,
T ∈ (fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π}) B₂ ∧
FactorsThruAlong T S f) }
X | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ∃ α_1 x X_1 π_1, ofArrows (fun i => Y i) π = ofArrows X_1 π_1 ∧ EffectiveEpiFamily X_1 π_1 | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π ∈
Coverage.covering
{ covering := fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π},
pullback :=
(_ :
∀ ⦃B₁ B₂ : C⦄ (f : B₂ ⟶ B₁) (S : Presieve B₁),
S ∈ (fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π}) B₁ →
∃ T,
T ∈ (fun B => {S | ∃ α x X π, S = ofArrows X π ∧ EffectiveEpiFamily X π}) B₂ ∧
FactorsThruAlong T S f) }
X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | use α, inferInstance, Y, π | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ∃ α_1 x X_1 π_1, ofArrows (fun i => Y i) π = ofArrows X_1 π_1 ∧ EffectiveEpiFamily X_1 π_1 | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π = ofArrows Y π ∧ EffectiveEpiFamily Y π | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ∃ α_1 x X_1 π_1, ofArrows (fun i => Y i) π = ofArrows X_1 π_1 ∧ EffectiveEpiFamily X_1 π_1
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | constructor | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π = ofArrows Y π ∧ EffectiveEpiFamily Y π | case left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π = ofArrows Y π
case right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ EffectiveEpiFamily Y π | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π = ofArrows Y π ∧ EffectiveEpiFamily Y π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | rfl | case left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π = ofArrows Y π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ ofArrows (fun i => Y i) π = ofArrows Y π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_of_has_EffectiveEpiFamily | [18, 1] | [42, 17] | exact hπ.1 | case right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ EffectiveEpiFamily Y π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
α : Type
h : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
hπ : EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
T : Sieve X := Sieve.generate (ofArrows (fun i => Y i) π)
h_le : T ≤ S
⊢ EffectiveEpiFamily Y π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
⊢ FamilyOfElements.Compatible (yonedaFamilyOfElements_fromCocone S s) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
⊢ (yoneda.obj s.pt).map g₁.op (yonedaFamilyOfElements_fromCocone S s f₁ hf₁) =
(yoneda.obj s.pt).map g₂.op (yonedaFamilyOfElements_fromCocone S s f₂ hf₂) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
⊢ FamilyOfElements.Compatible (yonedaFamilyOfElements_fromCocone S s)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | have := s.ι.naturality | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
⊢ (yoneda.obj s.pt).map g₁.op (yonedaFamilyOfElements_fromCocone S s f₁ hf₁) =
(yoneda.obj s.pt).map g₂.op (yonedaFamilyOfElements_fromCocone S s f₂ hf₂) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ (yoneda.obj s.pt).map g₁.op (yonedaFamilyOfElements_fromCocone S s f₁ hf₁) =
(yoneda.obj s.pt).map g₂.op (yonedaFamilyOfElements_fromCocone S s f₂ hf₂) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
⊢ (yoneda.obj s.pt).map g₁.op (yonedaFamilyOfElements_fromCocone S s f₁ hf₁) =
(yoneda.obj s.pt).map g₂.op (yonedaFamilyOfElements_fromCocone S s f₂ hf₂)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | simp | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ (yoneda.obj s.pt).map g₁.op (yonedaFamilyOfElements_fromCocone S s f₁ hf₁) =
(yoneda.obj s.pt).map g₂.op (yonedaFamilyOfElements_fromCocone S s f₂ hf₂) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ g₁ ≫ yonedaFamilyOfElements_fromCocone S s f₁ hf₁ = g₂ ≫ yonedaFamilyOfElements_fromCocone S s f₂ hf₂ | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ (yoneda.obj s.pt).map g₁.op (yonedaFamilyOfElements_fromCocone S s f₁ hf₁) =
(yoneda.obj s.pt).map g₂.op (yonedaFamilyOfElements_fromCocone S s f₂ hf₂)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | dsimp [yonedaFamilyOfElements_fromCocone] | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ g₁ ≫ yonedaFamilyOfElements_fromCocone S s f₁ hf₁ = g₂ ≫ yonedaFamilyOfElements_fromCocone S s f₂ hf₂ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ g₁ ≫ yonedaFamilyOfElements_fromCocone S s f₁ hf₁ = g₂ ≫ yonedaFamilyOfElements_fromCocone S s f₂ hf₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact Sieve.downward_closed S hf₂ g₂ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | let F : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk (g₂ ≫ f₂) : Over X) := (Over.homMk (𝟙 Z) ) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | let F₁ : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk f₁ : Over X) := (Over.homMk g₁) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | let F₂ : (Over.mk (g₂ ≫ f₂) : Over X) ⟶ (Over.mk f₂ : Over X) := (Over.homMk g₂) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | have hF := @this ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk (g₂ ≫ f₂), hgf₂⟩ F | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | have hF₁ := @this ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk f₁, hf₁⟩ F₁ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | have hF₂ := @this ⟨Over.mk (g₂ ≫ f₂), hgf₂⟩ ⟨Over.mk f₂, hf₂⟩ F₂ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
hF₂ :
(diagram S.arrows).map F₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } =
s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₂
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | simp at this ⊢ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
hF₂ :
(diagram S.arrows).map F₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } =
s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₂
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
hF₂ :
(diagram S.arrows).map F₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } =
s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₂
this : ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y), f.left ≫ s.ι.app Y = s.ι.app X_1
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
hF₂ :
(diagram S.arrows).map F₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } =
s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₂
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | aesop_cat | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
hF₂ :
(diagram S.arrows).map F₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } =
s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₂
this : ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y), f.left ≫ s.ι.app Y = s.ι.app X_1
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
(diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
(diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
hF₂ :
(diagram S.arrows).map F₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } =
s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } ≫
((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₂
this : ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y), f.left ≫ s.ι.app Y = s.ι.app X_1
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | exact Sieve.downward_closed S hf₁ g₁ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ S.arrows (g₁ ≫ f₁) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
⊢ S.arrows (g₁ ≫ f₁)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.yonedaFamily_fromCocone_compatible | [75, 1] | [93, 12] | exact Sieve.downward_closed S hf₂ g₂ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
⊢ S.arrows (g₂ ≫ f₂) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
this :
∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
(diagram S.arrows).map f ≫ s.ι.app Y =
s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
⊢ S.arrows (g₂ ≫ f₂)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | constructor | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) ↔ Nonempty (IsColimit (cocone S.arrows)) | case mp
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) → Nonempty (IsColimit (cocone S.arrows))
case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ Nonempty (IsColimit (cocone S.arrows)) → ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) ↔ Nonempty (IsColimit (cocone S.arrows))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | intro H | case mp
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) → Nonempty (IsColimit (cocone S.arrows)) | case mp
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ Nonempty (IsColimit (cocone S.arrows)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) → Nonempty (IsColimit (cocone S.arrows))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | refine Nonempty.intro ?mp.val | case mp
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ Nonempty (IsColimit (cocone S.arrows)) | case mp.val
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ IsColimit (cocone S.arrows) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ Nonempty (IsColimit (cocone S.arrows))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | exact {
desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S s)
(yonedaFamily_fromCocone_compatible S s) |>.choose
fac := by
intro s f
replace H := H s.pt (yonedaFamilyOfElements_fromCocone S s)
(yonedaFamily_fromCocone_compatible S s)
have ht := H.choose_spec.1 f.obj.hom f.property
aesop_cat
uniq := by
intro s Fs HFs
replace H := H s.pt (yonedaFamilyOfElements_fromCocone S s)
(yonedaFamily_fromCocone_compatible S s)
apply H.choose_spec.2 Fs
exact fun _ f hf => HFs ⟨Over.mk f, hf⟩
} | case mp.val
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ IsColimit (cocone S.arrows) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.val
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ IsColimit (cocone S.arrows)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | intro s f | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ ∀ (s : Cocone (diagram S.arrows)) (j : FullSubcategory fun f => S.arrows f.hom),
(cocone S.arrows).ι.app j ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t))
s =
s.ι.app j | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ ∀ (s : Cocone (diagram S.arrows)) (j : FullSubcategory fun f => S.arrows f.hom),
(cocone S.arrows).ι.app j ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t))
s =
s.ι.app j
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | replace H := H s.pt (yonedaFamilyOfElements_fromCocone S s)
(yonedaFamily_fromCocone_compatible S s) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | have ht := H.choose_spec.1 f.obj.hom f.property | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
ht :
(yoneda.obj s.pt).map f.obj.hom.op (Exists.choose H) =
yonedaFamilyOfElements_fromCocone S s f.obj.hom (_ : S.arrows f.obj.hom)
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | aesop_cat | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
ht :
(yoneda.obj s.pt).map f.obj.hom.op (Exists.choose H) =
yonedaFamilyOfElements_fromCocone S s f.obj.hom (_ : S.arrows f.obj.hom)
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
ht :
(yoneda.obj s.pt).map f.obj.hom.op (Exists.choose H) =
yonedaFamilyOfElements_fromCocone S s f.obj.hom (_ : S.arrows f.obj.hom)
⊢ (cocone S.arrows).ι.app f ≫
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s =
s.ι.app f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | intro s Fs HFs | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ ∀ (s : Cocone (diagram S.arrows)) (m : (cocone S.arrows).pt ⟶ s.pt),
(∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ m = s.ι.app j) →
m =
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
⊢ Fs = (fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ ∀ (s : Cocone (diagram S.arrows)) (m : (cocone S.arrows).pt ⟶ s.pt),
(∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ m = s.ι.app j) →
m =
(fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | replace H := H s.pt (yonedaFamilyOfElements_fromCocone S s)
(yonedaFamily_fromCocone_compatible S s) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
⊢ Fs = (fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ Fs = (fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
⊢ Fs = (fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | apply H.choose_spec.2 Fs | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ Fs = (fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) Fs | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ Fs = (fun s => Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t)) s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | exact fun _ f hf => HFs ⟨Over.mk f, hf⟩ | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) Fs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) t
⊢ FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S s) Fs
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | intro H W x hx | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ Nonempty (IsColimit (cocone S.arrows)) → ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : Nonempty (IsColimit (cocone S.arrows))
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ Nonempty (IsColimit (cocone S.arrows)) → ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | replace H := Classical.choice H | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : Nonempty (IsColimit (cocone S.arrows))
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
H : Nonempty (IsColimit (cocone S.arrows))
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | let s := Sieve.yonedafamily_toCocone W S x hx | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | use H.desc s | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (IsColimit.desc H s) ∧
∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = IsColimit.desc H s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | constructor | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (IsColimit.desc H s) ∧
∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = IsColimit.desc H s | case mpr.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (IsColimit.desc H s)
case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = IsColimit.desc H s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (IsColimit.desc H s) ∧
∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = IsColimit.desc H s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | exact fun _ f hf => (H.fac s) ⟨Over.mk f, hf⟩ | case mpr.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (IsColimit.desc H s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (IsColimit.desc H s)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | intro g hg | case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = IsColimit.desc H s | case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
⊢ g = IsColimit.desc H s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
⊢ ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = IsColimit.desc H s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | apply H.uniq s g | case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
⊢ g = IsColimit.desc H s | case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
⊢ ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ g = s.ι.app j | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
⊢ g = IsColimit.desc H s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | rintro ⟨⟨f, _, hom⟩, hf⟩ | case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
⊢ ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ g = s.ι.app j | case mpr.right.mk.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
f : C
right✝ : Discrete PUnit
hom : (𝟭 C).obj f ⟶ (Functor.fromPUnit X).obj right✝
hf : S.arrows { left := f, right := right✝, hom := hom }.hom
⊢ (cocone S.arrows).ι.app { obj := { left := f, right := right✝, hom := hom }, property := hf } ≫ g =
s.ι.app { obj := { left := f, right := right✝, hom := hom }, property := hf } | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
⊢ ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ g = s.ι.app j
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | Sieve.Yoneda_sheaf_iff_colimit | [97, 1] | [128, 22] | apply hg hom hf | case mpr.right.mk.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
f : C
right✝ : Discrete PUnit
hom : (𝟭 C).obj f ⟶ (Functor.fromPUnit X).obj right✝
hf : S.arrows { left := f, right := right✝, hom := hom }.hom
⊢ (cocone S.arrows).ι.app { obj := { left := f, right := right✝, hom := hom }, property := hf } ≫ g =
s.ι.app { obj := { left := f, right := right✝, hom := hom }, property := hf } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right.mk.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
W : C
x : FamilyOfElements (yoneda.obj W) S.arrows
hx : FamilyOfElements.Compatible x
H : IsColimit (cocone S.arrows)
s : Cocone (diagram S.arrows) := yonedafamily_toCocone W S.arrows x hx
g : (yoneda.obj W).obj X.op
hg : FamilyOfElements.IsAmalgamation x g
f : C
right✝ : Discrete PUnit
hom : (𝟭 C).obj f ⟶ (Functor.fromPUnit X).obj right✝
hf : S.arrows { left := f, right := right✝, hom := hom }.hom
⊢ (cocone S.arrows).ι.app { obj := { left := f, right := right✝, hom := hom }, property := hf } ≫ g =
s.ι.app { obj := { left := f, right := right✝, hom := hom }, property := hf }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | rw [isSheaf_coherent] | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X W : C
⊢ IsSheaf (coherentTopology C) (yoneda.obj W) | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X W : C
⊢ ∀ (B : C) (α : Type) [inst : Fintype α] (X : α → C) (π : (a : α) → X a ⟶ B),
EffectiveEpiFamily X π → IsSheafFor (yoneda.obj W) (ofArrows X π) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X W : C
⊢ IsSheaf (coherentTopology C) (yoneda.obj W)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | intro X α _ Y π H | C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X W : C
⊢ ∀ (B : C) (α : Type) [inst : Fintype α] (X : α → C) (π : (a : α) → X a ⟶ B),
EffectiveEpiFamily X π → IsSheafFor (yoneda.obj W) (ofArrows X π) | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X W : C
⊢ ∀ (B : C) (α : Type) [inst : Fintype α] (X : α → C) (π : (a : α) → X a ⟶ B),
EffectiveEpiFamily X π → IsSheafFor (yoneda.obj W) (ofArrows X π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | have h_colim:= isColimitOfEffectiveEpiFamilyStruct Y π H.effectiveEpiFamily.some | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π) | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generateFamily Y π).arrows)
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | rw [←Sieve.generateFamily_eq] at h_colim | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generateFamily Y π).arrows)
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π) | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generateFamily Y π).arrows)
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | intro x hx | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π) | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
⊢ IsSheafFor (yoneda.obj W) (ofArrows Y π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | let x_ext := FamilyOfElements.sieveExtend x | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | have hx_ext := FamilyOfElements.Compatible.sieveExtend hx | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | let S := Sieve.generate (ofArrows Y π) | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | have := (Sieve.Yoneda_sheaf_iff_colimit S).mpr ⟨h_colim⟩ W x_ext hx_ext | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
this : ∃! t, FamilyOfElements.IsAmalgamation x_ext t
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | rcases this with ⟨t, t_amalg, t_uniq⟩ | C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
this : ∃! t, FamilyOfElements.IsAmalgamation x_ext t
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | case intro.intro
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
this : ∃! t, FamilyOfElements.IsAmalgamation x_ext t
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | use t | case intro.intro
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t | case intro.intro
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) t ∧
∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ ∃! t, FamilyOfElements.IsAmalgamation x t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | constructor | case intro.intro
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) t ∧
∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = t | case intro.intro.left
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) t
case intro.intro.right
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) t ∧
∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | convert Presieve.isAmalgamation_restrict (Sieve.le_generate (ofArrows Y π)) _ _ t_amalg | case intro.intro.left
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) t | case h.e.h.e'_6.h.h.h
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ x = FamilyOfElements.restrict (_ : ofArrows Y π ≤ (Sieve.generate (ofArrows Y π)).arrows) x_ext | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.left
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ (fun t => FamilyOfElements.IsAmalgamation x t) t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | refine Eq.symm (restrict_extend hx) | case h.e.h.e'_6.h.h.h
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ x = FamilyOfElements.restrict (_ : ofArrows Y π ≤ (Sieve.generate (ofArrows Y π)).arrows) x_ext | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e.h.e'_6.h.h.h
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ x = FamilyOfElements.restrict (_ : ofArrows Y π ≤ (Sieve.generate (ofArrows Y π)).arrows) x_ext
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.isSheaf_Yoneda | [131, 1] | [149, 68] | exact fun y hy => t_uniq y <| isAmalgamation_sieveExtend x y hy | case intro.intro.right
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.right
C : Type u
inst✝² : Category C
inst✝¹ : Precoherent C
X✝ W X : C
α : Type
inst✝ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : IsColimit (cocone (Sieve.generate (ofArrows Y π)).arrows)
x : FamilyOfElements (yoneda.obj W) (ofArrows Y π)
hx : FamilyOfElements.Compatible x
x_ext : FamilyOfElements (yoneda.obj W) (Sieve.generate (ofArrows Y π)).arrows := FamilyOfElements.sieveExtend x
hx_ext : FamilyOfElements.Compatible (FamilyOfElements.sieveExtend x)
S : Sieve X := Sieve.generate (ofArrows Y π)
t : (yoneda.obj W).obj X.op
t_amalg : FamilyOfElements.IsAmalgamation x_ext t
t_uniq : ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x_ext t) y → y = t
⊢ ∀ (y : (yoneda.obj W).obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | rw [← Sieve.effectiveEpimorphic_family] | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
⊢ EffectiveEpiFamily (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
⊢ EffectiveEpimorphic (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
⊢ EffectiveEpiFamily (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | suffices h₂ : (Sieve.generate (Presieve.ofArrows (fun (⟨a, b⟩ : Σ _, β _) => Y_n a b)
(fun ⟨a,b⟩ => π_n a b ≫ π a))) ∈ GrothendieckTopology.sieves (coherentTopology C) X by
change Nonempty _
rw [← Sieve.Yoneda_sheaf_iff_colimit]
intro W
apply coherentTopology.isSheaf_Yoneda
exact h₂ | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
⊢ EffectiveEpimorphic (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst) | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
⊢ EffectiveEpimorphic (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | let h' := h | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpiFamily Y π := h
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | rw [← Sieve.effectiveEpimorphic_family] at h' | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpiFamily Y π := h
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpiFamily Y π := h
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | let H' := H | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a) := H
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | conv at H' =>
intro a
rw [← Sieve.effectiveEpimorphic_family] | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a) := H
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a) := H
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | apply Coverage.saturate.transitive X (Sieve.generate (Presieve.ofArrows Y π)) | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ Coverage.saturate (coherentCoverage C) X (Sieve.generate (ofArrows Y π))
case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ X⦄,
(Sieve.generate (ofArrows Y π)).arrows f →
Coverage.saturate (coherentCoverage C) Y_1
(Sieve.pullback f
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | change Nonempty _ | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
⊢ EffectiveEpimorphic (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst) | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
⊢ Nonempty
(IsColimit
(cocone (Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows)) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
⊢ EffectiveEpimorphic (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | rw [← Sieve.Yoneda_sheaf_iff_colimit] | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
⊢ Nonempty
(IsColimit
(cocone (Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows)) | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
⊢ ∀ (W : C),
IsSheafFor (yoneda.obj W)
(Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
⊢ Nonempty
(IsColimit
(cocone (Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | intro W | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
⊢ ∀ (W : C),
IsSheafFor (yoneda.obj W)
(Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
W : C
⊢ IsSheafFor (yoneda.obj W)
(Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
⊢ ∀ (W : C),
IsSheafFor (yoneda.obj W)
(Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | apply coherentTopology.isSheaf_Yoneda | C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
W : C
⊢ IsSheafFor (yoneda.obj W)
(Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
W : C
⊢ Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst) ∈
GrothendieckTopology.sieves (coherentTopology C) X | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
W : C
⊢ IsSheafFor (yoneda.obj W)
(Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst)).arrows
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | exact h₂ | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
W : C
⊢ Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst) ∈
GrothendieckTopology.sieves (coherentTopology C) X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h₂ :
Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a) ∈
GrothendieckTopology.sieves (coherentTopology C) X
W : C
⊢ Sieve.generate (ofArrows (fun c => Y_n c.fst c.snd) fun c => π_n c.fst c.snd ≫ π c.fst) ∈
GrothendieckTopology.sieves (coherentTopology C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | apply Coverage.saturate.of | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ Coverage.saturate (coherentCoverage C) X (Sieve.generate (ofArrows Y π)) | case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ofArrows Y π ∈ Coverage.covering (coherentCoverage C) X | Please generate a tactic in lean4 to solve the state.
STATE:
case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ Coverage.saturate (coherentCoverage C) X (Sieve.generate (ofArrows Y π))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | use α, inferInstance, Y, π | case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ofArrows Y π ∈ Coverage.covering (coherentCoverage C) X | case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ofArrows Y π = ofArrows Y π ∧ EffectiveEpiFamily Y π | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ofArrows Y π ∈ Coverage.covering (coherentCoverage C) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | simp only [true_and] | case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ofArrows Y π = ofArrows Y π ∧ EffectiveEpiFamily Y π | case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ EffectiveEpiFamily Y π | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ofArrows Y π = ofArrows Y π ∧ EffectiveEpiFamily Y π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | exact Iff.mp (Sieve.effectiveEpimorphic_family Y π) h' | case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ EffectiveEpiFamily Y π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hS
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ EffectiveEpiFamily Y π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | intro V f ⟨Y₁, h, g, ⟨hY, hf⟩⟩ | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ X⦄,
(Sieve.generate (ofArrows Y π)).arrows f →
Coverage.saturate (coherentCoverage C) Y_1
(Sieve.pullback f
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))) | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Coverage.saturate (coherentCoverage C) V
(Sieve.pullback f
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
⊢ ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ X⦄,
(Sieve.generate (ofArrows Y π)).arrows f →
Coverage.saturate (coherentCoverage C) Y_1
(Sieve.pullback f
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | rw [← hf, Sieve.pullback_comp] | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Coverage.saturate (coherentCoverage C) V
(Sieve.pullback f
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))) | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Coverage.saturate (coherentCoverage C) V
(Sieve.pullback h
(Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)))) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Coverage.saturate (coherentCoverage C) V
(Sieve.pullback f
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | apply (coherentTopology C).pullback_stable' | case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Coverage.saturate (coherentCoverage C) V
(Sieve.pullback h
(Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)))) | case a.a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)) ∈
GrothendieckTopology.sieves (coherentTopology C) Y₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Coverage.saturate (coherentCoverage C) V
(Sieve.pullback h
(Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | apply coherentTopology.Sieve_of_has_EffectiveEpiFamily | case a.a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)) ∈
GrothendieckTopology.sieves (coherentTopology C) Y₁ | case a.a.a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ ∃ α_1 x Y_1 π_1,
EffectiveEpiFamily Y_1 π_1 ∧
∀ (a : α_1),
(Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_1 a) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)) ∈
GrothendieckTopology.sieves (coherentTopology C) Y₁
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | rcases hY with ⟨i⟩ | case a.a.a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ ∃ α_1 x Y_1 π_1,
EffectiveEpiFamily Y_1 π_1 ∧
∀ (a : α_1),
(Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_1 a) | case a.a.a.mk
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ ∃ α_1 x Y π_1,
EffectiveEpiFamily Y π_1 ∧
∀ (a : α_1),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_1 a) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y : α → C
π : (a : α) → Y a ⟶ X
h✝ : EffectiveEpiFamily Y π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y₁ : C
h : V ⟶ Y₁
g : Y₁ ⟶ X
hY : ofArrows Y π g
hf : h ≫ g = f
⊢ ∃ α_1 x Y_1 π_1,
EffectiveEpiFamily Y_1 π_1 ∧
∀ (a : α_1),
(Sieve.pullback g
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_1 a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | use β i, inferInstance, Y_n i, π_n i | case a.a.a.mk
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ ∃ α_1 x Y π_1,
EffectiveEpiFamily Y π_1 ∧
∀ (a : α_1),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_1 a) | case a.a.a.mk
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ EffectiveEpiFamily (Y_n i) (π_n i) ∧
∀ (a : β i),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i a) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.mk
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ ∃ α_1 x Y π_1,
EffectiveEpiFamily Y π_1 ∧
∀ (a : α_1),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_1 a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | constructor | case a.a.a.mk
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ EffectiveEpiFamily (Y_n i) (π_n i) ∧
∀ (a : β i),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i a) | case a.a.a.mk.left
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ EffectiveEpiFamily (Y_n i) (π_n i)
case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ ∀ (a : β i),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i a) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.mk
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ EffectiveEpiFamily (Y_n i) (π_n i) ∧
∀ (a : β i),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | exact H i | case a.a.a.mk.left
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ EffectiveEpiFamily (Y_n i) (π_n i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.mk.left
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ EffectiveEpiFamily (Y_n i) (π_n i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | intro b | case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ ∀ (a : β i),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i a) | case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
b : β i
⊢ (Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i b) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
⊢ ∀ (a : β i),
(Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | use Y_n i b, (𝟙 _), π_n i b ≫ π i | case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
b : β i
⊢ (Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i b) | case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
b : β i
⊢ ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
(fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
(π_n i b ≫ π i) ∧
𝟙 (Y_n i b) ≫ π_n i b ≫ π i = π_n i b ≫ π i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
b : β i
⊢ (Sieve.pullback (π i)
(Sieve.generate
(ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a))).arrows
(π_n i b)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | constructor | case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
b : β i
⊢ ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
(fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
(π_n i b ≫ π i) ∧
𝟙 (Y_n i b) ≫ π_n i b ≫ π i = π_n i b ≫ π i | case a.a.a.mk.right.left
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
b : β i
⊢ ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
(fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
(π_n i b ≫ π i)
case a.a.a.mk.right.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
b : β i
⊢ 𝟙 (Y_n i b) ≫ π_n i b ≫ π i = π_n i b ≫ π i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.a.mk.right
C : Type u
inst✝³ : Category C
inst✝² : Precoherent C
X : C
α : Type
inst✝¹ : Fintype α
Y✝ : α → C
π : (a : α) → Y✝ a ⟶ X
h✝ : EffectiveEpiFamily Y✝ π
β : α → Type
inst✝ : (a : α) → Fintype (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (b : β a) → Y_n a b ⟶ Y✝ a
H : ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
h' : EffectiveEpimorphic (ofArrows Y✝ π)
H' : ∀ (a : α), EffectiveEpimorphic (ofArrows (Y_n a) (π_n a))
V : C
f : V ⟶ X
Y : C
i : α
h : V ⟶ Y✝ i
hf : h ≫ π i = f
b : β i
⊢ ofArrows
(fun x =>
match x with
| { fst := a, snd := b } => Y_n a b)
(fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
(π_n i b ≫ π i) ∧
𝟙 (Y_n i b) ≫ π_n i b ≫ π i = π_n i b ≫ π i
TACTIC:
|
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