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Added iota to delay
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3 changed files with 126 additions and 70 deletions
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.gitignore
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@ -1,4 +1,5 @@
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*.agdai
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*.pdf
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*.log
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Everything.agda
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Everything.agda
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public/
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Makefile
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Makefile
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@ -20,7 +20,7 @@ open:
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push: all
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mv public/Everything.html public/index.html
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scp -r public hy84coky@cip2a7.cip.cs.fau.de:.www/public
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scp -r public hy84coky@cip2a7.cip.cs.fau.de:.www/bsc-thesis
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Everything.agda:
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@ -10,10 +10,12 @@ open import Categories.Category.Extensive
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open import Categories.Category.BinaryProducts
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open import Categories.Category.Cocartesian
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open import Categories.Category.Cartesian
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open import Categories.Category.Cartesian
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open import Categories.Category.Cartesian.Bundle
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open import Categories.Object.Terminal
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open import Categories.Object.Initial
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open import Categories.Object.Coproduct
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open import Categories.Category.Construction.F-Coalgebras
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open import Categories.Category.Construction.F-Algebras
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open import Categories.Functor.Coalgebra
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open import Categories.Functor
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open import Categories.Functor.Algebra
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@ -41,11 +43,18 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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open Cartesian (ExtensiveDistributiveCategory.cartesian ED)
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open BinaryProducts products
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CC : CartesianCategory o ℓ e
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CC = record { U = C ; cartesian = (ExtensiveDistributiveCategory.cartesian ED) }
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open import Categories.Object.NaturalNumbers.Parametrized CC
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open import Categories.Object.NaturalNumbers.Properties.F-Algebras using (PNNO⇒Initial₂; PNNO-Algebra)
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open M C
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open MR C
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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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```
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### *Proposition 1*: Characterization of the delay monad ***D***
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@ -66,7 +75,7 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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module D A = Functor (delayF A)
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module _ (X : Obj) where
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open Terminal (algebras X) using (⊤; !)
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open Terminal (algebras X) using (⊤; !; !-unique)
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open F-Coalgebra ⊤ renaming (A to DX)
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D₀ = DX
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@ -93,6 +102,17 @@ 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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module _ (ℕ : ParametrizedNNO) where
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open ParametrizedNNO ℕ
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iso : X × N ≅ X + X × N
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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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monad : Monad C
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monad = Kleisli⇒Monad C (record
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{ F₀ = D₀
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@ -117,8 +137,75 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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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 = {! !}
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; sym-assoc = {! !}
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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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@ -129,29 +216,47 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
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extend g ∎
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})
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where
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open Terminal
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alg' : ∀ {X Y} → F-Coalgebra (delayF Y)
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alg' {X} {Y} = record { A = D₀ X ; α = i₂ }
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module _ {X Y : Obj} (f : X ⇒ D₀ Y) where
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open Terminal (algebras Y) using (!; ⊤-id)
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-- open Terminal (algebras Y) using (!; ⊤-id)
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alg : F-Coalgebra (delayF Y)
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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)
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extend : D₀ X ⇒ D₀ Y
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extend = F-Coalgebra-Morphism.f (! {A = alg}) ∘ i₁ {B = D₀ Y}
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!∘i₂ : F-Coalgebra-Morphism.f (! {A = alg}) ∘ i₂ ≈ idC
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!∘i₂ = ⊤-id (F-Coalgebras (delayF Y) [ ! ∘ record { f = i₂ ; commutes = inject₂ } ] )
|
||||
extend = F-Coalgebra-Morphism.f (! (algebras Y) {A = alg}) ∘ i₁ {B = D₀ Y}
|
||||
!∘i₂ : F-Coalgebra-Morphism.f (! (algebras Y) {A = alg}) ∘ i₂ ≈ idC
|
||||
!∘i₂ = ⊤-id (algebras Y) (F-Coalgebras (delayF Y) [ ! (algebras Y) ∘ record { f = i₂ ; commutes = inject₂ } ] )
|
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extendlaw : out Y ∘ extend ≈ [ out Y ∘ f , i₂ ∘ extend ] ∘ out X
|
||||
extendlaw = begin
|
||||
out Y ∘ extend ≈⟨ pullˡ (F-Coalgebra-Morphism.commutes (! {A = alg})) ⟩
|
||||
((idC +₁ (F-Coalgebra-Morphism.f !)) ∘ F-Coalgebra.α alg) ∘ coproduct.i₁ ≈⟨ pullʳ inject₁ ⟩
|
||||
(idC +₁ (F-Coalgebra-Morphism.f !)) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X ≈⟨ pullˡ ∘[] ⟩
|
||||
[ (idC +₁ (F-Coalgebra-Morphism.f !)) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (idC +₁ (F-Coalgebra-Morphism.f !)) ∘ i₂ ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
|
||||
[ [ (idC +₁ (F-Coalgebra-Morphism.f !)) ∘ i₁
|
||||
, (idC +₁ (F-Coalgebra-Morphism.f !)) ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (i₂ ∘ (F-Coalgebra-Morphism.f !)) ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ (([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl ⟩
|
||||
[ [ i₁ ∘ idC , (i₂ ∘ (F-Coalgebra-Morphism.f !)) ∘ i₂ ] ∘ (out Y ∘ f)
|
||||
, (i₂ ∘ (F-Coalgebra-Morphism.f !)) ∘ i₁ ] ∘ out X ≈⟨ ([]-cong₂ (elimˡ (([]-cong₂ identityʳ (cancelʳ !∘i₂)) ○ +-η)) assoc) ⟩∘⟨refl ⟩
|
||||
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 ∘ 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)
|
||||
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)
|
||||
|
|
@ -173,56 +278,6 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
|
|||
}
|
||||
```
|
||||
|
||||
### Old definitions:
|
||||
|
||||
```agda
|
||||
record DelayMonad : Set (o ⊔ ℓ ⊔ e) where
|
||||
field
|
||||
D₀ : Obj → Obj
|
||||
|
||||
field
|
||||
now : ∀ {X} → X ⇒ D₀ X
|
||||
later : ∀ {X} → D₀ X ⇒ D₀ X
|
||||
isIso : ∀ {X} → IsIso ([ now {X} , later {X} ])
|
||||
|
||||
out : ∀ {X} → D₀ X ⇒ X + D₀ X
|
||||
out {X} = IsIso.inv (isIso {X})
|
||||
|
||||
field
|
||||
coit : ∀ {X Y} → Y ⇒ X + Y → Y ⇒ D₀ X
|
||||
coit-law : ∀ {X Y} {f : Y ⇒ X + Y} → out ∘ (coit f) ≈ (idC +₁ (coit f)) ∘ f
|
||||
|
||||
field
|
||||
_* : ∀ {X Y} → X ⇒ D₀ Y → D₀ X ⇒ D₀ Y
|
||||
*-law : ∀ {X Y} {f : X ⇒ D₀ Y} → out ∘ (f *) ≈ [ out ∘ f , i₂ ∘ (f *) ] ∘ out
|
||||
*-unique : ∀ {X Y} (f : X ⇒ D₀ Y) (h : D₀ X ⇒ D₀ Y) → h ≈ f *
|
||||
*-resp-≈ : ∀ {X Y} {f h : X ⇒ D₀ Y} → f ≈ h → f * ≈ h *
|
||||
|
||||
unitLaw : ∀ {X} → out {X} ∘ now {X} ≈ i₁
|
||||
unitLaw = begin
|
||||
out ∘ now ≈⟨ refl⟩∘⟨ sym inject₁ ⟩
|
||||
out ∘ [ now , later ] ∘ i₁ ≈⟨ cancelˡ (IsIso.isoˡ isIso) ⟩
|
||||
i₁ ∎
|
||||
|
||||
toMonad : KleisliTriple C
|
||||
toMonad = record
|
||||
{ F₀ = D₀
|
||||
; unit = now
|
||||
; extend = _*
|
||||
; identityʳ = λ {X} {Y} {k} → begin
|
||||
k * ∘ now ≈⟨ introˡ (IsIso.isoʳ isIso) ⟩∘⟨refl ⟩
|
||||
(([ now , later ] ∘ out) ∘ k *) ∘ now ≈⟨ pullʳ *-law ⟩∘⟨refl ⟩
|
||||
([ now , later ] ∘ [ out ∘ k , i₂ ∘ (k *) ] ∘ out) ∘ now ≈⟨ pullʳ (pullʳ unitLaw) ⟩
|
||||
[ now , later ] ∘ [ out ∘ k , i₂ ∘ (k *) ] ∘ i₁ ≈⟨ refl⟩∘⟨ inject₁ ⟩
|
||||
[ now , later ] ∘ out ∘ k ≈⟨ cancelˡ (IsIso.isoʳ isIso) ⟩
|
||||
k ∎
|
||||
; identityˡ = λ {X} → sym (*-unique now idC)
|
||||
; assoc = λ {X} {Y} {Z} {f} {g} → sym (*-unique ((g *) ∘ f) ((g *) ∘ (f *)))
|
||||
; sym-assoc = λ {X} {Y} {Z} {f} {g} → *-unique ((g *) ∘ f) ((g *) ∘ (f *))
|
||||
; extend-≈ = *-resp-≈
|
||||
}
|
||||
```
|
||||
|
||||
### Definition 30: Search-Algebras
|
||||
|
||||
TODO
|
||||
|
|
|
|||
Loading…
Reference in a new issue