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/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | dsimp [IsChain, Pairwise] at hCh ⊢ | case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ IsChain (fun x y => x ⊇ y) Ch | case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : Set.Pairwise Ch fun x y => x ⊆ y ∨ y ⊆ x
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ Set.Pairwise Ch fun x y => x ⊇ y ∨ y ⊇ x | Please generate a tactic in lean4 to solve the state.
STATE:
case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ IsChain (fun x y => x ⊇ y) Ch
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact fun ⦃x⦄ a ⦃y⦄ a_1 a_2 => Or.comm.mp (hCh a a_1 a_2) | case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : Set.Pairwise Ch fun x y => x ⊆ y ∨ y ⊆ x
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ Set.Pairwise Ch fun x y => x ⊇ y ∨ y ⊇ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : Set.Pairwise Ch fun x y => x ⊆ y ∨ y ⊆ x
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ Set.Pairwise Ch fun x y => x ⊇ y ∨ y ⊇ x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | convert this | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ Set.Nonempty {x_1 | x_1 ∈ M ∧ π₁ φ f x_1 = x} | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ {x_1 | x_1 ∈ M ∧ π₁ φ f x_1 = x} = ⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x} | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ Set.Nonempty {x_1 | x_1 ∈ M ∧ π₁ φ f x_1 = x}
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [← iInter_inter, ← sInter_eq_iInter] | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ {x_1 | x_1 ∈ M ∧ π₁ φ f x_1 = x} = ⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x} | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ {x_1 | x_1 ∈ M ∧ π₁ φ f x_1 = x} = ⋂₀ Ch ∩ π₁ φ f ⁻¹' {x} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ {x_1 | x_1 ∈ M ∧ π₁ φ f x_1 = x} = ⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x}
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [←setOf_inter_eq_sep, inter_comm] | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ {x_1 | x_1 ∈ M ∧ π₁ φ f x_1 = x} = ⋂₀ Ch ∩ π₁ φ f ⁻¹' {x} | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ M ∩ {a | π₁ φ f a = x} = ⋂₀ Ch ∩ π₁ φ f ⁻¹' {x} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ {x_1 | x_1 ∈ M ∧ π₁ φ f x_1 = x} = ⋂₀ Ch ∩ π₁ φ f ⁻¹' {x}
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | congr | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ M ∩ {a | π₁ φ f a = x} = ⋂₀ Ch ∩ π₁ φ f ⁻¹' {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
this : Set.Nonempty (⋂ (X : ↑Ch), ↑X ∩ π₁ φ f ⁻¹' {x})
⊢ M ∩ {a | π₁ φ f a = x} = ⋂₀ Ch ∩ π₁ φ f ⁻¹' {x}
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | intro X | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
⊢ ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
⊢ Set.Nonempty (Z X) ∧ IsCompact (Z X) ∧ IsClosed (Z X) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
⊢ ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | simp | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
⊢ Set.Nonempty (Z X) ∧ IsCompact (Z X) ∧ IsClosed (Z X) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
⊢ Set.Nonempty (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
⊢ Set.Nonempty (Z X) ∧ IsCompact (Z X) ∧ IsClosed (Z X)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | have h₃ := (h (Subtype.mem X)).2 | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
⊢ Set.Nonempty (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ Set.Nonempty (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
⊢ Set.Nonempty (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x})
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [←image_inter_nonempty_iff, h₃] | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ Set.Nonempty (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ Set.Nonempty (univ ∩ {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ Set.Nonempty (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x})
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | simp | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ Set.Nonempty (univ ∩ {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ Set.Nonempty (univ ∩ {x}) ∧ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x})
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | have := (h (Subtype.mem X)).1 | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : CompactSpace ↑↑X
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x})
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [←isCompact_iff_compactSpace] at this | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : CompactSpace ↑↑X
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : IsCompact ↑X
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : CompactSpace ↑↑X
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x})
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | have h_cl := IsClosed.inter (IsCompact.isClosed this)
(IsClosed.preimage (Continuous.comp continuous_fst continuous_subtype_val) <| T1Space.t1 x) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : IsCompact ↑X
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : IsCompact ↑X
h_cl : IsClosed (↑X ∩ Prod.fst ∘ Subtype.val ⁻¹' {x})
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : IsCompact ↑X
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x})
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact And.intro (IsClosed.isCompact h_cl) h_cl | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : IsCompact ↑X
h_cl : IsClosed (↑X ∩ Prod.fst ∘ Subtype.val ⁻¹' {x})
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x}) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this✝ : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
X : ↑Ch
h₃ : π₁ φ f '' ↑X = univ
this : IsCompact ↑X
h_cl : IsClosed (↑X ∩ Prod.fst ∘ Subtype.val ⁻¹' {x})
⊢ IsCompact (↑X ∩ π₁ φ f ⁻¹' {x}) ∧ IsClosed (↑X ∩ π₁ φ f ⁻¹' {x})
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact fun x y => x ⊇ y | case refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ Set ↑(D' φ f) → Set ↑(D' φ f) → Prop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ Set ↑(D' φ f) → Set ↑(D' φ f) → Prop
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact fun ⦃x y⦄ a => inter_subset_inter_left _ a | case refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ ∀ ⦃x_1 y : Set ↑(D' φ f)⦄, x_1 ⊇ y → x_1 ∩ π₁ φ f ⁻¹' {x} ⊇ y ∩ π₁ φ f ⁻¹' {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
Z : ↑Ch → Set ↑(D' φ f) := fun X => ↑X ∩ π₁ φ f ⁻¹' {x}
assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i)
⊢ ∀ ⦃x_1 y : Set ↑(D' φ f)⦄, x_1 ⊇ y → x_1 ∩ π₁ φ f ⁻¹' {x} ⊇ y ∩ π₁ φ f ⁻¹' {x}
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | intro X hX | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
⊢ ∀ (s : Set ↑(D' φ f)), s ∈ Ch → M ⊆ s | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
X : Set ↑(D' φ f)
hX : X ∈ Ch
⊢ M ⊆ X | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
⊢ ∀ (s : Set ↑(D' φ f)), s ∈ Ch → M ⊆ s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact sInter_subset_of_mem hX | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
X : Set ↑(D' φ f)
hX : X ∈ Ch
⊢ M ⊆ X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
X : Set ↑(D' φ f)
hX : X ∈ Ch
⊢ M ⊆ X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact hE₁ | case intro.intro.intro.left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
⊢ CompactSpace ↑E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
⊢ CompactSpace ↑E
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact hE₂ | case intro.intro.intro.right.left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
⊢ π₁ φ f '' E = univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
⊢ π₁ φ f '' E = univ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | intro E₀ h₁ h₂ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
⊢ ∀ (E₀ : Set ↑(D' φ f)), E₀ ⊂ E → CompactSpace ↑E₀ → π₁ φ f '' E₀ ≠ univ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
⊢ π₁ φ f '' E₀ ≠ univ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
⊢ ∀ (E₀ : Set ↑(D' φ f)), E₀ ⊂ E → CompactSpace ↑E₀ → π₁ φ f '' E₀ ≠ univ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | replace hE₃ := hE₃ E₀ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
⊢ π₁ φ f '' E₀ ≠ univ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
hE₃ : E₀ ∈ S → E₀ ⊆ E → E₀ = E
⊢ π₁ φ f '' E₀ ≠ univ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₃ : ∀ (a : Set ↑(D' φ f)), a ∈ S → a ⊆ E → a = E
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
⊢ π₁ φ f '' E₀ ≠ univ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | by_contra h₃ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
hE₃ : E₀ ∈ S → E₀ ⊆ E → E₀ = E
⊢ π₁ φ f '' E₀ ≠ univ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
hE₃ : E₀ ∈ S → E₀ ⊆ E → E₀ = E
h₃ : π₁ φ f '' E₀ = univ
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
hE₃ : E₀ ∈ S → E₀ ⊆ E → E₀ = E
⊢ π₁ φ f '' E₀ ≠ univ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | replace hE₃ := hE₃ (And.intro h₂ h₃) (subset_of_ssubset h₁) | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
hE₃ : E₀ ∈ S → E₀ ⊆ E → E₀ = E
h₃ : π₁ φ f '' E₀ = univ
⊢ False | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
h₃ : π₁ φ f '' E₀ = univ
hE₃ : E₀ = E
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
hE₃ : E₀ ∈ S → E₀ ⊆ E → E₀ = E
h₃ : π₁ φ f '' E₀ = univ
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact (ne_of_ssubset h₁) hE₃ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
h₃ : π₁ φ f '' E₀ = univ
hE₃ : E₀ = E
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E✝), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
chain_cond :
∀ (c : Set (Set ↑(D' φ f))),
c ⊆ S → IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb, lb ∈ S ∧ ∀ (s : Set ↑(D' φ f)), s ∈ c → lb ⊆ s
E : Set ↑(D' φ f)
hE₁ : CompactSpace ↑E
hE₂ : π₁ φ f '' E = univ
E₀ : Set ↑(D' φ f)
h₁ : E₀ ⊂ E
h₂ : CompactSpace ↑E₀
h₃ : π₁ φ f '' E₀ = univ
hE₃ : E₀ = E
⊢ False
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | constructor | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun) ∧
f ∘ restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ | case left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun)
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun) ∧
f ∘ restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | refine' Continuous.comp _ _ | case left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun) | case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (restrict (E' hφ hf hf') (π₂ φ f))
case left.refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (ρ' hφ hf hf').toEquiv.invFun | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | refine' ContinuousOn.restrict _ | case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (restrict (E' hφ hf hf') (π₂ φ f)) | case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ ContinuousOn (π₂ φ f) (E' hφ hf hf') | Please generate a tactic in lean4 to solve the state.
STATE:
case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (restrict (E' hφ hf hf') (π₂ φ f))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | apply Continuous.continuousOn | case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ ContinuousOn (π₂ φ f) (E' hφ hf hf') | case left.refine'_1.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (π₂ φ f) | Please generate a tactic in lean4 to solve the state.
STATE:
case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ ContinuousOn (π₂ φ f) (E' hφ hf hf')
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | exact Continuous.comp continuous_snd continuous_subtype_val | case left.refine'_1.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (π₂ φ f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.refine'_1.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (π₂ φ f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | exact (Homeomorph.continuous (Homeomorph.symm (ρ' hφ hf hf'))) | case left.refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (ρ' hφ hf hf').toEquiv.invFun | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ Continuous (ρ' hφ hf hf').toEquiv.invFun
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | suffices : f ∘ (E' hφ hf hf').restrict (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toFun | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ
case this
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | ext x | case this
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ (f ∘ restrict (E' hφ hf hf') (π₂ φ f)) x = (φ ∘ (ρ' hφ hf hf').toEquiv.toFun) x | Please generate a tactic in lean4 to solve the state.
STATE:
case this
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | simp only [Function.comp_apply, restrict_apply, Equiv.toFun_as_coe, Homeomorph.coe_toEquiv] | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ (f ∘ restrict (E' hφ hf hf') (π₂ φ f)) x = (φ ∘ (ρ' hφ hf hf').toEquiv.toFun) x | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ f (π₂ φ f ↑x) = φ (↑(ρ' hφ hf hf') x) | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ (f ∘ restrict (E' hφ hf hf') (π₂ φ f)) x = (φ ∘ (ρ' hφ hf hf').toEquiv.toFun) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | dsimp [π₂, ρ', compact_to_T2_homeomorph, Continuous.homeoOfEquivCompactToT2_toEquiv,
Continuous.homeoOfEquivCompactToT2, π₁] | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ f (π₂ φ f ↑x) = φ (↑(ρ' hφ hf hf') x) | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ f (↑↑x).snd = φ (↑↑x).fst | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ f (π₂ φ f ↑x) = φ (↑(ρ' hφ hf hf') x)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | have := x.val.prop | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ f (↑↑x).snd = φ (↑↑x).fst | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
this : ↑↑x ∈ D' φ f
⊢ f (↑↑x).snd = φ (↑↑x).fst | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
⊢ f (↑↑x).snd = φ (↑↑x).fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | dsimp [D'] at this | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
this : ↑↑x ∈ D' φ f
⊢ f (↑↑x).snd = φ (↑↑x).fst | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
this : φ (↑↑x).fst = f (↑↑x).snd
⊢ f (↑↑x).snd = φ (↑↑x).fst | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
this : ↑↑x ∈ D' φ f
⊢ f (↑↑x).snd = φ (↑↑x).fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | exact this.symm | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
this : φ (↑↑x).fst = f (↑↑x).snd
⊢ f (↑↑x).snd = φ (↑↑x).fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
x : ↑(E' hφ hf hf')
this : φ (↑↑x).fst = f (↑↑x).snd
⊢ f (↑↑x).snd = φ (↑↑x).fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | simp only [← Function.comp.assoc] | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ (f ∘ restrict (E' hφ hf hf') (π₂ φ f)) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ f ∘ restrict (E' hφ hf hf') (π₂ φ f) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | rw [this] | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ (f ∘ restrict (E' hφ hf hf') (π₂ φ f)) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ (φ ∘ (ρ' hφ hf hf').toEquiv.toFun) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ (f ∘ restrict (E' hφ hf hf') (π₂ φ f)) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | simp only [Function.comp.assoc, Equiv.toFun_as_coe, Homeomorph.coe_toEquiv,
Equiv.invFun_as_coe, Homeomorph.coe_symm_toEquiv,
Homeomorph.self_comp_symm, Function.comp.right_id] | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ (φ ∘ (ρ' hφ hf hf').toEquiv.toFun) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁵ : T2Space A
inst✝⁴ : CompactSpace A
inst✝³ : T2Space B
inst✝² : CompactSpace B
inst✝¹ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
inst✝ : ExtremallyDisconnected A
this : f ∘ restrict (E' hφ hf hf') (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toEquiv.toFun
⊢ (φ ∘ (ρ' hφ hf hf').toEquiv.toFun) ∘ (ρ' hφ hf hf').toEquiv.invFun = φ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | intro a | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
⊢ ∀ (a : α), OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
⊢ ∀ (a : α), OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | have h₁ : OpenEmbedding (Sigma.desc i) :=
(ExtrDisc.homeoOfIso (asIso (Sigma.desc i))).openEmbedding | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | have h₂ : OpenEmbedding (Sigma.ι Z a) := openEmbedding_ι _ _ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | have := OpenEmbedding.comp h₁ h₂ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding (↑(Sigma.desc i) ∘ ↑(Sigma.ι Z a))
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | erw [← CategoryTheory.coe_comp (Sigma.ι Z a) (Sigma.desc i)] at this | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding (↑(Sigma.desc i) ∘ ↑(Sigma.ι Z a))
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(Sigma.ι Z a ≫ Sigma.desc i)
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding (↑(Sigma.desc i) ∘ ↑(Sigma.ι Z a))
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | simp only [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app] at this | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(Sigma.ι Z a ≫ Sigma.desc i)
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(i a)
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(Sigma.ι Z a ≫ Sigma.desc i)
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | assumption | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(i a)
⊢ OpenEmbedding ↑(i a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(i a)
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isIso_of_bijective | [33, 1] | [36, 41] | exact CompHaus.isIso_of_bijective _ hf | X Y : ExtrDisc
f : X ⟶ Y
hf : Function.Bijective ↑f
⊢ IsIso (toCompHaus.map f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : ExtrDisc
f : X ⟶ Y
hf : Function.Bijective ↑f
⊢ IsIso (toCompHaus.map f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isIso_of_bijective | [33, 1] | [36, 41] | apply isIso_of_fully_faithful toCompHaus | X Y : ExtrDisc
f : X ⟶ Y
hf : Function.Bijective ↑f
this : IsIso (toCompHaus.map f)
⊢ IsIso f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : ExtrDisc
f : X ⟶ Y
hf : Function.Bijective ↑f
this : IsIso (toCompHaus.map f)
⊢ IsIso f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | let ζ := finiteCoproduct.desc _ (fun a => pullback.snd f (hOpen a) ≫ finiteCoproduct.ι Z a ) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | let α := finiteCoproduct.desc _ ((fun a => pullback.fst f (hOpen a))) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | let β := finiteCoproduct.desc _ π | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | have comm : ζ ≫ β = α ≫ f := by
refine' finiteCoproduct.hom_ext _ _ _ (fun a => _)
simp [← Category.assoc, finiteCoproduct.ι_desc, ExtrDisc.pullback.condition] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | intro R₁ R₂ hR | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | have himage : (ζ ≫ β) R₁ = (ζ ≫ β) R₂ := by
rw [comm]; change f (α R₁) = f (α R₂); rw [hR] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ R₁ = R₂ | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | replace himage := congr_arg (inv β) himage | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂
⊢ R₁ = R₂ | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₁) = ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₂)
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | change ((ζ ≫ β ≫ inv β) R₁) = ((ζ ≫ β ≫ inv β) R₂) at himage | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₁) = ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₂)
⊢ R₁ = R₂ | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₁ = ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₁) = ↑(CategoryTheory.inv β) (↑(ζ ≫ β) R₂)
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | rw [IsIso.hom_inv_id, Category.comp_id] at himage | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₁ = ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₂
⊢ R₁ = R₂ | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑ζ R₁ = ↑ζ R₂
⊢ R₁ = R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₁ = ↑(ζ ≫ β ≫ CategoryTheory.inv β) R₂
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | exact Sigma.subtype_ext Hfst hR | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
⊢ R₁ = R₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑ζ R₁ = ↑ζ R₂
Hfst : R₁.fst = R₂.fst
⊢ R₁ = R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ ζ ≫ β = α ≫ f | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
a : α✝
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) a ≫ ζ ≫ β =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) a ≫ α ≫ f | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ ζ ≫ β = α ≫ f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | simp [← Category.assoc, finiteCoproduct.ι_desc, ExtrDisc.pullback.condition] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
a : α✝
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) a ≫ ζ ≫ β =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) a ≫ α ≫ f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
a : α✝
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) a ≫ ζ ≫ β =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) a ≫ α ≫ f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | rw [comm] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂ | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ ↑(α ≫ f) R₁ = ↑(α ≫ f) R₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ ↑(ζ ≫ β) R₁ = ↑(ζ ≫ β) R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | change f (α R₁) = f (α R₂) | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ ↑(α ≫ f) R₁ = ↑(α ≫ f) R₂ | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ ↑f (↑α R₁) = ↑f (↑α R₂) | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ ↑(α ≫ f) R₁ = ↑(α ≫ f) R₂
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | rw [hR] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ ↑f (↑α R₁) = ↑f (↑α R₂) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
⊢ ↑f (↑α R₁) = ↑f (↑α R₂)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | constructor <;> rfl | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑ζ R₁ = ↑ζ R₂
⊢ (↑ζ R₁).fst = R₁.fst ∧ (↑ζ R₂).fst = R₂.fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑ζ R₁ = ↑ζ R₂
⊢ (↑ζ R₁).fst = R₁.fst ∧ (↑ζ R₂).fst = R₂.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | rw [← this.1, ← this.2, himage] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑ζ R₁ = ↑ζ R₂
this : (↑ζ R₁).fst = R₁.fst ∧ (↑ζ R₂).fst = R₂.fst
⊢ R₁.fst = R₂.fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.snd f (_ : OpenEmbedding ↑(π a)) ≫ finiteCoproduct.ι Z a
α : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ Y :=
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
comm : ζ ≫ β = α ≫ f
R₁ R₂ : CoeSort.coe (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt)
hR :
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₁ =
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
R₂
himage : ↑ζ R₁ = ↑ζ R₂
this : (↑ζ R₁).fst = R₁.fst ∧ (↑ζ R₂).fst = R₂.fst
⊢ R₁.fst = R₂.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | let β := finiteCoproduct.desc _ π | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
⊢ IsIso
(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso
(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
⊢ IsIso
(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | refine' isIso_of_bijective ⟨extensivity_injective f HIso hOpen, fun y => _⟩ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso
(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a))) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ∃ a,
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
a =
y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso
(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | refine' ⟨⟨(inv β (f y)).1, ⟨y, (inv β (f y)).2, _⟩⟩, rfl⟩ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ∃ a,
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
a =
y | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd = ↑f y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ∃ a,
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(π a)))
a =
y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | have inj : Function.Injective (inv β) := by intros r s hrs
convert congr_arg β hrs <;> change _ = (inv β ≫ β) _<;> simp only [IsIso.inv_hom_id]<;> rfl | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd = ↑f y | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd = ↑f y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd = ↑f y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | apply inj | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd = ↑f y | case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd) =
↑(CategoryTheory.inv β) (↑f y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd = ↑f y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | intro a | case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a | case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
a : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
⊢ ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | simp only [IsIso.comp_inv_eq, finiteCoproduct.ι_desc] | case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
a : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
a : α
⊢ π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | intros r s hrs | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ Function.Injective ↑(CategoryTheory.inv β) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
r s : CoeSort.coe X
hrs : ↑(CategoryTheory.inv β) r = ↑(CategoryTheory.inv β) s
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ Function.Injective ↑(CategoryTheory.inv β)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | convert congr_arg β hrs <;> change _ = (inv β ≫ β) _<;> simp only [IsIso.inv_hom_id]<;> rfl | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
r s : CoeSort.coe X
hrs : ↑(CategoryTheory.inv β) r = ↑(CategoryTheory.inv β) s
⊢ r = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
r s : CoeSort.coe X
hrs : ↑(CategoryTheory.inv β) r = ↑(CategoryTheory.inv β) s
⊢ r = s
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | change (_ ≫ inv β) _ = _ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd) =
↑(CategoryTheory.inv β) (↑f y) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst ≫ CategoryTheory.inv β) (↑(CategoryTheory.inv β) (↑f y)).snd =
↑(CategoryTheory.inv β) (↑f y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(CategoryTheory.inv β) (↑(π (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd) =
↑(CategoryTheory.inv β) (↑f y)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | rw [this] | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst ≫ CategoryTheory.inv β) (↑(CategoryTheory.inv β) (↑f y)).snd =
↑(CategoryTheory.inv β) (↑f y) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(finiteCoproduct.ι Z (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd =
↑(CategoryTheory.inv β) (↑f y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(π (↑(CategoryTheory.inv β) (↑f y)).fst ≫ CategoryTheory.inv β) (↑(CategoryTheory.inv β) (↑f y)).snd =
↑(CategoryTheory.inv β) (↑f y)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | rfl | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(finiteCoproduct.ι Z (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd =
↑(CategoryTheory.inv β) (↑f y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(CategoryTheory.inv β)
this : ∀ (a : α), π a ≫ CategoryTheory.inv β = finiteCoproduct.ι Z a
⊢ ↑(finiteCoproduct.ι Z (↑(CategoryTheory.inv β) (↑f y)).fst) (↑(CategoryTheory.inv β) (↑f y)).snd =
↑(CategoryTheory.inv β) (↑f y)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | have hOpen := openEmbedding_of_sigma_desc_iso H | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | let θ := Sigma.mapIso (fun a => fromExplicitIso f (hOpen a)) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | let δ := FromFiniteCoproductIso (fun a => (OpenEmbeddingCone f (hOpen a)).pt) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
⊢ IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
⊢ IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
⊢ IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | convert extensivity_explicit f HIso hOpen | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ IsIso (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(i a)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ IsIso (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(i a)) | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
a : α
⊢ (finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(i a)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
⊢ (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(i a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | simp [← Category.assoc, finiteCoproduct.ι_desc, fromExplicit] | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
a : α
⊢ (finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(i a)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) i)
a : α
⊢ (finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) fun a =>
ExtrDisc.pullback.fst f (_ : OpenEmbedding ↑(i a))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | apply IsIso.of_isIso_comp_left θ.hom | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
this : IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
this : IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | apply IsIso.of_isIso_comp_left δ.hom | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
this : IsIso (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
⊢ IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
this : IsIso (δ.hom ≫ θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
⊢ IsIso (θ.hom ≫ Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | let ε := ToFiniteCoproductIso Z | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | convert H | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ IsIso (ε.hom ≫ finiteCoproduct.desc Z i) | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.desc i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ IsIso (ε.hom ≫ finiteCoproduct.desc Z i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | refine' Sigma.hom_ext _ _ (fun a => _) | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.desc i | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
a : α
⊢ Sigma.ι Z a ≫ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.ι Z a ≫ Sigma.desc i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
⊢ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.desc i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | simp [← Category.assoc] | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
a : α
⊢ Sigma.ι Z a ≫ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.ι Z a ≫ Sigma.desc i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
a : α
⊢ Sigma.ι Z a ≫ ε.hom ≫ finiteCoproduct.desc Z i = Sigma.ι Z a ≫ Sigma.desc i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | apply IsIso.of_isIso_comp_left ε.hom | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
this : IsIso (ε.hom ≫ finiteCoproduct.desc Z i)
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmbedding ↑(i a))
δ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅
∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt :=
FromFiniteCoproductIso fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt
ε : ∐ Z ≅ finiteCoproduct Z := ToFiniteCoproductIso Z
this : IsIso (ε.hom ≫ finiteCoproduct.desc Z i)
⊢ IsIso (finiteCoproduct.desc (fun a => Z a) i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | refine' fun P => ⟨(@fun X Y f e he => _)⟩ | ⊢ EverythingIsProjective ExtrDisc | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
⊢ ∃ f', f' ≫ e = f | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ EverythingIsProjective ExtrDisc
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | have proj : Projective (toCompHaus.obj P) := inferInstanceAs (Projective P.compHaus) | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
⊢ ∃ f', f' ≫ e = f | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ ∃ f', f' ≫ e = f | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
⊢ ∃ f', f' ≫ e = f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | have : Epi (toCompHaus.map e) := by rw [CompHaus.epi_iff_surjective]
change Function.Surjective e
rwa [← ExtrDisc.epi_iff_surjective] | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ ∃ f', f' ≫ e = f | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
⊢ ∃ f', f' ≫ e = f | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ ∃ f', f' ≫ e = f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | set g := toCompHaus.preimage <| Projective.factorThru (toCompHaus.map f) (toCompHaus.map e) with hg | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
⊢ ∃ f', f' ≫ e = f | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ ∃ f', f' ≫ e = f | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
⊢ ∃ f', f' ≫ e = f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | refine' ⟨g, toCompHaus.map_injective _⟩ | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ ∃ f', f' ≫ e = f | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ toCompHaus.map (g ≫ e) = toCompHaus.map f | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ ∃ f', f' ≫ e = f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | rw [map_comp, hg, image_preimage, Projective.factorThru_comp] | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ toCompHaus.map (g ≫ e) = toCompHaus.map f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ toCompHaus.map (g ≫ e) = toCompHaus.map f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | rw [CompHaus.epi_iff_surjective] | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ Epi (toCompHaus.map e) | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ Function.Surjective ↑(toCompHaus.map e) | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ Epi (toCompHaus.map e)
TACTIC:
|
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