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Leon Vatthauer 2023-07-25 17:23:36 +02:00
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13
ElgotAlgebra.agda
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@ -37,7 +37,9 @@ module _ (C𝒞 : CocartesianCategory o ℓ e) where
→ f # ≈ g # ∘ h
#-Compositionality : ∀ {X Y} {f : X ⇒ A + F₀ X} {h : Y ⇒ X + F₀ Y}
→ (((f #) +₁ idC) ∘ h)# ≈ ([ (idC +₁ (F₁ i₁)) ∘ f , i₂ ∘ (F₁ i₂) ] ∘ [ i₁ , h ])# ∘ i₂
#-resp-≈ : ∀ {X} {f g : X ⇒ A + F₀ X} → f ≈ g → (f #) ≈ (g #)
#-resp-≈ : ∀ {X} {f g : X ⇒ A + F₀ X}
→ f ≈ g
→ (f #) ≈ (g #)
--*
-- (unguarded) Elgot-Algebra
@ -100,7 +102,8 @@ module _ (C𝒞 : CocartesianCategory o ℓ e) where
[ (idC +₁ i₁) ∘ f , ([ (idC +₁ i₁) ∘ f , i₂ ∘ ([ i₁ , h ] ∘ i₂) ]) ∘ h ] ≈⟨ []-congˡ (∘-resp-≈ˡ ([]-congˡ (∘-resp-≈ʳ inject₂))) ⟩
[ (idC +₁ i₁) ∘ f , [ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ h ] ≈⟨ []-congʳ (sym (inject₁)) ⟩
[ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ i₁ , [ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ h ] ≈⟨ sym ∘[] ⟩
[ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ [ i₁ , h ] ∎))) ⟩
[ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ [ i₁ , h ] ∎))
)⟩
([ (idC +₁ i₁) ∘ f , i₂ ∘ i₂ ] ∘ [ i₁ , h ])# ∘ i₂ ∎
-- every elgot-algebra comes with a divergence constant
@ -185,7 +188,8 @@ module _ (C𝒞 : CocartesianCategory o ℓ e) where
(idC +₁ h) ∘ ((f #) +₁ idC) ∘ h ≈⟨ sym-assoc ⟩
(((idC +₁ h) ∘ ((f #) +₁ idC)) ∘ h) ≈⟨ ∘-resp-≈ˡ +₁∘+₁ ⟩
(((idC ∘ (f #)) +₁ (h ∘ idC)) ∘ h) ≈⟨ ∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ) ⟩
(f # +₁ h) ∘ h ∎)) ⟩
(f # +₁ h) ∘ h ∎)
)⟩
((((f #) +₁ idC) ∘ h) #) ≈⟨ #-Compositionality ⟩
(([ (idC +₁ i₁) ∘ f , i₂ ∘ i₂ ] ∘ [ i₁ , h ])# ∘ i₂) ≈⟨ ∘-resp-≈ˡ (#-Uniformity {f = [ (idC +₁ i₁) ∘ f , i₂ ∘ i₂ ] ∘ [ i₁ , h ]} {g = [ (idC +₁ i₁) ∘ f , i₂ ∘ h ]} {h = [ i₁ , h ]} (
begin
@ -200,7 +204,8 @@ module _ (C𝒞 : CocartesianCategory o ℓ e) where
[ (idC +₁ i₁) ∘ f , ([ (idC +₁ i₁) ∘ f , i₂ ∘ ([ i₁ , h ] ∘ i₂) ]) ∘ h ] ≈⟨ []-congˡ (∘-resp-≈ˡ ([]-congˡ (∘-resp-≈ʳ inject₂))) ⟩
[ (idC +₁ i₁) ∘ f , [ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ h ] ≈⟨ []-congʳ (sym (inject₁)) ⟩
[ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ i₁ , [ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ h ] ≈⟨ sym ∘[] ⟩
[ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ [ i₁ , h ] ∎)) ⟩
[ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ [ i₁ , h ] ∎)
)⟩
(([ (idC +₁ i₁) ∘ f , i₂ ∘ h ] # ∘ [ i₁ , h ]) ∘ i₂) ≈⟨ assoc ⟩
([ (idC +₁ i₁) ∘ f , i₂ ∘ h ] # ∘ ([ i₁ , h ] ∘ i₂)) ≈⟨ (∘-resp-≈ʳ $ inject₂) ⟩
([ (idC +₁ i₁) ∘ f , i₂ ∘ h ] # ∘ h) ≈⟨ sym $ ∘-resp-≈ˡ inject₂ ⟩

3
ElgotAlgebras.agda
View file

@ -126,7 +126,8 @@ module _ (CC : CocartesianCategory o ℓ e) where
[ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ≈⟨ []-cong₂ identityˡ identityʳ ⟩
[ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ] ≈⟨ sym ([]-cong₂ identityʳ project₂) ⟩
[ π₂ ∘ idC , π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ≈⟨ sym ∘[] ⟩
π₂ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∎) )⟩
π₂ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∎)
)⟩
([ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∘ f) ∎
; #-Uniformity = λ {X Y f g h} uni → unique′ (
begin