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Finished extend congruence proof
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src/FinalCoalgebras.lagda.md
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41
src/FinalCoalgebras.lagda.md
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```agda
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open import Level
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open import Categories.Category
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open import Categories.Functor
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import Categories.Morphism as M
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import Categories.Morphism.Reasoning as MR
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module FinalCoalgebras {o ℓ e} {C : Category o ℓ e} (F : Endofunctor C) where
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open Category C renaming (id to idC)
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open import Categories.Object.Terminal
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open import Categories.Functor.Coalgebra
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open import Categories.Category.Construction.F-Coalgebras
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open Functor F
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open MR C
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open HomReasoning
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open Equiv
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lemma : ∀ {X} → {f g : X ⇒ F₀ X} → f ≈ g → (terminal : Terminal (F-Coalgebras F)) → F-Coalgebra-Morphism.f (Terminal.! terminal {A = (to-Coalgebra f)}) ≈ F-Coalgebra-Morphism.f (Terminal.! terminal {A = (to-Coalgebra g)})
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lemma {X} {f} {g} eq terminal = begin
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F-Coalgebra-Morphism.f (Terminal.! terminal) ≈⟨ Terminal.!-unique terminal (record
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{ f = F-Coalgebra-Morphism.f (Terminal.! terminal {A = (to-Coalgebra g)}) ∘ F-Coalgebra-Morphism.f from
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; commutes = begin
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F-Coalgebra.α (Terminal.⊤ terminal) ∘ F-Coalgebra-Morphism.f (IsTerminal.! (Terminal.⊤-is-terminal terminal)) ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ○ F-Coalgebra-Morphism.commutes (Terminal.! terminal) ⟩
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F₁ (F-Coalgebra-Morphism.f (Terminal.! terminal)) ∘ g ≈⟨ refl⟩∘⟨ ⟺ eq ⟩
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F₁ (F-Coalgebra-Morphism.f (Terminal.! terminal)) ∘ f ≈˘⟨ F-resp-≈ identityʳ ⟩∘⟨refl ⟩
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F₁ (F-Coalgebra-Morphism.f (Terminal.! terminal) ∘ idC) ∘ f ∎
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}) ⟩
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F-Coalgebra-Morphism.f (Terminal.! terminal) ∘ idC ≈⟨ identityʳ ⟩
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F-Coalgebra-Morphism.f (Terminal.! terminal) ∎
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where
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eq' : M._≅_ (F-Coalgebras F) (to-Coalgebra f) (to-Coalgebra g)
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eq' = record
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{ from = record { f = idC ; commutes = id-comm ○ ⟺ identity ⟩∘⟨ ⟺ eq }
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; to = record { f = idC ; commutes = id-comm ○ ⟺ identity ⟩∘⟨ eq }
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; iso = record
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{ isoˡ = identity²
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; isoʳ = identity²
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}
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}
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open M._≅_ eq'
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```
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@ -20,6 +20,7 @@ open import Categories.Functor.Algebra
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open import Categories.Monad.Construction.Kleisli
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open import Categories.Category.Construction.F-Coalgebras
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open import Categories.NaturalTransformation
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open import FinalCoalgebras
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import Categories.Morphism as M
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import Categories.Morphism.Reasoning as MR
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```
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@ -92,7 +93,6 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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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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monad : Monad C
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monad = Kleisli⇒Monad C (record
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{ F₀ = D₀
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@ -119,31 +119,14 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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(idC +₁ extend (now X)) ∘ out X ∎ })
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; assoc = {! !}
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; sym-assoc = {! !}
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; extend-≈ = λ {X} {Y} {f} {g} eq → {! !}
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-- begin
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-- extend f ≈⟨ sym (Terminal.!-unique (algebras Y) (record { f = extend f ; commutes = F-Coalgebra-Morphism.commutes {! !} })) ⟩
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-- F-Coalgebra-Morphism.f ((Terminal.! (algebras Y) {A = alg' {X} {Y}})) ≈˘⟨ sym (Terminal.!-unique (algebras Y) (record { f = extend g ; commutes = {! !} })) ⟩
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-- extend g ∎
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-- let
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-- h : F-Coalgebra-Morphism (alg f) (alg g)
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-- h = record { f = idC ; commutes = begin
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-- F-Coalgebra.α (alg g) ∘ idC ≈⟨ id-comm ⟩
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-- idC ∘ F-Coalgebra.α (alg g) ≈⟨ refl⟩∘⟨ []-cong₂ (([]-cong₂ (refl⟩∘⟨ (refl⟩∘⟨ (sym eq))) refl) ⟩∘⟨refl) refl ⟩
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-- idC ∘ F-Coalgebra.α (alg f) ≈˘⟨ ([]-cong₂ identityʳ identityʳ ○ +-η) ⟩∘⟨refl ⟩
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-- (idC +₁ idC) ∘ F-Coalgebra.α (alg f) ∎ }
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-- x : F-Coalgebra-Morphism (alg f) (Terminal.⊤ (algebras Y))
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-- x = (F-Coalgebras (delayF Y)) [ Terminal.! (algebras Y) ∘ h ]
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-- in Terminal.!-unique₂ (algebras Y) {f = Terminal.! (algebras Y)} {g = {! !} } ⟩∘⟨refl
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-- extend f ≈⟨ insertˡ (_≅_.isoˡ (out-≅ Y)) ⟩
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-- out⁻¹ Y ∘ out Y ∘ extend f ≈⟨ refl⟩∘⟨ pullˡ (F-Coalgebra-Morphism.commutes (Terminal.! (algebras Y) {A = alg f})) ⟩
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-- out⁻¹ Y ∘ ((idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras Y)))) ∘ [ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X , (idC +₁ i₂) ∘ out Y ]) ∘ i₁ ≈⟨ refl⟩∘⟨ ((+₁-cong₂ refl (Terminal.!-unique₂ (algebras Y) {f = Terminal.! (algebras Y) {A = alg f}} {g = Terminal.! (algebras Y) {A = alg f}}) ⟩∘⟨ ([]-cong₂ (([]-cong₂ (refl⟩∘⟨ (refl⟩∘⟨ eq)) refl) ⟩∘⟨refl) refl)) ⟩∘⟨refl) ⟩
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-- out⁻¹ Y ∘ ((idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras Y)))) ∘ [ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ g) , i₂ ∘ i₁ ] ∘ out X , (idC +₁ i₂) ∘ out Y ]) ∘ i₁ ≈⟨ refl⟩∘⟨ (((+₁-cong₂ refl (Terminal.!-unique₂ (algebras Y) {f = Terminal.! (algebras Y)} {g = record { f = F-Coalgebra-Morphism.f (Terminal.! (algebras Y)) ; commutes = {! !} }})) ⟩∘⟨refl) ⟩∘⟨refl) ⟩
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-- out⁻¹ Y ∘ ((idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras Y)))) ∘ [ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ g) , i₂ ∘ i₁ ] ∘ out X , (idC +₁ i₂) ∘ out Y ]) ∘ i₁ ≈˘⟨ refl⟩∘⟨ pullˡ (F-Coalgebra-Morphism.commutes (Terminal.! (algebras Y) {A = alg g})) ⟩
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-- out⁻¹ Y ∘ out Y ∘ extend g ≈˘⟨ insertˡ (_≅_.isoˡ (out-≅ Y)) ⟩
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-- extend g ∎
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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 ⟩
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(F-Coalgebra-Morphism.f (Terminal.! (algebras Y) {A = alg g }) ∘ idC) ∘ i₁ {B = D₀ Y} ≈⟨ identityʳ ⟩∘⟨refl ⟩
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extend g ∎
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})
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where
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alg' : ∀ {X Y} → F-Coalgebra (delayF Y)
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@ -169,6 +152,25 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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[ [ i₁ ∘ idC , (i₂ ∘ (F-Coalgebra-Morphism.f !)) ∘ i₂ ] ∘ (out Y ∘ f)
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, (i₂ ∘ (F-Coalgebra-Morphism.f !)) ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ (elimˡ (([]-cong₂ identityʳ (cancelʳ !∘i₂)) ○ +-η)) assoc) ⟩∘⟨refl ⟩
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[ out Y ∘ f , i₂ ∘ extend ] ∘ out X ∎
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αf≈αg : ∀ {X Y} {f g : X ⇒ D₀ Y} → f ≈ g → F-Coalgebra.α (alg f) ≈ F-Coalgebra.α (alg g)
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αf≈αg {X} {Y} {f} {g} eq = []-cong₂ ([]-cong₂ (refl⟩∘⟨ refl⟩∘⟨ eq) refl ⟩∘⟨refl) refl
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alg-f≈alg-g : ∀ {X Y} {f g : X ⇒ D₀ Y} → f ≈ g → M._≅_ (F-Coalgebras (delayF Y)) (alg f) (alg g)
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alg-f≈alg-g {X} {Y} {f} {g} eq = record
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{ from = record { f = idC ; commutes = begin
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F-Coalgebra.α (alg g) ∘ idC ≈⟨ identityʳ ⟩
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F-Coalgebra.α (alg g) ≈⟨ ⟺ (αf≈αg eq) ⟩
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F-Coalgebra.α (alg f) ≈˘⟨ elimˡ (Functor.identity (delayF Y)) ⟩
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Functor.F₁ (delayF Y) idC ∘ F-Coalgebra.α (alg f) ∎ }
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; to = record { f = idC ; commutes = begin
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F-Coalgebra.α (alg f) ∘ idC ≈⟨ identityʳ ⟩
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F-Coalgebra.α (alg f) ≈⟨ αf≈αg eq ⟩
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F-Coalgebra.α (alg g) ≈˘⟨ elimˡ (Functor.identity (delayF Y)) ⟩
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Functor.F₁ (delayF Y) idC ∘ F-Coalgebra.α (alg g) ∎ }
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; iso = record
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{ isoˡ = identity²
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; isoʳ = identity²
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}
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}
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```
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### Old definitions:
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