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 | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | exact ⟨(⟨i, b⟩ : Σ (i : α), β 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) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | EffectiveEpiFamily_transitive | [161, 1] | [201, 47] | exact Category.id_comp (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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | constructor | 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 mp
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 mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology C) X → ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
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_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | exact coherentTopology.Sieve_of_has_EffectiveEpiFamily X S | case mp
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 | no goals | 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
⊢ (∃ 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_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | intro h | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology C) X → ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
h : S ∈ GrothendieckTopology.sieves (coherentTopology C) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology C) X → ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | induction' h with Y T hS Y Y R S _ _ a b | case mpr
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
h : S ∈ GrothendieckTopology.sieves (coherentTopology C) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | case mpr.of
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
hS : T ∈ Coverage.covering (coherentCoverage C) Y
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), (Sieve.generate T).arrows (π a)
case mpr.top
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), ⊤.arrows (π a)
case mpr.transitive
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
a : ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), R.arrows (π a)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | 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 : S ∈ GrothendieckTopology.sieves (coherentTopology C) X
⊢ ∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rcases hS with ⟨a, h, Y', π, h'⟩ | case mpr.of
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
hS : T ∈ Coverage.covering (coherentCoverage C) Y
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), (Sieve.generate T).arrows (π a) | case mpr.of.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), (Sieve.generate T).arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
hS : T ∈ Coverage.covering (coherentCoverage C) Y
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), (Sieve.generate T).arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | use a, h, Y', π | case mpr.of.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), (Sieve.generate T).arrows (π a) | case mpr.of.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ EffectiveEpiFamily Y' π ∧ ∀ (a : a), (Sieve.generate T).arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), (Sieve.generate T).arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | constructor | case mpr.of.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ EffectiveEpiFamily Y' π ∧ ∀ (a : a), (Sieve.generate T).arrows (π a) | case mpr.of.intro.intro.intro.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ EffectiveEpiFamily Y' π
case mpr.of.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ ∀ (a : a), (Sieve.generate T).arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ EffectiveEpiFamily Y' π ∧ ∀ (a : a), (Sieve.generate T).arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | tauto | case mpr.of.intro.intro.intro.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ EffectiveEpiFamily Y' π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ EffectiveEpiFamily Y' π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | intro a' | case mpr.of.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ ∀ (a : a), (Sieve.generate T).arrows (π a) | case mpr.of.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
a' : a
⊢ (Sieve.generate T).arrows (π a') | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
⊢ ∀ (a : a), (Sieve.generate T).arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | cases' h' with h_left h_right | case mpr.of.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
a' : a
⊢ (Sieve.generate T).arrows (π a') | case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ (Sieve.generate T).arrows (π a') | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
h' : T = ofArrows Y' π ∧ EffectiveEpiFamily Y' π
a' : a
⊢ (Sieve.generate T).arrows (π a')
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | simp only [Sieve.generate_apply] | case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ (Sieve.generate T).arrows (π a') | case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ ∃ Y_1 h g, T g ∧ h ≫ g = π a' | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ (Sieve.generate T).arrows (π a')
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | use Y' a', 𝟙 Y' a', π a' | case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ ∃ Y_1 h g, T g ∧ h ≫ g = π a' | case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ T (π a') ∧ 𝟙 Y' a' ≫ π a' = π a' | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ ∃ Y_1 h g, T g ∧ h ≫ g = π a'
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | constructor | case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ T (π a') ∧ 𝟙 Y' a' ≫ π a' = π a' | case mpr.of.intro.intro.intro.intro.right.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ T (π a')
case mpr.of.intro.intro.intro.intro.right.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ 𝟙 Y' a' ≫ π a' = π a' | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.right.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ T (π a') ∧ 𝟙 Y' a' ≫ π a' = π a'
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rw [h_left] | case mpr.of.intro.intro.intro.intro.right.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ T (π a') | case mpr.of.intro.intro.intro.intro.right.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ ofArrows Y' π (π a') | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.right.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ T (π a')
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | exact Presieve.ofArrows.mk a' | case mpr.of.intro.intro.intro.intro.right.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ ofArrows Y' π (π a') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.right.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ ofArrows Y' π (π a')
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | apply Category.id_comp | case mpr.of.intro.intro.intro.intro.right.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ 𝟙 Y' a' ≫ π a' = π a' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.of.intro.intro.intro.intro.right.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
T : Presieve Y
a : Type
h : Fintype a
Y' : a → C
π : (a : a) → Y' a ⟶ Y
a' : a
h_left : T = ofArrows Y' π
h_right : EffectiveEpiFamily Y' π
⊢ 𝟙 Y' a' ≫ π a' = π a'
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | use Unit, Unit.fintype, fun _ => Y, fun _ => (𝟙 Y) | case mpr.top
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), ⊤.arrows (π a) | case mpr.top
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
⊢ (EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y) ∧ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.top
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), ⊤.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | cases' S with arrows downward_closed | case mpr.top
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
⊢ (EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y) ∧ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a) | case mpr.top.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ (EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y) ∧ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.top
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S : Sieve X
Y : C
⊢ (EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y) ∧ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | constructor | case mpr.top.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ (EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y) ∧ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a) | case mpr.top.mk.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y
case mpr.top.mk.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.top.mk
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ (EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y) ∧ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | exact inferInstance | case mpr.top.mk.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.top.mk.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ EffectiveEpiFamily (fun x => Y) fun x => 𝟙 Y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | simp only [Sieve.top_apply, forall_const] | case mpr.top.mk.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.top.mk.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X Y : C
arrows : Presieve X
downward_closed : ∀ {Y Z : C} {f : Y ⟶ X}, arrows f → ∀ (g : Z ⟶ Y), arrows (g ≫ f)
⊢ ∀ (a : Unit), ⊤.arrows ((fun x => 𝟙 Y) a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rcases a with ⟨α, w, Y₁, π, ⟨h₁,h₂⟩⟩ | case mpr.transitive
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
a : ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), R.arrows (π a)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
a : ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), R.arrows (π a)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | have H : ∀ a : α, ∃ (β : Type) (_ : Fintype β) (Y_n : β → C)
(π_n: (b : β) → (Y_n b)⟶ Y₁ a),
EffectiveEpiFamily Y_n π_n ∧ (∀ b : β, (S.pullback (π a)).arrows (π_n b)) :=
fun a => b (h₂ a) | case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
H : ∀ (a : α), ∃ β x Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β), (Sieve.pullback (π a) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rw [Classical.skolem] at H | case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
H : ∀ (a : α), ∃ β x Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β), (Sieve.pullback (π a) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
H : ∃ f, ∀ (x : α), ∃ x_1 Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : f x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
H : ∀ (a : α), ∃ β x Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β), (Sieve.pullback (π a) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rcases H with ⟨β, H⟩ | case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
H : ∃ f, ∀ (x : α), ∃ x_1 Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : f x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
H : ∀ (x : α), ∃ x_1 Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
H : ∃ f, ∀ (x : α), ∃ x_1 Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : f x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rw [Classical.skolem] at H | case mpr.transitive.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
H : ∀ (x : α), ∃ x_1 Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
H : ∃ f, ∀ (x : α), ∃ Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
H : ∀ (x : α), ∃ x_1 Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rcases H with ⟨_, H⟩ | case mpr.transitive.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
H : ∃ f, ∀ (x : α), ∃ Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
H : ∀ (x : α), ∃ Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
H : ∃ f, ∀ (x : α), ∃ Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rw [Classical.skolem] at H | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
H : ∀ (x : α), ∃ Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
H : ∃ f, ∀ (x : α), ∃ π_n, EffectiveEpiFamily (f x) π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
H : ∀ (x : α), ∃ Y_n π_n, EffectiveEpiFamily Y_n π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rcases H with ⟨Y_n, H⟩ | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
H : ∃ f, ∀ (x : α), ∃ π_n, EffectiveEpiFamily (f x) π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
H : ∀ (x : α), ∃ π_n, EffectiveEpiFamily (Y_n x) π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
H : ∃ f, ∀ (x : α), ∃ π_n, EffectiveEpiFamily (f x) π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rw [Classical.skolem] at H | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
H : ∀ (x : α), ∃ π_n, EffectiveEpiFamily (Y_n x) π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
H : ∃ f, ∀ (x : α), EffectiveEpiFamily (Y_n x) (f x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (f x b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
H : ∀ (x : α), ∃ π_n, EffectiveEpiFamily (Y_n x) π_n ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | rcases H with ⟨π_n, H⟩ | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
H : ∃ f, ∀ (x : α), EffectiveEpiFamily (Y_n x) (f x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (f x b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
H : ∃ f, ∀ (x : α), EffectiveEpiFamily (Y_n x) (f x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (f x b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | use Σ x, β x, inferInstance, fun ⟨a,b⟩ => Y_n a b, fun ⟨a, b⟩ => (π_n a b) ≫ (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a) | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ (EffectiveEpiFamily
(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) ∧
∀ (a : (x : α) × β x),
S.arrows
((fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∃ α x Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | constructor | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ (EffectiveEpiFamily
(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) ∧
∀ (a : (x : α) × β x),
S.arrows
((fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
a) | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ EffectiveEpiFamily
(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 mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∀ (a : (x : α) × β x),
S.arrows
((fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ (EffectiveEpiFamily
(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) ∧
∀ (a : (x : α) × β x),
S.arrows
((fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | apply EffectiveEpiFamily_transitive | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ EffectiveEpiFamily
(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 mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left.h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ EffectiveEpiFamily (fun a => Y₁ a) π
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left.H
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ EffectiveEpiFamily
(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 | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | exact h₁ | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left.h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ EffectiveEpiFamily (fun a => Y₁ a) π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left.h
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ EffectiveEpiFamily (fun a => Y₁ a) π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | exact fun a => (H a).1 | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left.H
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.left.H
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∀ (a : α), EffectiveEpiFamily (Y_n a) (π_n a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Topologies.lean | coherentTopology.Sieve_iff_hasEffectiveEpiFamily | [205, 1] | [248, 41] | exact fun c => (H c.fst).2 c.snd | case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∀ (a : (x : α) × β x),
S.arrows
((fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.transitive.intro.intro.intro.intro.intro.intro.intro.intro.intro.right
C : Type u
inst✝¹ : Category C
inst✝ : Precoherent C
X : C
S✝ : Sieve X
Y : C
R S : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y R
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback f S)
b :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
R.arrows f → ∃ α x Y_2 π, EffectiveEpiFamily Y_2 π ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Fintype α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ Y
h₁ : EffectiveEpiFamily Y₁ π
h₂ : ∀ (a : α), R.arrows (π a)
β : α → Type
w✝ : (x : α) → Fintype (β x)
Y_n : (x : α) → β x → C
π_n : (x : α) → (b : β x) → Y_n x b ⟶ Y₁ x
H : ∀ (x : α), EffectiveEpiFamily (Y_n x) (π_n x) ∧ ∀ (b : β x), (Sieve.pullback (π x) S).arrows (π_n x b)
⊢ ∀ (a : (x : α) × β x),
S.arrows
((fun x =>
match x with
| { fst := a, snd := b } => π_n a b ≫ π a)
a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Coherent.lean | CategoryTheory.EffectiveEpiFamily.toCompHaus | [24, 1] | [28, 70] | refine' ((CompHaus.effectiveEpiFamily_tfae _ _).out 0 2).2 (fun b => _) | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamily X π
⊢ EffectiveEpiFamily (fun a => ExtrDisc.toCompHaus.obj (X a)) fun a => ExtrDisc.toCompHaus.map (π a) | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamily X π
b : ↑(ExtrDisc.toCompHaus.obj B).toTop
⊢ ∃ a x, ↑(ExtrDisc.toCompHaus.map (π a)) x = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamily X π
⊢ EffectiveEpiFamily (fun a => ExtrDisc.toCompHaus.obj (X a)) fun a => ExtrDisc.toCompHaus.map (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Coherent.lean | CategoryTheory.EffectiveEpiFamily.toCompHaus | [24, 1] | [28, 70] | exact (((effectiveEpiFamily_tfae _ _).out 0 2).1 H : ∀ _, ∃ _, _) _ | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamily X π
b : ↑(ExtrDisc.toCompHaus.obj B).toTop
⊢ ∃ a x, ↑(ExtrDisc.toCompHaus.map (π a)) x = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamily X π
b : ↑(ExtrDisc.toCompHaus.obj B).toTop
⊢ ∃ a x, ↑(ExtrDisc.toCompHaus.map (π a)) x = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | constructor | α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Epi f ↔ Function.Surjective ↑f | case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Epi f → Function.Surjective ↑f
case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Function.Surjective ↑f → Epi f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Epi f ↔ Function.Surjective ↑f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | dsimp [Function.Surjective] | case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Epi f → Function.Surjective ↑f | case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Epi f → ∀ (b : CoeSort.coe Y), ∃ a, ↑f a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Epi f → Function.Surjective ↑f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | contrapose! | case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Epi f → ∀ (b : CoeSort.coe Y), ∃ a, ↑f a = b | case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ (∃ b, ∀ (a : CoeSort.coe X), ↑f a ≠ b) → ¬Epi f | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Epi f → ∀ (b : CoeSort.coe Y), ∃ a, ↑f a = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | rintro ⟨y,hy⟩ h | case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ (∃ b, ∀ (a : CoeSort.coe X), ↑f a ≠ b) → ¬Epi f | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ (∃ b, ∀ (a : CoeSort.coe X), ↑f a ≠ b) → ¬Epi f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | let C := Set.range f | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
⊢ False | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | have hC : IsClosed C := (isCompact_range f.continuous).isClosed | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
⊢ False | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | let U := Cᶜ | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
⊢ False | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | have hyU : y ∈ U := by
refine' Set.mem_compl _
rintro ⟨y', hy'⟩
exact hy y' hy' | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
⊢ False | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | have hUy : U ∈ nhds y := hC.compl_mem_nhds hyU | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
⊢ False | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | haveI : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y)) :=
show TotallyDisconnectedSpace Y from inferInstance | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
⊢ False | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | obtain ⟨V, hV, hyV, hVU⟩ := isTopologicalBasis_clopen.mem_nhds_iff.mp hUy | case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
⊢ False | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | classical
let g : Y ⟶ ExtrDisc.two :=
⟨(LocallyConstant.ofClopen hV).map ULift.up, LocallyConstant.continuous _⟩
let h : Y ⟶ ExtrDisc.two := ⟨fun _ => ⟨1⟩, continuous_const⟩
have H : h = g := by
rw [← cancel_epi f]
apply ContinuousMap.ext ; intro x
apply ULift.ext
change 1 = _
dsimp [LocallyConstant.ofClopen]
rw [comp_apply, @ContinuousMap.coe_mk _ _ (ExtrDisc.instTopologicalSpace Y),
Function.comp_apply, if_neg]
refine' mt (fun α => hVU α) _
simp only [Set.mem_compl_iff, Set.mem_range, not_exists, not_forall, not_not]
exact ⟨x, rfl⟩
apply_fun fun e => (e y).down at H
dsimp [LocallyConstant.ofClopen] at H
change 1 = ite _ _ _ at H
rw [if_pos hyV] at H
exact top_ne_bot H | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | refine' Set.mem_compl _ | α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
⊢ y ∈ U | α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
⊢ ¬y ∈ C | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
⊢ y ∈ U
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | rintro ⟨y', hy'⟩ | α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
⊢ ¬y ∈ C | case intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
y' : CoeSort.coe X
hy' : ↑f y' = y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
⊢ ¬y ∈ C
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | exact hy y' hy' | case intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
y' : CoeSort.coe X
hy' : ↑f y' = y
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
y' : CoeSort.coe X
hy' : ↑f y' = y
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | let g : Y ⟶ ExtrDisc.two :=
⟨(LocallyConstant.ofClopen hV).map ULift.up, LocallyConstant.continuous _⟩ | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
⊢ False | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | let h : Y ⟶ ExtrDisc.two := ⟨fun _ => ⟨1⟩, continuous_const⟩ | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
⊢ False | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | have H : h = g := by
rw [← cancel_epi f]
apply ContinuousMap.ext ; intro x
apply ULift.ext
change 1 = _
dsimp [LocallyConstant.ofClopen]
rw [comp_apply, @ContinuousMap.coe_mk _ _ (ExtrDisc.instTopologicalSpace Y),
Function.comp_apply, if_neg]
refine' mt (fun α => hVU α) _
simp only [Set.mem_compl_iff, Set.mem_range, not_exists, not_forall, not_not]
exact ⟨x, rfl⟩ | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ False | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : h = g
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | apply_fun fun e => (e y).down at H | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : h = g
⊢ False | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : (↑h y).down = (↑g y).down
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : h = g
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | dsimp [LocallyConstant.ofClopen] at H | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : (↑h y).down = (↑g y).down
⊢ False | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H :
(↑(ContinuousMap.mk fun x => { down := 1 }) y).down =
(↑(ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) y).down
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : (↑h y).down = (↑g y).down
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | change 1 = ite _ _ _ at H | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H :
(↑(ContinuousMap.mk fun x => { down := 1 }) y).down =
(↑(ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) y).down
⊢ False | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : 1 = if y ∈ V then 0 else 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H :
(↑(ContinuousMap.mk fun x => { down := 1 }) y).down =
(↑(ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) y).down
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | rw [if_pos hyV] at H | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : 1 = if y ∈ V then 0 else 1
⊢ False | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : 1 = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : 1 = if y ∈ V then 0 else 1
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | exact top_ne_bot H | case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : 1 = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
H : 1 = 0
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | rw [← cancel_epi f] | α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ h = g | α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ f ≫ h = f ≫ g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ h = g
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | apply ContinuousMap.ext | α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ f ≫ h = f ≫ g | case h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ ∀ (a : ↑((fun x => x.compHaus) X).toTop), ↑(f ≫ h) a = ↑(f ≫ g) a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ f ≫ h = f ≫ g
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | intro x | case h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ ∀ (a : ↑((fun x => x.compHaus) X).toTop), ↑(f ≫ h) a = ↑(f ≫ g) a | case h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ↑(f ≫ h) x = ↑(f ≫ g) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
⊢ ∀ (a : ↑((fun x => x.compHaus) X).toTop), ↑(f ≫ h) a = ↑(f ≫ g) a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | apply ULift.ext | case h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ↑(f ≫ h) x = ↑(f ≫ g) x | case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ (↑(f ≫ h) x).down = (↑(f ≫ g) x).down | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ↑(f ≫ h) x = ↑(f ≫ g) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | change 1 = _ | case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ (↑(f ≫ h) x).down = (↑(f ≫ g) x).down | case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ 1 = (↑(f ≫ g) x).down | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ (↑(f ≫ h) x).down = (↑(f ≫ g) x).down
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | dsimp [LocallyConstant.ofClopen] | case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ 1 = (↑(f ≫ g) x).down | case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ 1 = (↑(f ≫ ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) x).down | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ 1 = (↑(f ≫ g) x).down
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | rw [comp_apply, @ContinuousMap.coe_mk _ _ (ExtrDisc.instTopologicalSpace Y),
Function.comp_apply, if_neg] | case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ 1 = (↑(f ≫ ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) x).down | case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ¬↑f x ∈ V | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ 1 = (↑(f ≫ ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) x).down
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | refine' mt (fun α => hVU α) _ | case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ¬↑f x ∈ V | case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ¬↑f x ∈ U | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ¬↑f x ∈ V
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | simp only [Set.mem_compl_iff, Set.mem_range, not_exists, not_forall, not_not] | case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ¬↑f x ∈ U | case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ∃ y, ↑f y = ↑f x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ¬↑f x ∈ U
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | exact ⟨x, rfl⟩ | case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ∃ y, ↑f y = ↑f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.hnc
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
y : CoeSort.coe Y
hy : ∀ (a : CoeSort.coe X), ↑f a ≠ y
h✝ : Epi f
C : Set (CoeSort.coe Y) := Set.range ↑f
hC : IsClosed C
U : Set (CoeSort.coe Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ nhds y
this : TotallyDisconnectedSpace ((forget CompHaus).obj (toCompHaus.obj Y))
V : Set (CoeSort.coe Y)
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
g : Y ⟶ ExtrDisc.two := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))
h : Y ⟶ ExtrDisc.two := ContinuousMap.mk fun x => { down := 1 }
x : ↑((fun x => x.compHaus) X).toTop
⊢ ∃ y, ↑f y = ↑f x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | intro (h : Function.Surjective (toCompHaus.map f)) | case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Function.Surjective ↑f → Epi f | case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Function.Surjective ↑(toCompHaus.map f)
⊢ Epi f | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
⊢ Function.Surjective ↑f → Epi f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | rw [← CompHaus.epi_iff_surjective] at h | case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Function.Surjective ↑(toCompHaus.map f)
⊢ Epi f | case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Epi (toCompHaus.map f)
⊢ Epi f | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Function.Surjective ↑(toCompHaus.map f)
⊢ Epi f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | constructor | case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Epi (toCompHaus.map f)
⊢ Epi f | case mpr.left_cancellation
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Epi (toCompHaus.map f)
⊢ ∀ {Z : ExtrDisc} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Epi (toCompHaus.map f)
⊢ Epi f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | intro W a b h | case mpr.left_cancellation
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Epi (toCompHaus.map f)
⊢ ∀ {Z : ExtrDisc} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h | case mpr.left_cancellation
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : f ≫ a = f ≫ b
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left_cancellation
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h : Epi (toCompHaus.map f)
⊢ ∀ {Z : ExtrDisc} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | apply Functor.map_injective toCompHaus | case mpr.left_cancellation
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : f ≫ a = f ≫ b
⊢ a = b | case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : f ≫ a = f ≫ b
⊢ toCompHaus.map a = toCompHaus.map b | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left_cancellation
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : f ≫ a = f ≫ b
⊢ a = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | apply_fun toCompHaus.map at h | case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : f ≫ a = f ≫ b
⊢ toCompHaus.map a = toCompHaus.map b | case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : toCompHaus.map (f ≫ a) = toCompHaus.map (f ≫ b)
⊢ toCompHaus.map a = toCompHaus.map b | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : f ≫ a = f ≫ b
⊢ toCompHaus.map a = toCompHaus.map b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | simp only [Functor.map_comp] at h | case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : toCompHaus.map (f ≫ a) = toCompHaus.map (f ≫ b)
⊢ toCompHaus.map a = toCompHaus.map b | case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : toCompHaus.map f ≫ toCompHaus.map a = toCompHaus.map f ≫ toCompHaus.map b
⊢ toCompHaus.map a = toCompHaus.map b | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : toCompHaus.map (f ≫ a) = toCompHaus.map (f ≫ b)
⊢ toCompHaus.map a = toCompHaus.map b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.epi_iff_surjective | [30, 1] | [75, 42] | rwa [← cancel_epi (toCompHaus.map f)] | case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : toCompHaus.map f ≫ toCompHaus.map a = toCompHaus.map f ≫ toCompHaus.map b
⊢ toCompHaus.map a = toCompHaus.map b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left_cancellation.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X✝ : α → ExtrDisc
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
X Y : ExtrDisc
f : X ⟶ Y
h✝ : Epi (toCompHaus.map f)
W : ExtrDisc
a b : Y ⟶ W
h : toCompHaus.map f ≫ toCompHaus.map a = toCompHaus.map f ≫ toCompHaus.map b
⊢ toCompHaus.map a = toCompHaus.map b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | intro Z a₁ a₂ g₁ g₂ hg | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
⊢ ∀ {Z : CompHaus} (a₁ a₂ : α) (g₁ : Z ⟶ F.obj (X a₁)) (g₂ : Z ⟶ F.obj (X a₂)),
g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂ | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
⊢ ∀ {Z : CompHaus} (a₁ a₂ : α) (g₁ : Z ⟶ F.obj (X a₁)) (g₂ : Z ⟶ F.obj (X a₂)),
g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | let βZ := Z.presentation | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂ | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | let g₁' := F.preimage (Z.presentationπ ≫ g₁ : F.obj βZ ⟶ F.obj (X a₁)) | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂ | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | let g₂' := F.preimage (Z.presentationπ ≫ g₂ : F.obj βZ ⟶ F.obj (X a₂)) | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂ | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | apply Epi.left_cancellation (f := Z.presentationπ) | α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂ | case a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ CompHaus.presentationπ Z ≫ g₁ ≫ e a₁ = CompHaus.presentationπ Z ≫ g₂ ≫ e a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁ ≫ e a₁ = g₂ ≫ e a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | change g₁' ≫ e a₁ = g₂' ≫ e a₂ | case a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ CompHaus.presentationπ Z ≫ g₁ ≫ e a₁ = CompHaus.presentationπ Z ≫ g₂ ≫ e a₂ | case a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁' ≫ e a₁ = g₂' ≫ e a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ CompHaus.presentationπ Z ≫ g₁ ≫ e a₁ = CompHaus.presentationπ Z ≫ g₂ ≫ e a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | apply h | case a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁' ≫ e a₁ = g₂' ≫ e a₂ | case a.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁' ≫ π a₁ = g₂' ≫ π a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁' ≫ e a₁ = g₂' ≫ e a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | change CompHaus.presentationπ Z ≫ g₁ ≫ π a₁ = CompHaus.presentationπ Z ≫ g₂ ≫ π a₂ | case a.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁' ≫ π a₁ = g₂' ≫ π a₂ | case a.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ CompHaus.presentationπ Z ≫ g₁ ≫ π a₁ = CompHaus.presentationπ Z ≫ g₂ ≫ π a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ g₁' ≫ π a₁ = g₂' ≫ π a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.EffectiveEpiFamily.helper | [88, 1] | [113, 12] | simp [hg] | case a.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ CompHaus.presentationπ Z ≫ g₁ ≫ π a₁ = CompHaus.presentationπ Z ≫ g₂ ≫ π a₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
surj : ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
W : ExtrDisc
e : (a : α) → X a ⟶ W
h : ∀ {Z : ExtrDisc} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : CompHaus
a₁ a₂ : α
g₁ : Z ⟶ F.obj (X a₁)
g₂ : Z ⟶ F.obj (X a₂)
hg : g₁ ≫ π a₁ = g₂ ≫ π a₂
βZ : ExtrDisc := CompHaus.presentation Z
g₁' : βZ ⟶ X a₁ := F.preimage (CompHaus.presentationπ Z ≫ g₁)
g₂' : βZ ⟶ X a₂ := F.preimage (CompHaus.presentationπ Z ≫ g₂)
⊢ CompHaus.presentationπ Z ≫ g₁ ≫ π a₁ = CompHaus.presentationπ Z ≫ g₂ ≫ π a₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | tfae_have 1 → 2 | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | tfae_have 1 → 2 | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | tfae_have 2 → 3 | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | tfae_have 3 → 1 | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | case tfae_3_to_1
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
⊢ (∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
tfae_3_to_1 : (∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | tfae_finish | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
tfae_3_to_1 : (∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b
tfae_3_to_1 : (∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | intro | case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) | case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π) | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | infer_instance | case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | intro | case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) | case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π) | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | infer_instance | case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_1_to_2
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π)
TACTIC:
|
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