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🎉 Finished proof that delay is a monad
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@ -54,7 +54,8 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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open Equiv
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open HomReasoning
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open CoLambek
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open F-Coalgebra-Morphism
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open F-Coalgebra-Morphism renaming (f to u)
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open F-Coalgebra
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```
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### *Proposition 1*: Characterization of the delay monad ***D***
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@ -92,16 +93,15 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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later : DX ⇒ DX
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later = out⁻¹ ∘ i₂
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-- TODO inline
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unitlaw : out ∘ now ≈ i₁
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unitlaw = cancelˡ (_≅_.isoʳ out-≅)
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module _ {Y : Obj} where
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coit : Y ⇒ X + Y → Y ⇒ DX
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coit f = F-Coalgebra-Morphism.f (! {A = record { A = Y ; α = f }})
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coit f = u (! {A = record { A = Y ; α = f }})
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coit-commutes : ∀ (f : Y ⇒ X + Y) → out ∘ (coit f) ≈ (idC +₁ coit f) ∘ f
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coit-commutes f = F-Coalgebra-Morphism.commutes (! {A = record { A = Y ; α = f }})
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coit-commutes f = commutes (! {A = record { A = Y ; α = f }})
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module _ (ℕ : ParametrizedNNO) where
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open ParametrizedNNO ℕ
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@ -110,7 +110,7 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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iso = Lambek.lambek (record { ⊥ = PNNO-Algebra CC coproducts X N z s ; ⊥-is-initial = PNNO⇒Initial₂ CC coproducts ℕ X })
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ι : X × N ⇒ DX
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ι = f (! {A = record { A = X × N ; α = _≅_.from iso }})
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ι = u (! {A = record { A = X × N ; α = _≅_.from iso }})
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monad : Monad C
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@ -118,164 +118,153 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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{ F₀ = D₀
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; unit = λ {X} → now X
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; extend = extend
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; identityʳ = λ {X} {Y} {f} → begin
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extend f ∘ now X ≈⟨ (insertˡ (_≅_.isoˡ (out-≅ Y))) ⟩∘⟨refl ⟩
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(out⁻¹ Y ∘ out Y ∘ extend f) ∘ now X ≈⟨ (refl⟩∘⟨ (extendlaw f)) ⟩∘⟨refl ⟩
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(out⁻¹ Y ∘ [ out Y ∘ f , i₂ ∘ extend f ] ∘ out X) ∘ now X ≈⟨ pullʳ (pullʳ (unitlaw X)) ⟩
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out⁻¹ Y ∘ [ out Y ∘ f , i₂ ∘ extend f ] ∘ i₁ ≈⟨ refl⟩∘⟨ inject₁ ⟩
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out⁻¹ Y ∘ out Y ∘ f ≈⟨ cancelˡ (_≅_.isoˡ (out-≅ Y)) ⟩
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f ∎
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; identityˡ = λ {X} → Terminal.⊤-id (algebras X) (record { f = extend (now X) ; commutes = begin
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out X ∘ extend (now X) ≈⟨ pullˡ ((F-Coalgebra-Morphism.commutes (Terminal.! (algebras X) {A = alg (now X)}))) ⟩
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((idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ F-Coalgebra.α (alg (now X))) ∘ i₁ ≈⟨ pullʳ inject₁ ⟩
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(idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)})))
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∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out X ∘ (now X)) , i₂ ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ []-cong₂ ((refl⟩∘⟨ (unitlaw X)) ○ inject₁) refl ⟩∘⟨refl ⟩
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(idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ [ i₁ , i₂ ∘ i₁ ] ∘ out X ≈⟨ pullˡ ∘[] ⟩
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[ (idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ i₁
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, (idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ i₂ ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
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[ i₁ ∘ idC , (i₂ ∘ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ refl assoc) ⟩∘⟨refl ⟩
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[ i₁ ∘ idC , i₂ ∘ (extend (now X)) ] ∘ out X ≈˘⟨ []∘+₁ ⟩∘⟨refl ⟩
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([ i₁ , i₂ ] ∘ (idC +₁ extend (now X))) ∘ out X ≈⟨ (elimˡ +-η) ⟩∘⟨refl ⟩
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(idC +₁ extend (now X)) ∘ out X ∎ })
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; assoc = λ {X} {Y} {Z} {g} {h} → {! !}
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-- begin
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-- extend (extend h ∘ g) ≈⟨ insertˡ (_≅_.isoˡ (out-≅ Z)) ⟩
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-- out⁻¹ Z ∘ out Z ∘ extend (extend h ∘ g) ≈⟨ refl⟩∘⟨ (pullˡ (commutes (! (algebras Z)))) ⟩
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-- out⁻¹ Z ∘ ((idC +₁ (f (! (algebras Z)))) ∘ F-Coalgebra.α (alg (extend h ∘ g))) ∘ i₁ ≈⟨ refl⟩∘⟨ (pullʳ inject₁) ⟩
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-- out⁻¹ Z ∘ (idC +₁ (f (! (algebras Z)))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ extend h ∘ g , i₂ ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ pullˡ ∘[] ⟩
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-- out⁻¹ Z ∘ [ (idC +₁ (f (! (algebras Z)))) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ extend h ∘ g , (idC +₁ (f (! (algebras Z)))) ∘ i₂ ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ [ (idC +₁ (f (! (algebras Z)))) ∘ i₁ , (idC +₁ (f (! (algebras Z)))) ∘ i₂ ∘ i₂ ] ∘ out Z ∘ extend h ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (([]-cong₂ (+₁∘i₁ ○ identityʳ) (pullˡ +₁∘i₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ [ i₁ , (i₂ ∘ (f (! (algebras Z)))) ∘ i₂ ] ∘ out Z ∘ extend h ∘ g
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-- , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁
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-- ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (refl⟩∘⟨ (pullˡ (pullˡ (commutes (! (algebras Z)))))) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ [ i₁ , (i₂ ∘ (f (! (algebras Z)))) ∘ i₂ ] ∘ (((idC +₁ f (! (algebras Z))) ∘ F-Coalgebra.α (alg h)) ∘ i₁) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (refl⟩∘⟨ ((pullʳ inject₁) ⟩∘⟨refl)) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ [ i₁ , (i₂ ∘ (f (! (algebras Z)))) ∘ i₂ ] ∘ ((idC +₁ f (! (algebras Z))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] ∘ out Y) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (pullˡ (pullˡ []∘+₁)) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ ([ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ ([ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] ∘ out Y)) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ ((pullˡ ∘[]) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ ([ [ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , [ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ i₂ ∘ i₁ ] ∘ out Y) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ ((([]-cong₂ (pullˡ ∘[]) (pullˡ inject₂)) ⟩∘⟨refl) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ ([ [ [ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ i₁ , [ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ i₂ ∘ i₂ ] ∘ out Z ∘ h , (((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z))) ∘ i₁ ] ∘ out Y) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ ((([]-cong₂ (([]-cong₂ inject₁ (pullˡ inject₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ ([ [ i₁ ∘ idC , (((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , (((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z))) ∘ i₁ ] ∘ out Y) ∘ g
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-- , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁
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-- ] ∘ out X ≈⟨ {! !} ⟩
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-- {! !} ≈⟨ {! !} ⟩
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-- {! !} ≈˘⟨ {! _○_ !} ⟩
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-- {! !} ≈˘⟨ {! !} ⟩
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-- {! !} ≈˘⟨ {! !} ⟩
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-- out⁻¹ Z ∘ [ [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , (((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y))) ∘ i₂ ] ∘ out Y ∘ g
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-- , (((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y))) ∘ i₁
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-- ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ (([]-cong₂ inject₁ (pullˡ inject₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ [ [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ i₁ , [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ i₂ ∘ i₂ ] ∘ out Y ∘ g
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-- , (((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y))) ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ (pullˡ ∘[]) (pullˡ inject₂)) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g
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-- , [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩
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-- out⁻¹ Z ∘ [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ (([]-cong₂ (+₁∘i₁ ○ identityʳ) (pullˡ +₁∘i₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ [ (idC +₁ f (! (algebras Z))) ∘ i₁ , (idC +₁ f (! (algebras Z))) ∘ i₂ ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ ((refl⟩∘⟨ identityʳ) ○ (pullˡ ∘[])) (pullˡ (pullˡ +₁∘i₂))) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ (idC +₁ f (! (algebras Z))) ∘ ([ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h) ∘ idC , (idC +₁ f (! (algebras Z))) ∘ (i₂ ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩
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-- out⁻¹ Z ∘ (idC +₁ f (! (algebras Z))) ∘ [ ([ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h) ∘ idC , (i₂ ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (refl⟩∘⟨ (pullˡ []∘+₁)) ⟩
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-- out⁻¹ Z ∘ (idC +₁ f (! (algebras Z))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ]
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-- ∘ (idC +₁ f (! (algebras Y))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ identityˡ)) ⟩
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-- out⁻¹ Z ∘ (idC +₁ f (! (algebras Z))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] ∘ idC
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-- ∘ (idC +₁ f (! (algebras Y))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ pullʳ (pullʳ (pullʳ (pullˡ (_≅_.isoʳ (out-≅ Y))))) ⟩
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-- (out⁻¹ Z ∘ (idC +₁ f (! (algebras Z))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] ∘ out Y)
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-- ∘ (out⁻¹ Y ∘ (idC +₁ f (! (algebras Y))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X) ≈˘⟨ (refl⟩∘⟨ (pullʳ inject₁)) ⟩∘⟨ (refl⟩∘⟨ (pullʳ inject₁)) ⟩
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-- (out⁻¹ Z ∘ ((idC +₁ (f (! (algebras Z)))) ∘ F-Coalgebra.α (alg h)) ∘ i₁) ∘ (out⁻¹ Y ∘ ((idC +₁ (f (! (algebras Y)))) ∘ F-Coalgebra.α (alg g)) ∘ i₁) ≈˘⟨ (refl⟩∘⟨ (pullˡ (commutes (! (algebras Z))))) ⟩∘⟨ refl⟩∘⟨ (pullˡ (commutes (! (algebras Y)))) ⟩
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-- (out⁻¹ Z ∘ out Z ∘ extend h) ∘ (out⁻¹ Y ∘ out Y ∘ extend g) ≈˘⟨ ((insertˡ (_≅_.isoˡ (out-≅ Z)))) ⟩∘⟨ ((insertˡ (_≅_.isoˡ (out-≅ Y)))) ⟩
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-- extend h ∘ extend g ∎
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-- begin
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-- extend (extend h ∘ g) ≈⟨ (insertˡ (_≅_.isoˡ (out-≅ Z))) ⟩
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-- out⁻¹ Z ∘ out Z ∘ extend (extend h ∘ g) ≈⟨ refl⟩∘⟨ extendlaw (extend h ∘ g) ⟩
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-- out⁻¹ Z ∘ [ out Z ∘ extend h ∘ g , i₂ ∘ extend (extend h ∘ g) ] ∘ out X ≈⟨ {! !} ⟩
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-- {! !} ≈⟨ {! !} ⟩
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-- {! !} ≈⟨ {! !} ⟩
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-- {! !} ≈⟨ {! !} ⟩
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-- {! !} ≈˘⟨ {! !} ⟩
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-- {! !} ≈˘⟨ {! !} ⟩
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-- out⁻¹ Z ∘ [ [ out Z ∘ h , i₂ ∘ extend h ] ∘ out Y ∘ g , (i₂ ∘ extend h) ∘ extend g ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ refl (pullˡ inject₂)) ⟩∘⟨refl) ⟩
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-- out⁻¹ Z ∘ [ [ out Z ∘ h , i₂ ∘ extend h ] ∘ out Y ∘ g , [ out Z ∘ h , i₂ ∘ extend h ] ∘ i₂ ∘ extend g ] ∘ out X ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩
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-- out⁻¹ Z ∘ [ out Z ∘ h , i₂ ∘ extend h ] ∘ [ out Y ∘ g , i₂ ∘ extend g ] ∘ out X ≈˘⟨ refl⟩∘⟨ (refl⟩∘⟨ identityˡ) ⟩
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-- out⁻¹ Z ∘ [ out Z ∘ h , i₂ ∘ extend h ] ∘ idC ∘ [ out Y ∘ g , i₂ ∘ extend g ] ∘ out X ≈˘⟨ pullʳ (pullʳ (pullˡ (_≅_.isoʳ (out-≅ Y)))) ⟩
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-- (out⁻¹ Z ∘ [ out Z ∘ h , i₂ ∘ extend h ] ∘ out Y) ∘ out⁻¹ Y ∘ [ out Y ∘ g , i₂ ∘ extend g ] ∘ out X ≈˘⟨ (refl⟩∘⟨ extendlaw h) ⟩∘⟨ (refl⟩∘⟨ extendlaw g) ⟩
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-- (out⁻¹ Z ∘ out Z ∘ extend h) ∘ (out⁻¹ Y ∘ out Y ∘ extend g) ≈˘⟨ ((insertˡ (_≅_.isoˡ (out-≅ Z)))) ⟩∘⟨ ((insertˡ (_≅_.isoˡ (out-≅ Y)))) ⟩
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-- extend h ∘ extend g ∎
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--begin
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-- f (! (algebras Z) {A = alg (extend h ∘ g)}) ∘ i₁ {A = D₀ X} {B = D₀ Z} ≈⟨ (!-unique (algebras Z) (record { f = {! (f (! (algebras Z) {A = alg h}) ∘ i₁) ∘ f (! (algebras Y) {A = alg g}) !} ; commutes = {! !} })) ⟩∘⟨refl ⟩
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-- {! !} ∘ i₁ ≈⟨ {! !} ⟩
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-- (f (! (algebras Z) {A = alg h}) ∘ i₁) ∘ f (! (algebras Y) {A = alg g}) ∘ i₁ ∎
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; sym-assoc = λ {X} {Y} {Z} {g} {h} → {! !}
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; extend-≈ = λ {X} {Y} {f} {g} eq → begin
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F-Coalgebra-Morphism.f (Terminal.! (algebras Y) {A = alg f }) ∘ i₁ {B = D₀ Y} ≈⟨ (Terminal.!-unique (algebras Y) (record { f = (F-Coalgebra-Morphism.f (Terminal.! (algebras Y) {A = alg g }) ∘ idC) ; commutes = begin
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F-Coalgebra.α (Terminal.⊤ (algebras Y)) ∘ F-Coalgebra-Morphism.f (Terminal.! (algebras Y)) ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩
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F-Coalgebra.α (Terminal.⊤ (algebras Y)) ∘ F-Coalgebra-Morphism.f (Terminal.! (algebras Y)) ≈⟨ F-Coalgebra-Morphism.commutes (Terminal.! (algebras Y)) ⟩
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Functor.F₁ (delayF Y) (F-Coalgebra-Morphism.f (Terminal.! (algebras Y))) ∘ F-Coalgebra.α (alg g) ≈˘⟨ (Functor.F-resp-≈ (delayF Y) identityʳ) ⟩∘⟨ (αf≈αg eq) ⟩
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Functor.F₁ (delayF Y) (F-Coalgebra-Morphism.f (Terminal.! (algebras Y)) ∘ idC) ∘ F-Coalgebra.α (alg f) ∎ })) ⟩∘⟨refl ⟩
|
||||
(F-Coalgebra-Morphism.f (Terminal.! (algebras Y) {A = alg g }) ∘ idC) ∘ i₁ {B = D₀ Y} ≈⟨ identityʳ ⟩∘⟨refl ⟩
|
||||
extend g ∎
|
||||
; identityʳ = identityʳ'
|
||||
; identityˡ = identityˡ'
|
||||
; assoc = assoc'
|
||||
; sym-assoc = sym assoc'
|
||||
; extend-≈ = extend-≈'
|
||||
})
|
||||
where
|
||||
open Terminal
|
||||
alg' : ∀ {X Y} → F-Coalgebra (delayF Y)
|
||||
alg' {X} {Y} = record { A = D₀ X ; α = i₂ }
|
||||
module _ {X Y : Obj} (f : X ⇒ D₀ Y) where
|
||||
-- open Terminal (algebras Y) using (!; ⊤-id)
|
||||
alg : F-Coalgebra (delayF Y)
|
||||
alg = record { A = D₀ X + D₀ Y ; α = [ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X , (idC +₁ i₂) ∘ out Y ] } -- (idC +₁ (idC +₁ [ idC , idC ]) ∘ _≅_.to +-assoc ∘ _≅_.to +-comm)
|
||||
alg = record { A = D₀ X + D₀ Y ; α = [ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X , (idC +₁ i₂) ∘ out Y ] }
|
||||
|
||||
extend : D₀ X ⇒ D₀ Y
|
||||
extend = F-Coalgebra-Morphism.f (! (algebras Y) {A = alg}) ∘ i₁ {B = D₀ Y}
|
||||
!∘i₂ : F-Coalgebra-Morphism.f (! (algebras Y) {A = alg}) ∘ i₂ ≈ idC
|
||||
extend = u (! (algebras Y) {A = alg}) ∘ i₁ {B = D₀ Y}
|
||||
|
||||
!∘i₂ : u (! (algebras Y) {A = alg}) ∘ i₂ ≈ idC
|
||||
!∘i₂ = ⊤-id (algebras Y) (F-Coalgebras (delayF Y) [ ! (algebras Y) ∘ record { f = i₂ ; commutes = inject₂ } ] )
|
||||
|
||||
extendlaw : out Y ∘ extend ≈ [ out Y ∘ f , i₂ ∘ extend ] ∘ out X
|
||||
extendlaw = begin
|
||||
out Y ∘ extend ≈⟨ pullˡ (F-Coalgebra-Morphism.commutes (! (algebras Y) {A = alg})) ⟩
|
||||
((idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ F-Coalgebra.α alg) ∘ coproduct.i₁ ≈⟨ pullʳ inject₁ ⟩
|
||||
(idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X ≈⟨ pullˡ ∘[] ⟩
|
||||
[ (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₂ ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
|
||||
[ [ (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₁
|
||||
, (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (i₂ ∘ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ (([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl ⟩
|
||||
[ [ i₁ ∘ idC , (i₂ ∘ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (i₂ ∘ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ (elimˡ (([]-cong₂ identityʳ (cancelʳ !∘i₂)) ○ +-η)) assoc) ⟩∘⟨refl ⟩
|
||||
out Y ∘ extend ≈⟨ pullˡ (commutes (! (algebras Y) {A = alg})) ⟩
|
||||
((idC +₁ (u (! (algebras Y)))) ∘ α alg) ∘ i₁ ≈⟨ pullʳ inject₁ ⟩
|
||||
(idC +₁ (u (! (algebras Y)))) ∘ [ [ i₁ , i₂ ∘ i₂ ]
|
||||
∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X ≈⟨ pullˡ ∘[] ⟩
|
||||
[ (idC +₁ (u (! (algebras Y)))) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (idC +₁ (u (! (algebras Y)))) ∘ i₂ ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
|
||||
[ [ (idC +₁ (u (! (algebras Y)))) ∘ i₁
|
||||
, (idC +₁ (u (! (algebras Y)))) ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (i₂ ∘ (u (! (algebras Y)))) ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂
|
||||
(([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl)
|
||||
refl) ⟩∘⟨refl ⟩
|
||||
[ [ i₁ ∘ idC , (i₂ ∘ (u (! (algebras Y)))) ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (i₂ ∘ (u (! (algebras Y)))) ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂
|
||||
(elimˡ (([]-cong₂ identityʳ (cancelʳ !∘i₂)) ○ +-η))
|
||||
assoc) ⟩∘⟨refl ⟩
|
||||
[ out Y ∘ f , i₂ ∘ extend ] ∘ out X ∎
|
||||
extend-unique : (g : D₀ X ⇒ D₀ Y) → extend ≈ g
|
||||
extend-unique g = {! !}
|
||||
-- begin
|
||||
-- F-Coalgebra-Morphism.f (! (algebras Y) {A = alg}) ∘ i₁ {B = D₀ Y} ≈⟨ (!-unique (algebras Y) (record { f = [ g , idC ] ; commutes = begin
|
||||
-- out Y ∘ [ g , idC ] ≈⟨ ∘[] ⟩
|
||||
-- [ out Y ∘ g , out Y ∘ idC ] ≈⟨ []-cong₂ {! !} identityʳ ⟩
|
||||
-- {! !} ≈˘⟨ {! !} ⟩
|
||||
-- [ ([ out Y , i₂ ] ∘ (f +₁ g)) ∘ out X , out Y ] ≈˘⟨ []-cong₂ (sym []∘+₁ ⟩∘⟨refl) refl ⟩
|
||||
-- [ [ out Y ∘ f , i₂ ∘ g ] ∘ out X , out Y ] ≈˘⟨ {! !} ⟩
|
||||
-- [ [ [ i₁ , i₂ ∘ idC ] ∘ (out Y ∘ f) , i₂ ∘ g ] ∘ out X , out Y ] ≈˘⟨ []-cong₂ (([]-cong₂ (([]-cong₂ identityʳ (pullʳ inject₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) refl ⟩
|
||||
-- [ [ [ i₁ ∘ idC , (i₂ ∘ [ g , idC ]) ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ g ] ∘ out X , out Y ] ≈˘⟨ []-cong₂ (([]-cong₂ (([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl) (pullʳ inject₁)) ⟩∘⟨refl) (elimˡ (Functor.identity (delayF Y))) ⟩
|
||||
-- [ [ [ (idC +₁ [ g , idC ]) ∘ i₁ , (idC +₁ [ g , idC ]) ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f) , (i₂ ∘ [ g , idC ]) ∘ i₁ ] ∘ out X , (idC +₁ idC) ∘ out Y ] ≈˘⟨ []-cong₂ (([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl) ((+₁-cong₂ identity² inject₂) ⟩∘⟨refl) ⟩
|
||||
-- [ [ (idC +₁ [ g , idC ]) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , (idC +₁ [ g , idC ]) ∘ i₂ ∘ i₁ ] ∘ out X , (idC ∘ idC +₁ [ g , idC ] ∘ i₂) ∘ out Y ] ≈˘⟨ []-cong₂ (pullˡ ∘[]) (pullˡ +₁∘+₁) ⟩
|
||||
-- [ (idC +₁ [ g , idC ]) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X , (idC +₁ [ g , idC ]) ∘ (idC +₁ i₂) ∘ out Y ] ≈˘⟨ ∘[] ⟩
|
||||
-- (idC +₁ [ g , idC ]) ∘ F-Coalgebra.α alg ∎ })) ⟩∘⟨refl ⟩
|
||||
-- [ g , idC ] ∘ i₁ ≈⟨ inject₁ ⟩
|
||||
-- g ∎
|
||||
αf≈αg : ∀ {X Y} {f g : X ⇒ D₀ Y} → f ≈ g → F-Coalgebra.α (alg f) ≈ F-Coalgebra.α (alg g)
|
||||
|
||||
extend-unique : (g : D₀ X ⇒ D₀ Y) → (out Y ∘ g ≈ [ out Y ∘ f , i₂ ∘ g ] ∘ out X) → extend ≈ g
|
||||
extend-unique g g-commutes = begin
|
||||
extend ≈⟨ (!-unique (algebras Y) (record { f = [ g , idC ] ; commutes = begin
|
||||
out Y ∘ [ g , idC ] ≈⟨ ∘[] ⟩
|
||||
[ out Y ∘ g , out Y ∘ idC ] ≈⟨ []-cong₂ g-commutes identityʳ ⟩
|
||||
[ [ out Y ∘ f , i₂ ∘ g ] ∘ out X , out Y ] ≈˘⟨ []-cong₂
|
||||
(([]-cong₂
|
||||
(([]-cong₂ refl identityʳ) ⟩∘⟨refl ○ (elimˡ +-η))
|
||||
refl)
|
||||
⟩∘⟨refl)
|
||||
refl ⟩
|
||||
[ [ [ i₁ , i₂ ∘ idC ] ∘ (out Y ∘ f)
|
||||
, i₂ ∘ g ] ∘ out X
|
||||
, out Y ] ≈˘⟨ []-cong₂
|
||||
(([]-cong₂
|
||||
(([]-cong₂ identityʳ (pullʳ inject₂)) ⟩∘⟨refl)
|
||||
refl)
|
||||
⟩∘⟨refl)
|
||||
refl
|
||||
⟩
|
||||
[ [ [ i₁ ∘ idC
|
||||
, (i₂ ∘ [ g , idC ]) ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, i₂ ∘ g ] ∘ out X
|
||||
, out Y ] ≈˘⟨ []-cong₂
|
||||
(([]-cong₂
|
||||
(([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl)
|
||||
(pullʳ inject₁))
|
||||
⟩∘⟨refl)
|
||||
(elimˡ (Functor.identity (delayF Y)))
|
||||
⟩
|
||||
[ [ [ (idC +₁ [ g , idC ]) ∘ i₁
|
||||
, (idC +₁ [ g , idC ]) ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (i₂ ∘ [ g , idC ]) ∘ i₁ ] ∘ out X
|
||||
, (idC +₁ idC) ∘ out Y ] ≈˘⟨ []-cong₂
|
||||
(([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl)
|
||||
((+₁-cong₂ identity² inject₂) ⟩∘⟨refl) ⟩
|
||||
[ [ (idC +₁ [ g , idC ]) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (idC +₁ [ g , idC ]) ∘ i₂ ∘ i₁ ] ∘ out X
|
||||
, (idC ∘ idC +₁ [ g , idC ] ∘ i₂) ∘ out Y ] ≈˘⟨ []-cong₂ (pullˡ ∘[]) (pullˡ +₁∘+₁) ⟩
|
||||
[ (idC +₁ [ g , idC ]) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X
|
||||
, (idC +₁ [ g , idC ]) ∘ (idC +₁ i₂) ∘ out Y ] ≈˘⟨ ∘[] ⟩
|
||||
(idC +₁ [ g , idC ]) ∘ α alg ∎ })) ⟩∘⟨refl ⟩
|
||||
[ g , idC ] ∘ i₁ ≈⟨ inject₁ ⟩
|
||||
g ∎
|
||||
|
||||
αf≈αg : ∀ {X Y} {f g : X ⇒ D₀ Y} → f ≈ g → α (alg f) ≈ α (alg g)
|
||||
αf≈αg {X} {Y} {f} {g} eq = []-cong₂ ([]-cong₂ (refl⟩∘⟨ refl⟩∘⟨ eq) refl ⟩∘⟨refl) refl
|
||||
|
||||
alg-f≈alg-g : ∀ {X Y} {f g : X ⇒ D₀ Y} → f ≈ g → M._≅_ (F-Coalgebras (delayF Y)) (alg f) (alg g)
|
||||
alg-f≈alg-g {X} {Y} {f} {g} eq = record
|
||||
{ from = record { f = idC ; commutes = begin
|
||||
F-Coalgebra.α (alg g) ∘ idC ≈⟨ identityʳ ⟩
|
||||
F-Coalgebra.α (alg g) ≈⟨ ⟺ (αf≈αg eq) ⟩
|
||||
F-Coalgebra.α (alg f) ≈˘⟨ elimˡ (Functor.identity (delayF Y)) ⟩
|
||||
Functor.F₁ (delayF Y) idC ∘ F-Coalgebra.α (alg f) ∎ }
|
||||
α (alg g) ∘ idC ≈⟨ identityʳ ⟩
|
||||
α (alg g) ≈⟨ ⟺ (αf≈αg eq) ⟩
|
||||
α (alg f) ≈˘⟨ elimˡ identity ⟩
|
||||
(idC +₁ idC) ∘ α (alg f) ∎ }
|
||||
; to = record { f = idC ; commutes = begin
|
||||
F-Coalgebra.α (alg f) ∘ idC ≈⟨ identityʳ ⟩
|
||||
F-Coalgebra.α (alg f) ≈⟨ αf≈αg eq ⟩
|
||||
F-Coalgebra.α (alg g) ≈˘⟨ elimˡ (Functor.identity (delayF Y)) ⟩
|
||||
Functor.F₁ (delayF Y) idC ∘ F-Coalgebra.α (alg g) ∎ }
|
||||
α (alg f) ∘ idC ≈⟨ identityʳ ⟩
|
||||
α (alg f) ≈⟨ αf≈αg eq ⟩
|
||||
α (alg g) ≈˘⟨ elimˡ identity ⟩
|
||||
(idC +₁ idC) ∘ α (alg g) ∎ }
|
||||
; iso = record
|
||||
{ isoˡ = identity²
|
||||
; isoʳ = identity²
|
||||
}
|
||||
}
|
||||
where open Functor (delayF Y) using (identity)
|
||||
|
||||
identityʳ' : ∀ {X} {Y} {f} → extend f ∘ now X ≈ f
|
||||
identityʳ' {X} {Y} {f} = begin
|
||||
extend f ∘ now X ≈⟨ insertˡ (_≅_.isoˡ (out-≅ Y)) ⟩∘⟨refl ⟩
|
||||
(out⁻¹ Y ∘ out Y ∘ extend f) ∘ now X ≈⟨ (refl⟩∘⟨ (extendlaw f)) ⟩∘⟨refl ⟩
|
||||
(out⁻¹ Y ∘ [ out Y ∘ f , i₂ ∘ extend f ] ∘ out X) ∘ now X ≈⟨ pullʳ (pullʳ (unitlaw X)) ⟩
|
||||
out⁻¹ Y ∘ [ out Y ∘ f , i₂ ∘ extend f ] ∘ i₁ ≈⟨ refl⟩∘⟨ inject₁ ⟩
|
||||
out⁻¹ Y ∘ out Y ∘ f ≈⟨ cancelˡ (_≅_.isoˡ (out-≅ Y)) ⟩
|
||||
f ∎
|
||||
|
||||
identityˡ' : ∀ {X} → extend (now X) ≈ idC
|
||||
identityˡ' {X} = Terminal.⊤-id (algebras X) (record { f = extend (now X) ; commutes = begin
|
||||
out X ∘ extend (now X) ≈⟨ pullˡ ((commutes (! (algebras X) {A = alg (now X)}))) ⟩
|
||||
((idC +₁ (u (! (algebras X) {A = alg (now X)}))) ∘ α (alg (now X))) ∘ i₁ ≈⟨ pullʳ inject₁ ⟩
|
||||
(idC +₁ (u (! (algebras X) {A = alg (now X)})))
|
||||
∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out X ∘ (now X)) , i₂ ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ []-cong₂ ((refl⟩∘⟨ unitlaw X) ○ inject₁) refl ⟩∘⟨refl ⟩
|
||||
(idC +₁ (u (! (algebras X) {A = alg (now X)}))) ∘ [ i₁ , i₂ ∘ i₁ ] ∘ out X ≈⟨ pullˡ ∘[] ⟩
|
||||
[ (idC +₁ (u (! (algebras X) {A = alg (now X)}))) ∘ i₁
|
||||
, (idC +₁ (u (! (algebras X) {A = alg (now X)}))) ∘ i₂ ∘ i₁ ] ∘ out X ≈⟨ []-cong₂ +₁∘i₁ (pullˡ +₁∘i₂) ⟩∘⟨refl ⟩
|
||||
[ i₁ ∘ idC , (i₂ ∘ (u (! (algebras X) {A = alg (now X)}))) ∘ i₁ ] ∘ out X ≈⟨ []-cong₂ refl assoc ⟩∘⟨refl ⟩
|
||||
[ i₁ ∘ idC , i₂ ∘ (extend (now X)) ] ∘ out X ≈˘⟨ []∘+₁ ⟩∘⟨refl ⟩
|
||||
([ i₁ , i₂ ] ∘ (idC +₁ extend (now X))) ∘ out X ≈⟨ elimˡ +-η ⟩∘⟨refl ⟩
|
||||
(idC +₁ extend (now X)) ∘ out X ∎ })
|
||||
|
||||
assoc' : ∀ {X} {Y} {Z} {g} {h} → extend (extend h ∘ g) ≈ extend h ∘ extend g
|
||||
assoc' {X} {Y} {Z} {g} {h} = extend-unique (extend h ∘ g) (extend h ∘ extend g) (begin
|
||||
out Z ∘ extend h ∘ extend g ≈⟨ pullˡ (extendlaw h) ⟩
|
||||
([ out Z ∘ h , i₂ ∘ extend h ] ∘ out Y) ∘ extend g ≈⟨ pullʳ (extendlaw g) ⟩
|
||||
[ out Z ∘ h , i₂ ∘ extend h ] ∘ [ out Y ∘ g , i₂ ∘ extend g ] ∘ out X ≈⟨ pullˡ ∘[] ⟩
|
||||
[ [ out Z ∘ h , i₂ ∘ extend h ] ∘ out Y ∘ g
|
||||
, [ out Z ∘ h , i₂ ∘ extend h ] ∘ i₂ ∘ extend g ] ∘ out X ≈⟨ []-cong₂ (pullˡ (⟺ (extendlaw h))) (pullˡ inject₂) ⟩∘⟨refl ⟩
|
||||
[ (out Z ∘ extend h) ∘ g , (i₂ ∘ extend h) ∘ extend g ] ∘ out X ≈⟨ ([]-cong₂ assoc assoc) ⟩∘⟨refl ⟩
|
||||
[ out Z ∘ extend h ∘ g , i₂ ∘ extend h ∘ extend g ] ∘ out X ∎)
|
||||
|
||||
extend-≈' : ∀ {X} {Y} {f g : X ⇒ D₀ Y} → f ≈ g → extend f ≈ extend g
|
||||
extend-≈' {X} {Y} {f} {g} eq = begin
|
||||
u (! (algebras Y) {A = alg f }) ∘ i₁ {B = D₀ Y} ≈⟨ (!-unique (algebras Y) (record { f = (u (! (algebras Y) {A = alg g }) ∘ idC) ; commutes = begin
|
||||
α (⊤ (algebras Y)) ∘ u (! (algebras Y)) ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩
|
||||
α (⊤ (algebras Y)) ∘ u (! (algebras Y)) ≈⟨ commutes (! (algebras Y)) ⟩
|
||||
(idC +₁ u (! (algebras Y))) ∘ α (alg g) ≈˘⟨ Functor.F-resp-≈ (delayF Y) identityʳ ⟩∘⟨ αf≈αg eq ⟩
|
||||
(idC +₁ u (! (algebras Y)) ∘ idC) ∘ α (alg f) ∎ }))
|
||||
⟩∘⟨refl ⟩
|
||||
(u (! (algebras Y) {A = alg g }) ∘ idC) ∘ i₁ {B = D₀ Y} ≈⟨ identityʳ ⟩∘⟨refl ⟩
|
||||
extend g ∎
|
||||
```
|
||||
|
||||
### Definition 30: Search-Algebras
|
||||
|
|
|
|||
Loading…
Reference in a new issue