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refactor: various improvements for doing Lemma 3.9 #51

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refactor: various improvements for doing Lemma 3.9 #51

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157 changes: 75 additions & 82 deletions src/Mugen/Cat/HierarchyTheory/Universality/EndomorphismEmbedding.agda
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Expand Up @@ -31,7 +31,7 @@ import Mugen.Order.Reasoning as Reasoning
-- The Universal Embedding Theorem
-- Section 3.4, Lemma 3.8
--
-- Given a hierarchy theory 'H', a strict order Δ, and a set Ψ, we can
-- Given a hierarchy theory 'H', a poset Δ, and a set Ψ, we can
-- construct a faithful functor 'T : Endos (Fᴴ Δ) → Endos Fᴹᴰ Ψ', where
-- 'Fᴴ' denotes the free H-algebra on Δ, and 'Fᴹᴰ Ψ' denotes the free McBride
-- Hierarchy theory over the endomorphism displacement algebra on 'H (◆ ⊕ Δ ⊕ Δ)'.
Expand All @@ -58,7 +58,7 @@ module Mugen.Cat.HierarchyTheory.Universality.EndomorphismEmbedding

H⟨Δ⁺⟩ : Poset o r
H⟨Δ⁺⟩ = H.M₀ Δ⁺
module H⟨Δ⁺⟩ = Poset H⟨Δ⁺⟩
module H⟨Δ⁺⟩ = Reasoning H⟨Δ⁺⟩

H⟨Δ⁺⟩→-poset : Poset (lsuc (o ⊔ r)) (o ⊔ r)
H⟨Δ⁺⟩→-poset = Endomorphism-poset H Δ⁺
Expand All @@ -84,124 +84,119 @@ module Mugen.Cat.HierarchyTheory.Universality.EndomorphismEmbedding
SOrdᴹᴰ = Eilenberg-Moore SOrd↑ (McBride H⟨Δ⁺⟩→)
module SOrdᴹᴰ = Cat SOrdᴹᴰ

Fᴴ⟨_⟩ : Poset o r → Algebra SOrd H
Fᴴ⟨_⟩ = Functor.F₀ (Free SOrd H)
Fᴴ₀ : Poset o r → Algebra SOrd H
Fᴴ₀ = Functor.F₀ (Free SOrd H)

Fᴴ₁ : {X Y : Poset o r} → Hom X Y → SOrdᴴ.Hom (Fᴴ₀ X) (Fᴴ₀ Y)
Fᴴ₁ = Functor.F₁ (Free SOrd H)

Endoᴴ⟨Δ⟩ : Type (o ⊔ r)
Endoᴴ⟨Δ⟩ = Hom (Fᴴ⟨ Δ ⟩ .fst) (Fᴴ⟨ Δ ⟩ .fst)
Endoᴴ⟨Δ⟩ = Hom (H.M₀ Δ) (H.M₀ Δ)

Fᴹᴰ⟨_⟩ : Poset (lsuc o ⊔ lsuc r) (lsuc o ⊔ lsuc r) → Algebra SOrd↑ (McBride H⟨Δ⁺⟩→)
Fᴹᴰ⟨_⟩ = Functor.F₀ (Free SOrd↑ (McBride H⟨Δ⁺⟩→))
Fᴹᴰ₀ : Poset (lsuc o ⊔ lsuc r) (lsuc o ⊔ lsuc r) → Algebra SOrd↑ (McBride H⟨Δ⁺⟩→)
Fᴹᴰ₀ = Functor.F₀ (Free SOrd↑ (McBride H⟨Δ⁺⟩→))

pattern ⋆ = lift tt
pattern ι₀ α = inl α
pattern ι₁ α = inr (inl α)
pattern ι₂ α = inr (inr α)
-- These patterns and definitions are exported for the naturality proof
-- in another file.

ι₁-inj : ∀ {x y : ⌞ Δ ⌟} → _≡_ {A = ⌞ Δ⁺ ⌟} (ι₁ x) (ι₁ y) → x ≡ y
ι₁-inj = inl-inj ⊙ inr-inj
pattern ⋆ = lift tt

ι₂-inj : ∀ {x y : ⌞ Δ ⌟} → _≡_ {A = ⌞ Δ⁺ ⌟} (ι₂ x) (ι₂ y) → x ≡ y
ι₂-inj = inr-inj ⊙ inr-inj
pattern ι₀ α = inl α

ι₀-hom : Hom ◆ Δ⁺
ι₀-hom .hom = ι₀
ι₀-hom .pres-≤[]-equal α≤β = lift α≤β , λ _ → refl

pattern ι₁ α = inr (inl α)

ι₁-inj : ∀ {x y : ⌞ Δ ⌟} → _≡_ {A = ⌞ Δ⁺ ⌟} (ι₁ x) (ι₁ y) → x ≡ y
ι₁-inj = inl-inj ⊙ inr-inj

ι₁-hom : Hom Δ Δ⁺
ι₁-hom .hom = ι₁
ι₁-hom .pres-≤[]-equal α≤β = lift (lift α≤β) , ι₁-inj

ι₁-monic : SOrd.is-monic ι₁-hom
ι₁-monic g h p = ext λ α → ι₁-inj (p #ₚ α)

pattern ι₂ α = inr (inr α)

ι₂-inj : ∀ {x y : ⌞ Δ ⌟} → _≡_ {A = ⌞ Δ⁺ ⌟} (ι₂ x) (ι₂ y) → x ≡ y
ι₂-inj = inr-inj ⊙ inr-inj


--------------------------------------------------------------------------------
-- Construction of the functor T
-- Section 3.4, Lemma 3.8

σ̅ : Algebra-hom SOrd H Fᴴ⟨ Δ ⟩ Fᴴ⟨ Δ ⟩ → Hom Δ⁺ H⟨Δ⁺⟩
σ̅ σ .hom (ι₀ ⋆) = H.unit.η Δ⁺ # ι₀ ⋆
σ̅ σ .hom (ι₁ α) = H.M₁ ι₁-hom # (σ # (H.unit.η Δ # α))
σ̅ σ .hom (ι₂ α) = H.unit.η _ # ι₂ α
σ̅ : SOrdᴴ.Hom (Fᴴ₀ Δ) (Fᴴ₀ Δ) → Hom Δ⁺ H⟨Δ⁺⟩
σ̅ σ .hom (ι₀ ⋆) = H.η Δ⁺ # ι₀ ⋆
σ̅ σ .hom (ι₁ α) = H.M₁ ι₁-hom # (σ # (H.η Δ # α))
σ̅ σ .hom (ι₂ α) = H.η Δ⁺ # ι₂ α
σ̅ σ .pres-≤[]-equal {ι₀ ⋆} {ι₀ ⋆} _ = H⟨Δ⁺⟩.≤-refl , λ _ → refl
σ̅ σ .pres-≤[]-equal {ι₁ α} {ι₁ β} (lift (lift α≤β)) = Σ-map₂ (ap ι₁ ⊙_) $ (H.M₁ ι₁-hom ∘ σ .morphism ∘ H.unit.η Δ) .pres-≤[]-equal α≤β
σ̅ σ .pres-≤[]-equal {ι₂ α} {ι₂ β} α≤β = H.unit.η Δ⁺ .pres-≤[]-equal α≤β
σ̅ σ .pres-≤[]-equal {ι₁ α} {ι₁ β} (lift (lift α≤β)) =
H⟨Δ⁺⟩.≤[]-map (ap ι₁) $ (H.M₁ ι₁-hom ∘ σ .morphism ∘ H.η Δ) .pres-≤[]-equal α≤β
σ̅ σ .pres-≤[]-equal {ι₂ α} {ι₂ β} α≤β = H.η Δ⁺ .pres-≤[]-equal α≤β

abstract
σ̅-id : σ̅ SOrdᴴ.id ≡ H.unit.η Δ⁺
σ̅-id : σ̅ SOrdᴴ.id ≡ H.η Δ⁺
σ̅-id = ext λ where
(ι₀ α) → refl
(ι₁ α) → sym (H.unit.is-natural Δ Δ⁺ ι₁-hom) #ₚ α
(ι₂ α) → refl

abstract
σ̅-ι
: ∀ (σ : Algebra-hom SOrd H Fᴴ⟨ Δ ⟩ Fᴴ⟨ Δ ⟩)
: ∀ (σ : SOrdᴴ.Hom (Fᴴ₀ Δ) (Fᴴ₀ Δ))
→ (α : ⌞ H.M₀ Δ ⌟)
→ H.M₁ (H.M₁ ι₁-hom) # (H.M₁ (σ .morphism) # (H.M₁ (H.unit.η Δ) # α))
→ H.M₁ (H.M₁ ι₁-hom ∘ σ .morphism ∘ H.η Δ) # α
≡ H.M₁ (σ̅ σ) # (H.M₁ ι₁-hom # α)
σ̅-ι σ α =
(H.M₁ (H.M₁ ι₁-hom) ∘ H.M₁ (σ .morphism) ∘ H.M₁ (H.unit.η Δ)) # α ≡˘⟨ ap (H.M₁ (H.M₁ ι₁-hom) ∘_) (H.M-∘ _ _) #ₚ α ⟩
(H.M₁ (H.M₁ ι₁-hom) ∘ H.M₁ (σ .morphism ∘ H.unit.η Δ)) # α ≡˘⟨ H.M-∘ _ _ #ₚ α ⟩
H.M₁ (H.M₁ ι₁-hom ∘ σ .morphism ∘ H.unit.η Δ) # α ≡⟨ ap H.M₁ lemma #ₚ α ⟩
H.M₁ (σ̅ σ ∘ ι₁-hom) # α ≡⟨ H.M-∘ _ _ #ₚ α ⟩
H.M₁ (σ̅ σ) # (H.M₁ ι₁-hom # α) ∎
where
lemma : H.M₁ ι₁-hom ∘ σ .morphism ∘ H.unit.η Δ ≡ σ̅ σ ∘ ι₁-hom
lemma = ext λ _ → refl
H.M₁ (H.M₁ ι₁-hom ∘ σ .morphism ∘ H.η Δ) # α ≡⟨ ap (λ m → H.M₁ m # α) $ ext (λ _ → refl) ⟩
H.M₁ (σ̅ σ ∘ ι₁-hom) # α ≡⟨ H.M-∘ _ _ #ₚ α ⟩
H.M₁ (σ̅ σ) # (H.M₁ ι₁-hom # α) ∎

abstract
σ̅-∘ : ∀ (σ δ : Algebra-hom SOrd H Fᴴ⟨ Δ ⟩ Fᴴ⟨ Δ ⟩) → σ̅ (σ SOrdᴴ.∘ δ) ≡ H.mult.η Δ⁺ ∘ H.M₁ (σ̅ σ) ∘ σ̅ δ
σ̅-∘ : ∀ (σ δ : SOrdᴴ.Hom (Fᴴ₀ Δ) (Fᴴ₀ Δ)) → σ̅ (σ SOrdᴴ.∘ δ) ≡ H.μ Δ⁺ ∘ H.M₁ (σ̅ σ) ∘ σ̅ δ
σ̅-∘ σ δ = ext λ where
(ι₀ α) →
H.unit.η Δ⁺ # ι₀ α ≡˘⟨ H.right-ident #ₚ (H.unit.η Δ⁺ # ι₀ α) ⟩
H.mult.η Δ⁺ # (H.unit.η (H.M₀ Δ⁺) # (H.unit.η Δ⁺ # ι₀ α)) ≡⟨ ap# (H.mult.η _) (H.unit.is-natural _ _ (σ̅ σ) #ₚ ι₀ α) ⟩
H.mult.η Δ⁺ # (H.M₁ (σ̅ σ) # (H.unit.η Δ⁺ # ι₀ α)) ∎
H.η Δ⁺ # ι₀ α ≡˘⟨ μ-η H (σ̅ σ) #ₚ ι₀ α ⟩
H.μ Δ⁺ # (H.M₁ (σ̅ σ) # (H.η Δ⁺ # ι₀ α)) ∎
(ι₁ α) →
H.M₁ ι₁-hom # (σ # (δ # (H.unit.η Δ # α))) ≡˘⟨ ap (λ e → H.M₁ ι₁-hom # (σ # e)) (H.left-ident #ₚ _) ⟩
H.M₁ ι₁-hom # (σ # (H.mult.η Δ # (H.M₁ (H.unit.η Δ) # (δ # (H.unit.η Δ # α))))) ≡⟨ ap# (H.M₁ ι₁-hom) (σ .commutes #ₚ _) ⟩
H.M₁ ι₁-hom # (H.mult.η _ # (H.M₁ (σ .morphism) # (H.M₁ (H.unit.η Δ) # (δ # (H.unit.η Δ # α))))) ≡˘⟨ H.mult.is-natural _ _ ι₁-hom #ₚ _ ⟩
H.mult.η _ # (H.M₁ (H.M₁ ι₁-hom) # (H.M₁ (σ .morphism) # (H.M₁ (H.unit.η Δ) # (δ # (H.unit.η Δ # α))))) ≡⟨ ap# (H.mult.η _) (σ̅-ι σ (δ # (H.unit.η _ # α))) ⟩
H.mult.η _ # (H.M₁ (σ̅ σ) # (H.M₁ ι₁-hom # (δ # (H.unit.η Δ # α))) ) ∎
H.M₁ ι₁-hom # (σ # (δ # (H.η Δ # α))) ≡˘⟨ ap# (H.M₁ ι₁-hom ∘ σ .morphism) $ H.left-ident #ₚ _ ⟩
H.M₁ ι₁-hom # (σ # (H.μ Δ # (H.M₁ (H.η Δ) # (δ # (H.η Δ # α))))) ≡˘⟨ ap# (H.M₁ ι₁-hom) $ μ-M-∘-Algebraic H σ (H.η Δ) #ₚ _ ⟩
H.M₁ ι₁-hom # (H.μ _ # (H.M₁ (σ .morphism ∘ H.η Δ) # (δ # (H.η Δ # α)))) ≡˘⟨ μ-M-∘-M H ι₁-hom (σ .morphism ∘ H.η Δ) #ₚ _ ⟩
H.μ _ # (H.M₁ (H.M₁ ι₁-hom ∘ σ .morphism ∘ H.η Δ) # (δ # (H.η Δ # α))) ≡⟨ ap# (H.μ _) (σ̅-ι σ (δ # (H.η _ # α))) ⟩
H.μ _ # (H.M₁ (σ̅ σ) # (H.M₁ ι₁-hom # (δ # (H.η Δ # α)))) ∎
(ι₂ α) →
H.unit.η Δ⁺ # ι₂ α ≡˘⟨ H.right-ident #ₚ _ ⟩
H.mult.η Δ⁺ # (H.unit.η (H.M₀ Δ⁺) # (H.unit.η Δ⁺ # ι₂ α)) ≡⟨ ap# (H.mult.η _) (H.unit.is-natural _ _ (σ̅ σ) #ₚ (ι₂ α)) ⟩
H.mult.η Δ⁺ # (H.M₁ (σ̅ σ) # (H.unit.η Δ⁺ # ι₂ α)) ∎

H.η Δ⁺ # ι₂ α ≡˘⟨ μ-η H (σ̅ σ) #ₚ ι₂ α ⟩
H.μ Δ⁺ # (H.M₁ (σ̅ σ) # (H.η Δ⁺ # ι₂ α)) ∎

T′ : (σ : Algebra-hom SOrd H Fᴴ⟨ Δ ⟩ Fᴴ⟨ Δ ⟩) → Algebra-hom SOrd H Fᴴ⟨ Δ⁺ ⟩ Fᴴ⟨ Δ⁺ ⟩
T′ σ .morphism = H.mult.η Δ⁺ ∘ H.M₁ (σ̅ σ)
T′ : (σ : SOrdᴴ.Hom (Fᴴ₀ Δ) (Fᴴ₀ Δ)) → SOrdᴴ.Hom (Fᴴ₀ Δ⁺) (Fᴴ₀ Δ⁺)
T′ σ .morphism = H.μ Δ⁺ ∘ H.M₁ (σ̅ σ)
T′ σ .commutes = ext λ α →
H.mult.η Δ⁺ # (H.M₁ (σ̅ σ) # (H.mult.η Δ⁺ # α)) ≡˘⟨ ap# (H.mult.η _) (H.mult.is-natural _ _ (σ̅ σ) #ₚ α) ⟩
H.mult.η Δ⁺ # (H.mult.η (H.M₀ Δ⁺) # (H.M₁ (H.M₁ (σ̅ σ)) # α)) ≡˘⟨ H.mult-assoc #ₚ ((H.M₁ (H.M₁ (σ̅ σ))) # α) ⟩
H.mult.η Δ⁺ # (H.M₁ (H.mult.η Δ⁺) # (H.M₁ (H.M₁ (σ̅ σ)) # α)) ≡˘⟨ ap# (H.mult.η _) (H.M-∘ (H.mult.η _) (H.M₁ (σ̅ σ)) #ₚ α) ⟩
H.mult.η Δ⁺ # (H.M₁ (H.mult.η Δ⁺ ∘ H.M₁ (σ̅ σ)) # α) ∎


T : Functor (Endos SOrdᴴ Fᴴ⟨ Δ ⟩) (Endos SOrdᴹᴰ Fᴹᴰ⟨ Disc Ψ ⟩)
T = functor
where
functor : Functor (Endos SOrdᴴ Fᴴ⟨ Δ ⟩) (Endos SOrdᴹᴰ Fᴹᴰ⟨ Disc Ψ ⟩)
functor .Functor.F₀ _ = tt
functor .Functor.F₁ σ .morphism .hom (α , d) = α , (T′ σ SOrdᴴ.∘ d)
functor .Functor.F₁ σ .morphism .pres-≤[]-equal {α , d1} {β , d2} p =
let d1≤d2 , injr = H⟨Δ⁺⟩→.left-strict-invariant {T′ σ} {d1} {d2} (⋉-snd-invariant p) in
inc (biased (⋉-fst-invariant p) d1≤d2) , λ q i → q i .fst , injr (ap snd q) i
functor .Functor.F₁ σ .commutes = trivial!
functor .Functor.F-id = ext λ α d →
refl , λ β →
H.mult.η _ # (H.M₁ (σ̅ SOrdᴴ.id) # (d # β)) ≡⟨ ap (λ e → H.mult.η _ # (H.M₁ e # (d # β))) σ̅-id ⟩
H.mult.η _ # (H.M₁ (H.unit.η _) # (d # β)) ≡⟨ H.left-ident #ₚ _ ⟩
d # β ∎
functor .Functor.F-∘ σ δ = ext λ α d →
refl , λ β →
H.mult.η _ # (H.M₁ (σ̅ (σ SOrdᴴ.∘ δ)) # (d # β)) ≡⟨ ap (λ e → H.mult.η _ # (H.M₁ e # (d # β))) (σ̅-∘ σ δ) ⟩
H.mult.η _ # (H.M₁ (H.mult.η _ ∘ H.M₁ (σ̅ σ) ∘ σ̅ δ) # (d # β)) ≡⟨ ap (λ e → H.mult.η _ # (e # (d # β))) (H.M-∘ _ _) ⟩
H.mult.η _ # ((H.M₁ (H.mult.η _) ∘ H.M₁ (H.M₁ (σ̅ σ) ∘ σ̅ δ)) # (d # β)) ≡⟨ ap (λ e → H.mult.η _ # ((H.M₁ (H.mult.η _) ∘ e) # (d # β))) (H.M-∘ _ _) ⟩
H.mult.η _ # (H.M₁ (H.mult.η _) # (H.M₁ (H.M₁ (σ̅ σ)) # (H.M₁ (σ̅ δ) # (d # β)))) ≡⟨ H.mult-assoc #ₚ _ ⟩
H.mult.η _ # (H.mult.η _ # (H.M₁ (H.M₁ (σ̅ σ)) # (H.M₁ (σ̅ δ) # (d # β)))) ≡⟨ ap# (H.mult.η _) (H.mult.is-natural _ _ (σ̅ σ) #ₚ _) ⟩
H.mult.η _ # (H.M₁ (σ̅ σ) # (H.mult.η _ # (H.M₁ (σ̅ δ) # (d # β)))) ∎
H.μ Δ⁺ # (H.M₁ (σ̅ σ) # (H.μ Δ⁺ # α)) ≡˘⟨ ap# (H.μ _) $ H.mult.is-natural _ _ (σ̅ σ) #ₚ α ⟩
H.μ Δ⁺ # (H.μ (H.M₀ Δ⁺) # (H.M₁ (H.M₁ (σ̅ σ)) # α)) ≡˘⟨ μ-M-∘-μ H (H.M₁ (σ̅ σ)) #ₚ α ⟩
H.μ Δ⁺ # (H.M₁ (H.μ Δ⁺ ∘ H.M₁ (σ̅ σ)) # α) ∎

T : Functor (Endos SOrdᴴ (Fᴴ₀ Δ)) (Endos SOrdᴹᴰ (Fᴹᴰ₀ (Disc Ψ)))
T .Functor.F₀ _ = tt
T .Functor.F₁ σ .morphism .hom (α , d) = α , (T′ σ SOrdᴴ.∘ d)
T .Functor.F₁ σ .morphism .pres-≤[]-equal {α , d1} {β , d2} p =
let d1≤d2 , injr = H⟨Δ⁺⟩→.left-strict-invariant {T′ σ} {d1} {d2} (⋉-snd-invariant p) in
inc (biased (⋉-fst-invariant p) d1≤d2) , λ q i → q i .fst , injr (ap snd q) i
T .Functor.F₁ σ .commutes = trivial!
T .Functor.F-id = ext λ α d →
refl , λ β →
H.μ _ # (H.M₁ (σ̅ SOrdᴴ.id) # (d # β)) ≡⟨ ap (λ m → H.μ _ # (H.M₁ m # (d # β))) σ̅-id ⟩
H.μ _ # (H.M₁ (H.η _) # (d # β)) ≡⟨ H.left-ident #ₚ _ ⟩
d # β ∎
T .Functor.F-∘ σ δ = ext λ α d →
refl , λ β →
H.μ _ # (H.M₁ (σ̅ (σ SOrdᴴ.∘ δ)) # (d # β)) ≡⟨ ap (λ m → H.μ _ # (H.M₁ m # (d # β))) (σ̅-∘ σ δ) ⟩
H.μ _ # (H.M₁ (H.μ _ ∘ H.M₁ (σ̅ σ) ∘ σ̅ δ) # (d # β)) ≡⟨ μ-M-∘-μ H (H.M₁ (σ̅ σ) ∘ σ̅ δ) #ₚ (d # β) ⟩
H.μ _ # (H.μ _ # (H.M₁ (H.M₁ (σ̅ σ) ∘ σ̅ δ) # (d # β))) ≡⟨ ap# (H.μ _) $ μ-M-∘-M H (σ̅ σ) (σ̅ δ) #ₚ (d # β) ⟩
H.μ _ # (H.M₁ (σ̅ σ) # (H.μ _ # (H.M₁ (σ̅ δ) # (d # β)))) ∎

--------------------------------------------------------------------------------
-- Faithfulness of T
Expand All @@ -211,9 +206,7 @@ module Mugen.Cat.HierarchyTheory.Universality.EndomorphismEmbedding
T-faithful : ∣ Ψ ∣ → preserves-monos H → is-faithful T
T-faithful pt H-preserves-monos {x} {y} {σ} {δ} p =
free-algebra-hom-path H $ H-preserves-monos ι₁-hom ι₁-monic _ _ $ ext λ α →
σ̅ σ # (ι₁ α) ≡˘⟨ H.right-ident #ₚ _ ⟩
H.mult.η _ # (H.unit.η _ # (σ̅ σ # (ι₁ α))) ≡⟨ ap# (H.mult.η _) (H.unit.is-natural _ _ (σ̅ σ) #ₚ _) ⟩
H.mult.η _ # (H.M₁ (σ̅ σ) # (H.unit.η _ # ι₁ α)) ≡⟨ ap snd (p #ₚ (pt , SOrdᴴ.id)) #ₚ _ ⟩
H.mult.η _ # (H.M₁ (σ̅ δ) # (H.unit.η _ # ι₁ α)) ≡˘⟨ ap# (H.mult.η _) (H.unit.is-natural _ _ (σ̅ δ) #ₚ _) ⟩
H.mult.η _ # (H.unit.η _ # (σ̅ δ # (ι₁ α))) ≡⟨ H.right-ident #ₚ _ ⟩
σ̅ δ # (ι₁ α) ∎
σ̅ σ # ι₁ α ≡˘⟨ μ-η H (σ̅ σ) #ₚ ι₁ α ⟩
H.μ _ # (H.M₁ (σ̅ σ) # (H.η _ # ι₁ α)) ≡⟨ ap snd (p #ₚ (pt , SOrdᴴ.id)) #ₚ _ ⟩
H.μ _ # (H.M₁ (σ̅ δ) # (H.η _ # ι₁ α)) ≡⟨ μ-η H (σ̅ δ) #ₚ ι₁ α ⟩
σ̅ δ # ι₁ α ∎
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