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small refactor
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17 changed files with 338 additions and 456 deletions
agda/src
Algebra/Elgot
Category/Construction
Monad/Instance
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@ -22,8 +22,7 @@ import Categories.Morphism.Reasoning as MR
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```agda
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module Algebra.Elgot.Free {o ℓ e} {C : Category o ℓ e} (cocartesian : Cocartesian C) where
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open import Algebra.Elgot cocartesian
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open import Category.Construction.ElgotAlgebras {C = C}
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open Cat cocartesian
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open import Category.Construction.ElgotAlgebras cocartesian
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open import Categories.Morphism.Properties C
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open Category C
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@ -1,5 +1,4 @@
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```agda
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{-# OPTIONS --allow-unsolved-metas #-}
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open import Level
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open import Categories.FreeObjects.Free
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open import Categories.Functor.Core
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@ -13,8 +12,6 @@ open import Categories.Category.CartesianClosed using (CartesianClosed)
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open import Categories.Category.Cocartesian using (Cocartesian)
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open import Categories.Category.Cartesian using (Cartesian)
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open import Categories.Category.BinaryProducts using (BinaryProducts)
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open import Categories.Category.Construction.EilenbergMoore
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open import Categories.Monad.Relative renaming (Monad to RMonad)
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import Categories.Morphism as M
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import Categories.Morphism.Reasoning as MR
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@ -39,9 +36,8 @@ module Algebra.Elgot.Properties {o ℓ e} {C : Category o ℓ e} (distributive :
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open CartesianClosed ccc hiding (exp; cartesian)
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open import Category.Construction.ElgotAlgebras {C = C}
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open Cat cocartesian
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open Exponentials distributive expos
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open import Category.Construction.ElgotAlgebras cocartesian
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open import Category.Construction.ElgotAlgebras.Exponentials distributive expos
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open import Algebra.Elgot cocartesian
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open import Algebra.Elgot.Free cocartesian
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open import Algebra.Elgot.Stable distributive
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@ -49,7 +45,7 @@ module Algebra.Elgot.Properties {o ℓ e} {C : Category o ℓ e} (distributive :
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open Elgot-Algebra-Morphism renaming (h to <_>)
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free-isStableˡ : ∀ {Y} (fe : FreeElgotAlgebra Y) → IsStableFreeElgotAlgebraˡ fe
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IsStableFreeElgotAlgebraˡ.[_,_]♯ˡ (free-isStableˡ {Y} fe) {X} A f = (eval′ ∘ (< FreeObject._* fe {B^A-Helper {A} {X}} (λg f) > ⁂ id))
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IsStableFreeElgotAlgebraˡ.[_,_]♯ˡ (free-isStableˡ {Y} fe) {X} A f = (eval′ ∘ (< FreeObject._* fe {Exponential-Elgot-Algebra {A} {X}} (λg f) > ⁂ id))
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where open FreeObject fe
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IsStableFreeElgotAlgebraˡ.♯ˡ-law (free-isStableˡ {Y} fe) {X} {A} f = sym (begin
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(eval′ ∘ (< (λg f)* > ⁂ id)) ∘ (η ⁂ id) ≈⟨ pullʳ ⁂∘⁂ ⟩
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@ -83,4 +79,7 @@ module Algebra.Elgot.Properties {o ℓ e} {C : Category o ℓ e} (distributive :
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open FreeObject fe renaming (FX to KY)
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open Elgot-Algebra B using () renaming (_# to _#ᵇ; #-resp-≈ to #ᵇ-resp-≈)
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open Elgot-Algebra KY using () renaming (_# to _#ʸ)
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free-isStable : ∀ {Y} (fe : FreeElgotAlgebra Y) → IsStableFreeElgotAlgebra fe
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free-isStable {Y} fe = isStableˡ⇒isStable fe (free-isStableˡ fe)
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```
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@ -24,8 +24,7 @@ module Algebra.Elgot.Stable {o ℓ e} {C : Category o ℓ e} (distributive : Dis
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open Distributive distributive
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open import Categories.Category.Distributive.Properties distributive
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open import Algebra.Elgot cocartesian
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open import Category.Construction.ElgotAlgebras {C = C}
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open Cat cocartesian
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open import Category.Construction.ElgotAlgebras cocartesian
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open import Categories.Morphism.Properties C
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open Category C
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@ -1,429 +1,67 @@
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<!--
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```agda
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{-# OPTIONS --allow-unsolved-metas #-}
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open import Level
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open import Categories.Category.Cocartesian using (Cocartesian)
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open import Categories.Category.Cartesian using (Cartesian)
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open import Categories.Category.CartesianClosed using (CartesianClosed)
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open import Categories.Category.CartesianClosed.Canonical renaming (CartesianClosed to CanonicalCartesianClosed)
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open import Categories.Category.BinaryProducts using (BinaryProducts)
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open import Categories.Functor using (Functor) renaming (id to idF)
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open import Categories.Object.Terminal using (Terminal)
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open import Categories.Object.Product using (Product)
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open import Categories.Object.Coproduct using (Coproduct)
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open import Categories.Object.Exponential using (Exponential)
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open import Categories.Category
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open import Categories.Category.Distributive
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open import Categories.Object.Exponential
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open import Categories.Category.Extensive.Bundle
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open import Categories.Category.Extensive
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open import Category.Ambient using (Ambient)
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import Categories.Morphism as M
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import Categories.Morphism.Reasoning as MR
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import Categories.Morphism.Properties as MP
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```
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-->
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# The Category of Elgot Algebras (and Constructions)
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# The Category of Elgot Algebras
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```agda
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module Category.Construction.ElgotAlgebras {o ℓ e} {C : Category o ℓ e} where
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module Category.Construction.ElgotAlgebras {o ℓ e} {C : Category o ℓ e} (cocartesian : Cocartesian C) where
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open Category C
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open M C
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open MR C
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open MP C
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open HomReasoning
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open Equiv
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open Cocartesian cocartesian
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open import Algebra.Elgot cocartesian
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-- iteration preversing morphism between two elgot-algebras
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module _ (E₁ E₂ : Elgot-Algebra) where
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open Elgot-Algebra E₁ renaming (_# to _#₁)
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open Elgot-Algebra E₂ renaming (_# to _#₂; A to B)
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record Elgot-Algebra-Morphism : Set (o ⊔ ℓ ⊔ e) where
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field
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h : A ⇒ B
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preserves : ∀ {X} {f : X ⇒ A + X} → h ∘ (f #₁) ≈ ((h +₁ id) ∘ f)#₂
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-- the category of elgot algebras for a given category
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Elgot-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
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Elgot-Algebras = record
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{ Obj = Elgot-Algebra
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; _⇒_ = Elgot-Algebra-Morphism
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; _≈_ = λ f g → Elgot-Algebra-Morphism.h f ≈ Elgot-Algebra-Morphism.h g
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; id = λ {EB} → let open Elgot-Algebra EB in
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record { h = id; preserves = λ {X : Obj} {f : X ⇒ A + X} → begin
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id ∘ f # ≈⟨ identityˡ ⟩
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f # ≈⟨ #-resp-≈ (introˡ (coproduct.unique id-comm-sym id-comm-sym)) ⟩
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((id +₁ id) ∘ f) # ∎ }
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; _∘_ = λ {EA} {EB} {EC} f g → let
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open Elgot-Algebra-Morphism f renaming (h to hᶠ; preserves to preservesᶠ)
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open Elgot-Algebra-Morphism g renaming (h to hᵍ; preserves to preservesᵍ)
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open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ)
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open Elgot-Algebra EB using () renaming (_# to _#ᵇ; A to B)
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open Elgot-Algebra EC using () renaming (_# to _#ᶜ; A to C; #-resp-≈ to #ᶜ-resp-≈)
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in record { h = hᶠ ∘ hᵍ; preserves = λ {X} {f : X ⇒ A + X} → begin
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(hᶠ ∘ hᵍ) ∘ (f #ᵃ) ≈⟨ pullʳ preservesᵍ ⟩
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(hᶠ ∘ (((hᵍ +₁ id) ∘ f) #ᵇ)) ≈⟨ preservesᶠ ⟩
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(((hᶠ +₁ id) ∘ (hᵍ +₁ id) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ (pullˡ (trans +₁∘+₁ (+₁-cong₂ refl (identity²)))) ⟩
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((hᶠ ∘ hᵍ +₁ id) ∘ f) #ᶜ ∎ }
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; identityˡ = identityˡ
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; identityʳ = identityʳ
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; identity² = identity²
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; assoc = assoc
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; sym-assoc = sym-assoc
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; equiv = record
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{ refl = refl
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; sym = sym
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; trans = trans
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}
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; ∘-resp-≈ = ∘-resp-≈
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}
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where open Elgot-Algebra-Morphism
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```
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## The Category of Elgot Algebras
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```agda
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module Cat (cocartesian : Cocartesian C) where
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open Cocartesian cocartesian
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open import Algebra.Elgot cocartesian
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-- iteration preversing morphism between two elgot-algebras
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module _ (E₁ E₂ : Elgot-Algebra) where
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open Elgot-Algebra E₁ renaming (_# to _#₁)
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open Elgot-Algebra E₂ renaming (_# to _#₂; A to B)
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record Elgot-Algebra-Morphism : Set (o ⊔ ℓ ⊔ e) where
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field
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h : A ⇒ B
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preserves : ∀ {X} {f : X ⇒ A + X} → h ∘ (f #₁) ≈ ((h +₁ id) ∘ f)#₂
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-- the category of elgot algebras for a given category
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Elgot-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
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Elgot-Algebras = record
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{ Obj = Elgot-Algebra
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; _⇒_ = Elgot-Algebra-Morphism
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; _≈_ = λ f g → Elgot-Algebra-Morphism.h f ≈ Elgot-Algebra-Morphism.h g
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; id = λ {EB} → let open Elgot-Algebra EB in
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record { h = id; preserves = λ {X : Obj} {f : X ⇒ A + X} → begin
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id ∘ f # ≈⟨ identityˡ ⟩
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f # ≈⟨ #-resp-≈ (introˡ (coproduct.unique id-comm-sym id-comm-sym)) ⟩
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((id +₁ id) ∘ f) # ∎ }
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; _∘_ = λ {EA} {EB} {EC} f g → let
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open Elgot-Algebra-Morphism f renaming (h to hᶠ; preserves to preservesᶠ)
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open Elgot-Algebra-Morphism g renaming (h to hᵍ; preserves to preservesᵍ)
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open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ)
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open Elgot-Algebra EB using () renaming (_# to _#ᵇ; A to B)
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open Elgot-Algebra EC using () renaming (_# to _#ᶜ; A to C; #-resp-≈ to #ᶜ-resp-≈)
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in record { h = hᶠ ∘ hᵍ; preserves = λ {X} {f : X ⇒ A + X} → begin
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(hᶠ ∘ hᵍ) ∘ (f #ᵃ) ≈⟨ pullʳ preservesᵍ ⟩
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(hᶠ ∘ (((hᵍ +₁ id) ∘ f) #ᵇ)) ≈⟨ preservesᶠ ⟩
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(((hᶠ +₁ id) ∘ (hᵍ +₁ id) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ (pullˡ (trans +₁∘+₁ (+₁-cong₂ refl (identity²)))) ⟩
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((hᶠ ∘ hᵍ +₁ id) ∘ f) #ᶜ ∎ }
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; identityˡ = identityˡ
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; identityʳ = identityʳ
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; identity² = identity²
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; assoc = assoc
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; sym-assoc = sym-assoc
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; equiv = record
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{ refl = refl
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; sym = sym
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; trans = trans
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}
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; ∘-resp-≈ = ∘-resp-≈
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}
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where open Elgot-Algebra-Morphism
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```
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## Finite Products of Elgot Algebras
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```agda
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module FiniteProducts (cocartesian : Cocartesian C) (cartesian : Cartesian C) where
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open Cartesian cartesian
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open Cocartesian cocartesian
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open import Algebra.Elgot cocartesian
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open Cat cocartesian
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open BinaryProducts products renaming (unique to ⟨⟩-unique; unique′ to ⟨⟩-unique′)
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-- if the carrier contains a terminal, so does elgot-algebras
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Terminal-Elgot-Algebras : Terminal Elgot-Algebras
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Terminal-Elgot-Algebras = record
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{ ⊤ = record
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{ A = ⊤
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; algebra = record
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{ _# = λ x → !
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; #-Fixpoint = λ {_ f} → !-unique ([ id , ! ] ∘ f)
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; #-Uniformity = λ {_ _ _ _ h} _ → !-unique (! ∘ h)
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; #-Folding = refl
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; #-resp-≈ = λ _ → refl
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}
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}
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; ⊤-is-terminal = record
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{ ! = λ {A} → record { h = ! ; preserves = λ {X} {f} → sym (!-unique (! ∘ (A Elgot-Algebra.#) f)) }
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; !-unique = λ {A} f → !-unique (Elgot-Algebra-Morphism.h f)
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}
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}
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where
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open Terminal terminal
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-- if the carriers of the algebra form a product, so do the algebras
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A×B-Helper : ∀ {EA EB : Elgot-Algebra} → Elgot-Algebra
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A×B-Helper {EA} {EB} = record
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{ A = A × B
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; algebra = record
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{ _# = λ {X : Obj} (h : X ⇒ A×B + X) → ⟨ ((π₁ +₁ id) ∘ h)#ᵃ , ((π₂ +₁ id) ∘ h)#ᵇ ⟩
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; #-Fixpoint = λ {X} {f} → begin
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⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ≈⟨ ⟨⟩-cong₂ #ᵃ-Fixpoint #ᵇ-Fixpoint ⟩
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⟨ [ id , ((π₁ +₁ id) ∘ f)#ᵃ ] ∘ ((π₁ +₁ id) ∘ f) , [ id , ((π₂ +₁ id) ∘ f)#ᵇ ] ∘ ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ ⟨⟩-cong₂ (pullˡ []∘+₁) (pullˡ []∘+₁) ⟩
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⟨ [ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] ∘ f , [ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ∘ f ⟩ ≈˘⟨ ⟨⟩∘ ⟩
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(⟨ [ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] , [ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ⟩ ∘ f) ≈⟨ ∘-resp-≈ˡ (⟨⟩-unique′
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(begin
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π₁ ∘ ⟨ [ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] , [ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ⟩ ≈⟨ project₁ ⟩
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[ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] ≈⟨ []-cong₂ id-comm-sym (trans identityʳ (sym project₁)) ⟩
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[ π₁ ∘ id , π₁ ∘ ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ≈˘⟨ ∘[] ⟩
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π₁ ∘ [ id , ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ∎)
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(begin
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π₂ ∘ ⟨ [ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] , [ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ⟩ ≈⟨ project₂ ⟩
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[ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ≈⟨ []-cong₂ id-comm-sym (trans identityʳ (sym project₂)) ⟩
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[ π₂ ∘ id , π₂ ∘ ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ≈˘⟨ ∘[] ⟩
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π₂ ∘ [ id , ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ∎)
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)⟩
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([ id , ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ∘ f) ∎
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; #-Uniformity = λ {X Y f g h} uni → ⟨⟩-unique′
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(begin
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π₁ ∘ ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ≈⟨ project₁ ⟩
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(((π₁ +₁ id) ∘ f)#ᵃ) ≈⟨ #ᵃ-Uniformity
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(begin
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-- TODO factor these out or adjust +₁ swap... maybe call it +₁-id-comm
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(id +₁ h) ∘ (π₁ +₁ id) ∘ f ≈⟨ pullˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm)) ⟩
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(π₁ ∘ id +₁ id ∘ h) ∘ f ≈˘⟨ pullˡ +₁∘+₁ ⟩
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(π₁ +₁ id) ∘ (id +₁ h) ∘ f ≈⟨ pushʳ uni ⟩
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((π₁ +₁ id) ∘ g) ∘ h ∎)⟩
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(((π₁ +₁ id) ∘ g)#ᵃ ∘ h) ≈˘⟨ pullˡ project₁ ⟩
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π₁ ∘ ⟨ ((π₁ +₁ id) ∘ g)#ᵃ , ((π₂ +₁ id) ∘ g)#ᵇ ⟩ ∘ h ∎)
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(begin
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π₂ ∘ ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ≈⟨ project₂ ⟩
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((π₂ +₁ id) ∘ f)#ᵇ ≈⟨ #ᵇ-Uniformity
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(begin
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(id +₁ h) ∘ (π₂ +₁ id) ∘ f ≈⟨ pullˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm))⟩
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((π₂ ∘ id +₁ id ∘ h) ∘ f) ≈˘⟨ pullˡ +₁∘+₁ ⟩
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(π₂ +₁ id) ∘ ((id +₁ h)) ∘ f ≈⟨ pushʳ uni ⟩
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((π₂ +₁ id) ∘ g) ∘ h ∎)⟩
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((π₂ +₁ id) ∘ g)#ᵇ ∘ h ≈˘⟨ pullˡ project₂ ⟩
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π₂ ∘ ⟨ ((π₁ +₁ id) ∘ g)#ᵃ , ((π₂ +₁ id) ∘ g)#ᵇ ⟩ ∘ h ∎)
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; #-Folding = λ {X} {Y} {f} {h} → ⟨⟩-cong₂ (foldingˡ {X} {Y}) (foldingʳ {X} {Y})
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; #-resp-≈ = λ fg → ⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ʳ fg)) (#ᵇ-resp-≈ (∘-resp-≈ʳ fg))
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}
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}
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where
|
||||
open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
|
||||
open Elgot-Algebra EB using () renaming (A to B; _# to _#ᵇ; #-Fixpoint to #ᵇ-Fixpoint; #-Uniformity to #ᵇ-Uniformity; #-Folding to #ᵇ-Folding; #-resp-≈ to #ᵇ-resp-≈)
|
||||
+₁-id-swap : ∀ {X Y C} {f : X ⇒ (A × B) + X} {h : Y ⇒ X + Y} (π : A × B ⇒ C) → [ (id +₁ i₁) ∘ ((π +₁ id) ∘ f) , i₂ ∘ h ] ≈ (π +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ]
|
||||
+₁-id-swap {X} {Y} {C} {f} {h} π = begin ([ (id +₁ i₁) ∘ ((π +₁ id) ∘ f) , i₂ ∘ h ] ) ≈⟨ ([]-congʳ sym-assoc) ⟩
|
||||
([ ((id +₁ i₁) ∘ (π +₁ id)) ∘ f , i₂ ∘ h ] ) ≈⟨ []-cong₂ (∘-resp-≈ˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm))) (∘-resp-≈ˡ (sym identityʳ)) ⟩
|
||||
(([ (π ∘ id +₁ id ∘ i₁) ∘ f , (i₂ ∘ id) ∘ h ])) ≈˘⟨ []-cong₂ (pullˡ +₁∘+₁) (pullˡ +₁∘i₂) ⟩
|
||||
(([ (π +₁ id) ∘ (id +₁ i₁) ∘ f , (π +₁ id) ∘ i₂ ∘ h ])) ≈˘⟨ ∘[] ⟩
|
||||
((π +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ]) ∎
|
||||
foldingˡ : ∀ {X} {Y} {f} {h} → (((π₁ +₁ id) ∘ (⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ) ≈ ((π₁ +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ
|
||||
foldingˡ {X} {Y} {f} {h} = begin
|
||||
((π₁ +₁ id) ∘ (⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ ≈⟨ #ᵃ-resp-≈ (trans +₁∘+₁ (+₁-cong₂ project₁ identityˡ)) ⟩
|
||||
((((π₁ +₁ id) ∘ f)#ᵃ +₁ h)#ᵃ) ≈⟨ #ᵃ-Folding ⟩
|
||||
([ (id +₁ i₁) ∘ ((π₁ +₁ id) ∘ f) , i₂ ∘ h ] #ᵃ) ≈⟨ #ᵃ-resp-≈ (+₁-id-swap π₁)⟩
|
||||
((π₁ +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ ∎
|
||||
foldingʳ : ∀ {X} {Y} {f} {h} → ((π₂ +₁ id) ∘ (⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈ ((π₂ +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ
|
||||
foldingʳ {X} {Y} {f} {h} = begin
|
||||
((π₂ +₁ id) ∘ (⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈⟨ #ᵇ-resp-≈ (trans +₁∘+₁ (+₁-cong₂ project₂ identityˡ)) ⟩
|
||||
((((π₂ +₁ id) ∘ f)#ᵇ +₁ h)#ᵇ) ≈⟨ #ᵇ-Folding ⟩
|
||||
[ (id +₁ i₁) ∘ ((π₂ +₁ id) ∘ f) , i₂ ∘ h ] #ᵇ ≈⟨ #ᵇ-resp-≈ (+₁-id-swap π₂) ⟩
|
||||
((π₂ +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ ∎
|
||||
|
||||
Product-Elgot-Algebras : ∀ (EA EB : Elgot-Algebra) → Product Elgot-Algebras EA EB
|
||||
Product-Elgot-Algebras EA EB = record
|
||||
{ A×B = A×B-Helper {EA} {EB}
|
||||
; π₁ = record { h = π₁ ; preserves = λ {X} {f} → project₁ }
|
||||
; π₂ = record { h = π₂ ; preserves = λ {X} {f} → project₂ }
|
||||
; ⟨_,_⟩ = λ {E} f g → let
|
||||
open Elgot-Algebra-Morphism f renaming (h to f′; preserves to preservesᶠ)
|
||||
open Elgot-Algebra-Morphism g renaming (h to g′; preserves to preservesᵍ)
|
||||
open Elgot-Algebra E renaming (_# to _#ᵉ) in record { h = ⟨ f′ , g′ ⟩ ; preserves = λ {X} {h} →
|
||||
begin
|
||||
⟨ f′ , g′ ⟩ ∘ (h #ᵉ) ≈⟨ ⟨⟩∘ ⟩
|
||||
⟨ f′ ∘ (h #ᵉ) , g′ ∘ (h #ᵉ) ⟩ ≈⟨ ⟨⟩-cong₂ preservesᶠ preservesᵍ ⟩
|
||||
⟨ ((f′ +₁ id) ∘ h) #ᵃ , ((g′ +₁ id) ∘ h) #ᵇ ⟩ ≈˘⟨ ⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ project₁ identity²))) (#ᵇ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ project₂ identity²))) ⟩
|
||||
⟨ ((π₁ ∘ ⟨ f′ , g′ ⟩ +₁ id ∘ id) ∘ h) #ᵃ , ((π₂ ∘ ⟨ f′ , g′ ⟩ +₁ id ∘ id) ∘ h) #ᵇ ⟩ ≈˘⟨ ⟨⟩-cong₂ (#ᵃ-resp-≈ (pullˡ +₁∘+₁)) (#ᵇ-resp-≈ (pullˡ +₁∘+₁)) ⟩
|
||||
⟨ ((π₁ +₁ id) ∘ (⟨ f′ , g′ ⟩ +₁ id) ∘ h) #ᵃ , ((π₂ +₁ id) ∘ (⟨ f′ , g′ ⟩ +₁ id) ∘ h) #ᵇ ⟩ ∎ }
|
||||
; project₁ = project₁
|
||||
; project₂ = project₂
|
||||
; unique = ⟨⟩-unique
|
||||
}
|
||||
where
|
||||
open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
|
||||
open Elgot-Algebra EB using () renaming (A to B; _# to _#ᵇ; #-Fixpoint to #ᵇ-Fixpoint; #-Uniformity to #ᵇ-Uniformity; #-Folding to #ᵇ-Folding; #-resp-≈ to #ᵇ-resp-≈)
|
||||
open Elgot-Algebra (A×B-Helper {EA} {EB}) using () renaming (_# to _#ᵖ)
|
||||
|
||||
|
||||
-- if the carrier is cartesian, so is the category of algebras
|
||||
Cartesian-Elgot-Algebras : Cartesian Elgot-Algebras
|
||||
Cartesian-Elgot-Algebras = record
|
||||
{ terminal = Terminal-Elgot-Algebras
|
||||
; products = record { product = λ {EA EB} → Product-Elgot-Algebras EA EB }
|
||||
}
|
||||
```
|
||||
|
||||
## Exponentials of Elgot Algebras TODO
|
||||
|
||||
```agda
|
||||
|
||||
module Exponentials (distributive : Distributive C) (exp : ∀ {A B : Obj} → Exponential C A B) where
|
||||
open Distributive distributive
|
||||
open import Categories.Category.Distributive.Properties distributive
|
||||
ccc : CartesianClosed C
|
||||
ccc = record { cartesian = cartesian ; exp = exp }
|
||||
open CartesianClosed ccc hiding (cartesian; exp)
|
||||
open Cocartesian cocartesian
|
||||
open Cartesian cartesian
|
||||
open BinaryProducts products renaming (unique to ⟨⟩-unique; unique′ to ⟨⟩-unique′)
|
||||
open import Algebra.Elgot cocartesian
|
||||
open Cat cocartesian
|
||||
open FiniteProducts cocartesian cartesian
|
||||
|
||||
B^A-Helper : ∀ {EA : Elgot-Algebra} {X : Obj} → Elgot-Algebra
|
||||
B^A-Helper {EA} {X} = record
|
||||
{ A = A ^ X
|
||||
; algebra = record
|
||||
{ _# = λ {Z} f → λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)
|
||||
; #-Fixpoint = λ {X} {f} → begin
|
||||
λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ≈⟨ λ-cong #ᵃ-Fixpoint ⟩
|
||||
λg ([ id , ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ ] ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) ≈⟨ λ-cong ((pullˡ []∘+₁) ○ ([]-cong₂ identityˡ identityʳ) ⟩∘⟨refl) ⟩
|
||||
λg ([ eval′ , ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ ] ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) ≈˘⟨ λ-unique′ (begin
|
||||
eval′ ∘ (([ id , λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ] ∘ f) ⁂ id) ≈˘⟨ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²) ⟩
|
||||
eval′ ∘ ([ id , λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ] ⁂ id) ∘ (f ⁂ id) ≈⟨ refl⟩∘⟨ (⟨⟩-unique (begin
|
||||
π₁ ∘ [ id , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ⁂ id ] ∘ distributeʳ⁻¹ ≈⟨ pullˡ (∘[] ○ []-cong₂ id-comm π₁∘⁂) ⟩
|
||||
[ id ∘ π₁ , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ∘ π₁ ] ∘ distributeʳ⁻¹ ≈˘⟨ pullˡ []∘+₁ ⟩
|
||||
[ id , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ] ∘ (π₁ +₁ π₁) ∘ distributeʳ⁻¹ ≈⟨ refl⟩∘⟨ distributeʳ⁻¹-π₁ ⟩
|
||||
[ id , λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ] ∘ π₁ ∎)
|
||||
(begin
|
||||
π₂ ∘ [ id , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ⁂ id ] ∘ distributeʳ⁻¹ ≈⟨ pullˡ (∘[] ○ []-cong₂ identityʳ (π₂∘⁂ ○ identityˡ)) ⟩
|
||||
[ π₂ , π₂ ] ∘ distributeʳ⁻¹ ≈⟨ distributeʳ⁻¹-π₂ ○ (sym identityˡ) ⟩
|
||||
id ∘ π₂ ∎)
|
||||
⟩∘⟨refl) ⟩
|
||||
eval′ ∘ ([ id , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ⁂ id ] ∘ distributeʳ⁻¹) ∘ (f ⁂ id) ≈⟨ pullˡ (pullˡ ∘[]) ⟩
|
||||
([ eval′ ∘ id , eval′ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ⁂ id) ] ∘ distributeʳ⁻¹) ∘ (f ⁂ id) ≈⟨ assoc ○ ([]-cong₂ identityʳ β′) ⟩∘⟨refl ⟩
|
||||
[ eval′ , ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ ] ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ∎) ⟩
|
||||
[ id , λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ] ∘ f ∎
|
||||
; #-Uniformity = #-Uniformity
|
||||
; #-Folding = #-Folding
|
||||
; #-resp-≈ = λ {Z} {f} {g} f≈g → λ-cong (#ᵃ-resp-≈ (refl⟩∘⟨ refl⟩∘⟨ ⁂-cong₂ f≈g refl))
|
||||
}
|
||||
}
|
||||
where
|
||||
open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
|
||||
#-Uniformity : ∀ {D E} {f : D ⇒ A ^ X + D} {g : E ⇒ A ^ X + E} {h : D ⇒ E} → (id +₁ h) ∘ f ≈ g ∘ h → λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ≈ λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᵃ) ∘ h
|
||||
#-Uniformity {D} {E} {f} {g} {h} eq = sym (λ-unique′ (begin
|
||||
eval′ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᵃ) ∘ h) ⁂ id) ≈˘⟨ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identityˡ) ⟩
|
||||
eval′ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᵃ)) ⁂ id) ∘ (h ⁂ id) ≈⟨ pullˡ β′ ⟩
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᵃ ∘ (h ⁂ id) ≈˘⟨ #ᵃ-Uniformity by-uni ⟩
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ ∎))
|
||||
where
|
||||
by-uni : (id +₁ (h ⁂ id)) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ≈ ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) ∘ (h ⁂ id)
|
||||
by-uni = begin
|
||||
(id +₁ (h ⁂ id)) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ identityʳ) ⟩
|
||||
(eval′ +₁ (h ⁂ id)) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ≈˘⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityʳ identityˡ) ⟩
|
||||
(eval′ +₁ id) ∘ (id +₁ (h ⁂ id)) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ≈⟨ refl⟩∘⟨ pullˡ ((+₁-cong₂ (sym (⟨⟩-unique id-comm id-comm)) refl) ⟩∘⟨refl ○ distributeʳ⁻¹-natural id id h) ⟩
|
||||
(eval′ +₁ id) ∘ (distributeʳ⁻¹ ∘ ((id +₁ h) ⁂ id)) ∘ (f ⁂ id) ≈⟨ refl⟩∘⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ eq identity²) ⟩
|
||||
(eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ∘ h ⁂ id) ≈˘⟨ pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ refl identity²)) ⟩
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) ∘ (h ⁂ id) ∎
|
||||
#-Folding : ∀ {D E} {f : D ⇒ A ^ X + D} {h : E ⇒ D + E} → λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) +₁ h) ⁂ id)) #ᵃ) ≈ λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) #ᵃ)
|
||||
#-Folding {D} {E} {f} {h} = λ-cong (begin
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) +₁ h) ⁂ id)) #ᵃ ≈⟨ #ᵃ-resp-≈ (refl⟩∘⟨ sym (distributeʳ⁻¹-natural id (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) h)) ⟩
|
||||
((eval′ +₁ id) ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ⁂ id) +₁ (h ⁂ id)) ∘ distributeʳ⁻¹) #ᵃ ≈⟨ #ᵃ-resp-≈ (pullˡ (+₁∘+₁ ○ +₁-cong₂ β′ identityˡ)) ⟩
|
||||
((((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ h ⁂ id) ∘ distributeʳ⁻¹) #ᵃ ≈⟨ #ᵃ-Uniformity by-uni₁ ⟩
|
||||
(((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ distributeʳ⁻¹ ∘ (h ⁂ id)) #ᵃ ∘ distributeʳ⁻¹ ≈⟨ #ᵃ-Folding ⟩∘⟨refl ⟩
|
||||
[ ((id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) , (i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id)) ] #ᵃ ∘ distributeʳ⁻¹ ≈˘⟨ #ᵃ-Uniformity by-uni₂ ⟩
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) #ᵃ ∎)
|
||||
where
|
||||
by-uni₁ : (id +₁ distributeʳ⁻¹) ∘ (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ h ⁂ id) ∘ distributeʳ⁻¹ ≈ ((((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ distributeʳ⁻¹ ∘ (h ⁂ id))) ∘ distributeʳ⁻¹
|
||||
by-uni₁ = begin
|
||||
(id +₁ distributeʳ⁻¹) ∘ (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ h ⁂ id) ∘ distributeʳ⁻¹ ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ refl) ⟩
|
||||
((((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ distributeʳ⁻¹ ∘ (h ⁂ id))) ∘ distributeʳ⁻¹ ∎
|
||||
by-uni₂ : (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id) ≈ [ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ∘ distributeʳ⁻¹
|
||||
by-uni₂ = Iso⇒Epi (IsIso.iso isIsoʳ) ((id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) ([ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ∘ distributeʳ⁻¹) (begin
|
||||
((id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) ∘ distributeʳ ≈⟨ ∘[] ⟩
|
||||
[ (((id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) ∘ (i₁ ⁂ id)) , (((id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) ∘ (i₂ ⁂ id)) ] ≈⟨ []-cong₂ (pullʳ (pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ inject₁ identity²)))) (pullʳ (pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ inject₂ identity²)))) ⟩
|
||||
[ (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((id +₁ i₁) ∘ f) ⁂ id) , (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((i₂ ∘ h) ⁂ id) ] ≈⟨ []-cong₂ (refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ sym (⁂∘⁂ ○ ⁂-cong₂ refl identity²)) (refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ sym (⁂∘⁂ ○ ⁂-cong₂ refl identity²)) ⟩
|
||||
[ (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((id +₁ i₁)) ⁂ id) ∘ (f ⁂ id) , (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (i₂ ⁂ id) ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (refl⟩∘⟨ refl⟩∘⟨ pullˡ (sym (distributeʳ⁻¹-natural id id i₁))) (refl⟩∘⟨ refl⟩∘⟨ pullˡ distributeʳ⁻¹-i₂) ⟩
|
||||
[ (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ ((id ⁂ id +₁ i₁ ⁂ id) ∘ distributeʳ⁻¹) ∘ (f ⁂ id) , (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ i₂ ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (pullˡ +₁∘+₁) (pullˡ +₁∘+₁) ⟩
|
||||
[ (id ∘ eval′ +₁ distributeʳ⁻¹ ∘ id) ∘ ((id ⁂ id +₁ i₁ ⁂ id) ∘ distributeʳ⁻¹) ∘ (f ⁂ id) , (id ∘ eval′ +₁ distributeʳ⁻¹ ∘ id) ∘ i₂ ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (pullˡ (pullˡ +₁∘+₁)) (pullˡ +₁∘i₂) ⟩
|
||||
[ ((((id ∘ eval′) ∘ (id ⁂ id)) +₁ ((distributeʳ⁻¹ ∘ id) ∘ (i₁ ⁂ id))) ∘ distributeʳ⁻¹) ∘ (f ⁂ id) , (i₂ ∘ (distributeʳ⁻¹ ∘ id)) ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (((+₁-cong₂ (identityˡ ⟩∘⟨refl) (identityʳ ⟩∘⟨refl ○ distributeʳ⁻¹-i₁)) ⟩∘⟨refl) ⟩∘⟨refl) (pullʳ (pullʳ identityˡ)) ⟩
|
||||
[ (((eval′ ∘ (id ⁂ id)) +₁ i₁) ∘ distributeʳ⁻¹) ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (assoc ○ (+₁-cong₂ (elimʳ (⟨⟩-unique id-comm id-comm)) refl) ⟩∘⟨refl) refl ⟩
|
||||
[ (eval′ +₁ i₁) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ≈˘⟨ []-cong₂ (pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ identityʳ)) refl ⟩
|
||||
[ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoʳ) ⟩
|
||||
([ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ∘ distributeʳ⁻¹) ∘ distributeʳ ∎)
|
||||
|
||||
open Elgot-Algebra-Morphism renaming (h to <_>)
|
||||
|
||||
-- cccc : CanonicalCartesianClosed Elgot-Algebras
|
||||
-- CanonicalCartesianClosed.⊤ cccc = Terminal.⊤ Terminal-Elgot-Algebras
|
||||
-- CanonicalCartesianClosed._×_ cccc X Y = A×B-Helper {X} {Y}
|
||||
-- CanonicalCartesianClosed.! cccc = Terminal.! Terminal-Elgot-Algebras
|
||||
-- CanonicalCartesianClosed.π₁ cccc {X} {Y} = Product.π₁ (Product-Elgot-Algebras X Y)
|
||||
-- CanonicalCartesianClosed.π₂ cccc {X} {Y} = Product.π₂ (Product-Elgot-Algebras X Y)
|
||||
-- CanonicalCartesianClosed.⟨_,_⟩ cccc {X} {Y} {Z} f g = Product.⟨_,_⟩ (Product-Elgot-Algebras Y Z) f g
|
||||
-- CanonicalCartesianClosed.!-unique cccc = Terminal.!-unique Terminal-Elgot-Algebras
|
||||
-- CanonicalCartesianClosed.π₁-comp cccc {X} {Y} {f} {Z} {g} = Product.project₁ (Product-Elgot-Algebras Y Z) {h = f} {i = g}
|
||||
-- CanonicalCartesianClosed.π₂-comp cccc {X} {Y} {f} {Z} {g} = Product.project₂ (Product-Elgot-Algebras Y Z) {h = f} {i = g}
|
||||
-- CanonicalCartesianClosed.⟨,⟩-unique cccc {C} {A} {B} {f} {g} {h} eq₁ eq₂ = Product.unique (Product-Elgot-Algebras A B) {h = h} {i = f} {j = g} eq₁ eq₂
|
||||
-- CanonicalCartesianClosed._^_ cccc A X = B^A-Helper {A} {Elgot-Algebra.A X}
|
||||
-- (Elgot-Algebra-Morphism.h) (CanonicalCartesianClosed.eval cccc {A} {B}) = eval′
|
||||
-- (Elgot-Algebra-Morphism.preserves) (CanonicalCartesianClosed.eval cccc {A} {B}) {X} {f} = begin
|
||||
-- eval′ ∘ ⟨ (λg ((Elgot-Algebra._# A) ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((π₁ +₁ id) ∘ f ⁂ id)))) , _ ⟩ ≈˘⟨ refl⟩∘⟨ (⁂∘⟨⟩ ○ ⟨⟩-cong₂ identityʳ identityˡ) ⟩
|
||||
-- eval′ ∘ ((λg ((Elgot-Algebra._# A) ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((π₁ +₁ id) ∘ f ⁂ id)))) ⁂ id) ∘ ⟨ id , _ ⟩ ≈⟨ pullˡ β′ ⟩
|
||||
-- ((Elgot-Algebra._# A) ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((π₁ +₁ id) ∘ f ⁂ id))) ∘ ⟨ id , _ ⟩ ≈˘⟨ Elgot-Algebra.#-Uniformity A by-uni ⟩
|
||||
-- (Elgot-Algebra._# A) ((eval′ +₁ id) ∘ f) ∎ -- ⟨ ((π₁ +₁ id) ∘ h)#ᵃ , ((π₂ +₁ id) ∘ h)#ᵇ ⟩
|
||||
-- where
|
||||
-- by-uni : (id +₁ ⟨ id , _ ⟩) ∘ ((eval′ +₁ id) ∘ f) ≈ ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((π₁ +₁ id) ∘ f ⁂ id)) ∘ ⟨ id , _ ⟩
|
||||
-- by-uni = begin
|
||||
-- (id +₁ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩) ∘ ((eval′ +₁ id) ∘ f) ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ identityʳ) ⟩
|
||||
-- (eval′ +₁ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩) ∘ f ≈˘⟨ sym-assoc ○ sym-assoc ○ (sym (+-unique by-inj₁ by-inj₂)) ⟩∘⟨refl ⟩
|
||||
-- (eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹) ∘ ⟨ id , [ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ] ∘ (π₂ +₁ id) ⟩ ∘ f ≈⟨ refl⟩∘⟨ refl⟩∘⟨ (⟨⟩∘ ○ ⟨⟩-cong₂ refl assoc) ⟩
|
||||
-- (eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹) ∘ ⟨ id ∘ f , [ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ] ∘ (π₂ +₁ id) ∘ f ⟩ ≈˘⟨ refl⟩∘⟨ (refl⟩∘⟨ (⟨⟩-cong₂ refl (Elgot-Algebra.#-Fixpoint B))) ⟩
|
||||
-- (eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹) ∘ ⟨ id ∘ f , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈˘⟨ pullʳ (pullʳ (⁂∘⟨⟩ ○ ⟨⟩-cong₂ id-comm identityˡ)) ⟩
|
||||
-- ((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹) ∘ (f ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈˘⟨ (refl⟩∘⟨ (pullˡ (sym (distributeʳ⁻¹-natural id π₁ id)))) ⟩∘⟨refl ⟩
|
||||
-- ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((π₁ +₁ id) ⁂ id) ∘ (f ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ (refl⟩∘⟨ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²)) ⟩∘⟨refl ⟩
|
||||
-- ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((π₁ +₁ id) ∘ f ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ∎
|
||||
-- where
|
||||
-- by-inj₁ : (((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ ⟨ id , [ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ] ∘ (π₂ +₁ id) ⟩) ∘ i₁ ≈ i₁ ∘ eval′
|
||||
-- by-inj₁ = begin
|
||||
-- (((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ ⟨ id , [ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ] ∘ (π₂ +₁ id) ⟩) ∘ i₁ ≈⟨ pullʳ ⟨⟩∘ ⟩
|
||||
-- ((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ ⟨ id ∘ i₁ , ([ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ] ∘ (π₂ +₁ id)) ∘ i₁ ⟩ ≈⟨ refl⟩∘⟨ (⟨⟩-cong₂ id-comm-sym (pullʳ inject₁ ○ pullˡ inject₁ ○ identityˡ)) ⟩
|
||||
-- ((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ ⟨ i₁ ∘ id , π₂ ⟩ ≈˘⟨ refl⟩∘⟨ (⁂∘⟨⟩ ○ ⟨⟩-cong₂ refl identityˡ) ⟩
|
||||
-- ((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ (i₁ ⁂ id) ∘ ⟨ id , π₂ ⟩ ≈⟨ pullʳ (pullʳ (pullˡ distributeʳ⁻¹-i₁)) ⟩
|
||||
-- (eval′ +₁ id) ∘ (π₁ ⁂ id +₁ id ⁂ id) ∘ i₁ ∘ ⟨ id , π₂ ⟩ ≈⟨ refl⟩∘⟨ (pullˡ +₁∘i₁) ⟩
|
||||
-- (eval′ +₁ id) ∘ (i₁ ∘ (π₁ ⁂ id)) ∘ ⟨ id , π₂ ⟩ ≈⟨ pullˡ (pullˡ +₁∘i₁) ⟩
|
||||
-- ((i₁ ∘ eval′) ∘ (π₁ ⁂ id)) ∘ ⟨ id , π₂ ⟩ ≈⟨ cancelʳ (⁂∘⟨⟩ ○ (⟨⟩-cong₂ identityʳ identityˡ) ○ ⟨⟩-unique identityʳ identityʳ) ⟩
|
||||
-- i₁ ∘ eval′ ∎
|
||||
-- by-inj₂ : (((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ ⟨ id , [ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ] ∘ (π₂ +₁ id) ⟩) ∘ i₂ ≈ i₂ ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩
|
||||
-- by-inj₂ = begin
|
||||
-- (((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ ⟨ id , [ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ] ∘ (π₂ +₁ id) ⟩) ∘ i₂ ≈⟨ pullʳ (⟨⟩∘ ○ ⟨⟩-cong₂ id-comm-sym (pullʳ inject₂)) ⟩
|
||||
-- ((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ ⟨ i₂ ∘ id , [ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ] ∘ i₂ ∘ id ⟩ ≈⟨ refl⟩∘⟨ (⟨⟩-cong₂ refl (refl⟩∘⟨ identityʳ ○ inject₂)) ⟩
|
||||
-- ((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ ⟨ i₂ ∘ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈˘⟨ refl⟩∘⟨ (⁂∘⟨⟩ ○ ⟨⟩-cong₂ refl identityˡ) ⟩
|
||||
-- ((eval′ +₁ id) ∘ ((π₁ ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹)) ∘ (i₂ ⁂ id) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ pullʳ (pullʳ (pullˡ distributeʳ⁻¹-i₂)) ⟩
|
||||
-- (eval′ +₁ id) ∘ (π₁ ⁂ id +₁ id ⁂ id) ∘ i₂ ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ refl⟩∘⟨ (pullˡ inject₂) ⟩
|
||||
-- (eval′ +₁ id) ∘ (i₂ ∘ (id ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ pullˡ (pullˡ inject₂) ⟩
|
||||
-- ((i₂ ∘ id) ∘ (id ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ (cancelʳ (identityˡ ○ ⟨⟩-unique id-comm id-comm)) ⟩∘⟨refl ⟩
|
||||
-- i₂ ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ∎
|
||||
-- < CanonicalCartesianClosed.curry cccc {A} {B} {C} f > = λg < f >
|
||||
-- preserves (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) {X} {g} = begin
|
||||
-- λg < f > ∘ g #ᵃ ≈⟨ subst ⟩
|
||||
-- λg (< f > ∘ (g #ᵃ ⁂ id)) ≈⟨ {! !} ⟩
|
||||
-- {! !} ≈⟨ {! !} ⟩
|
||||
-- {! !} ≈⟨ {! !} ⟩
|
||||
-- λg (< f > ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩) ≈⟨ λ-cong (preserves f) ⟩
|
||||
-- λg (((< f > +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᶜ) ≈⟨ {! !} ⟩
|
||||
-- {! !} ≈⟨ {! !} ⟩
|
||||
-- {! !} ≈⟨ {! !} ⟩
|
||||
-- λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((λg < f > +₁ id) ∘ g) ⁂ id)) #ᶜ) ∎
|
||||
-- where
|
||||
-- open Elgot-Algebra A using () renaming (_# to _#ᵃ; #-Uniformity to #ᵃ-Uniformity)
|
||||
-- open Elgot-Algebra B using () renaming (_# to _#ᵇ; #-Uniformity to #ᵇ-Uniformity)
|
||||
-- open Elgot-Algebra C using () renaming (_# to _#ᶜ; #-resp-≈ to #ᶜ-resp-≈)
|
||||
-- -- uniqueness : ∀
|
||||
-- -- Elgot-Algebra-Morphism.preserves (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) {X} {g} = λ-unique′ (begin
|
||||
-- -- eval′ ∘ ((λg < f > ∘ g #ᵃ) ⁂ id) ≈⟨ refl⟩∘⟨ ⁂-cong₂ subst refl ⟩
|
||||
-- -- eval′ ∘ ((λg (< f > ∘ (g #ᵃ ⁂ id))) ⁂ id) ≈⟨ β′ ⟩
|
||||
-- -- < f > ∘ (g #ᵃ ⁂ id) ≈⟨ refl⟩∘⟨ (⟨⟩-unique by-π₁ by-π₂) ⟩
|
||||
-- -- < f > ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩ ≈⟨ Elgot-Algebra-Morphism.preserves f ⟩
|
||||
-- -- ((< f > +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᶜ ≈˘⟨ #ᶜ-resp-≈ (refl⟩∘⟨ assoc ○ pullˡ (+₁∘+₁ ○ +₁-cong₂ β′ (elimʳ (⟨⟩-unique id-comm id-comm)))) ⟩
|
||||
-- -- ((eval′ +₁ id) ∘ (((λg < f > ⁂ id) +₁ (id ⁂ id)) ∘ distributeʳ⁻¹) ∘ (g ⁂ id)) #ᶜ ≈˘⟨ Elgot-Algebra.#-resp-≈ C (refl⟩∘⟨ (pullˡ (sym (distributeʳ⁻¹-natural id (λg < f >) id)))) ⟩
|
||||
-- -- ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((λg < f > +₁ id) ⁂ id) ∘ (g ⁂ id)) #ᶜ ≈⟨ Elgot-Algebra.#-resp-≈ C (refl⟩∘⟨ (refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²))) ⟩
|
||||
-- -- ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((λg < f > +₁ id) ∘ g) ⁂ id)) #ᶜ ∎)
|
||||
-- -- where
|
||||
-- -- open Elgot-Algebra A using () renaming (_# to _#ᵃ; #-Uniformity to #ᵃ-Uniformity)
|
||||
-- -- open Elgot-Algebra B using () renaming (_# to _#ᵇ; #-Uniformity to #ᵇ-Uniformity)
|
||||
-- -- open Elgot-Algebra C using () renaming (_# to _#ᶜ; #-resp-≈ to #ᶜ-resp-≈)
|
||||
-- -- by-π₁ : π₁ ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩ ≈ g #ᵃ ∘ π₁
|
||||
-- -- by-π₁ = project₁ ○ #ᵃ-Uniformity by-uni
|
||||
-- -- where
|
||||
-- -- by-uni : (id +₁ π₁) ∘ (π₁ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id) ≈ g ∘ π₁
|
||||
-- -- by-uni = begin
|
||||
-- -- (id +₁ π₁) ∘ (π₁ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id) ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ identityʳ) ⟩
|
||||
-- -- (π₁ +₁ π₁) ∘ distributeʳ⁻¹ ∘ (g ⁂ id) ≈⟨ pullˡ distributeʳ⁻¹-π₁ ⟩
|
||||
-- -- π₁ ∘ (g ⁂ id) ≈⟨ project₁ ⟩
|
||||
-- -- g ∘ π₁ ∎
|
||||
-- -- by-π₂ : π₂ ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩ ≈ id ∘ π₂
|
||||
-- -- by-π₂ = begin
|
||||
-- -- π₂ ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩ ≈⟨ project₂ ⟩
|
||||
-- -- ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ≈⟨ #ᵇ-Uniformity by-uni ⟩
|
||||
-- -- {! !} #ᵇ ∘ π₂ ≈⟨ {! !} ⟩
|
||||
-- -- {! !} ≈⟨ {! !} ⟩
|
||||
-- -- {! !} ≈⟨ {! !} ⟩
|
||||
-- -- {! !} ≈⟨ {! !} ⟩
|
||||
-- -- id ∘ π₂ ∎
|
||||
-- -- where
|
||||
-- -- by-uni : (id +₁ π₂) ∘ (π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)) ≈ {! !} ∘ π₂
|
||||
-- -- by-uni = begin
|
||||
-- -- {! !} ≈⟨ {! !} ⟩
|
||||
-- -- {! !} ≈⟨ {! !} ⟩
|
||||
-- -- {! !} ≈⟨ {! !} ⟩
|
||||
-- -- {! !} ≈⟨ {! !} ⟩
|
||||
-- -- {! !} ∎
|
||||
-- CanonicalCartesianClosed.eval-comp cccc {A} {B} {C} {f} = β′
|
||||
-- CanonicalCartesianClosed.curry-unique cccc {A} {B} {C} {f} {g} eq = λ-unique′ eq
|
||||
|
||||
-- Elgot-CCC : CartesianClosed Elgot-Algebras
|
||||
-- Elgot-CCC = Equivalence.fromCanonical Elgot-Algebras cccc
|
||||
```
|
||||
|
|
|
|||
|
|
@ -0,0 +1,115 @@
|
|||
<!--
|
||||
```agda
|
||||
open import Level
|
||||
open import Categories.Category.Cocartesian using (Cocartesian)
|
||||
open import Categories.Category.Cartesian using (Cartesian)
|
||||
open import Categories.Category.CartesianClosed using (CartesianClosed)
|
||||
open import Categories.Category.BinaryProducts using (BinaryProducts)
|
||||
open import Categories.Object.Terminal using (Terminal)
|
||||
open import Categories.Object.Product using (Product)
|
||||
open import Categories.Object.Coproduct using (Coproduct)
|
||||
open import Categories.Object.Exponential using (Exponential)
|
||||
open import Categories.Category
|
||||
open import Categories.Category.Distributive
|
||||
open import Categories.Object.Exponential
|
||||
open import Category.Ambient using (Ambient)
|
||||
|
||||
import Categories.Morphism as M
|
||||
import Categories.Morphism.Reasoning as MR
|
||||
import Categories.Morphism.Properties as MP
|
||||
```
|
||||
-->
|
||||
|
||||
```agda
|
||||
module Category.Construction.ElgotAlgebras.Exponentials {o ℓ e} {C : Category o ℓ e} (distributive : Distributive C) (exp : ∀ {A B} → Exponential C A B) where
|
||||
open Category C
|
||||
open HomReasoning
|
||||
open M C
|
||||
open MR C
|
||||
open MP C
|
||||
open Equiv
|
||||
open Distributive distributive
|
||||
open import Categories.Category.Distributive.Properties distributive
|
||||
ccc : CartesianClosed C
|
||||
ccc = record { cartesian = cartesian ; exp = exp }
|
||||
open CartesianClosed ccc hiding (cartesian; exp)
|
||||
open Cocartesian cocartesian
|
||||
open Cartesian cartesian
|
||||
open BinaryProducts products renaming (unique to ⟨⟩-unique; unique′ to ⟨⟩-unique′)
|
||||
open import Algebra.Elgot cocartesian
|
||||
open import Category.Construction.ElgotAlgebras cocartesian
|
||||
open import Category.Construction.ElgotAlgebras.Products cocartesian cartesian
|
||||
|
||||
Exponential-Elgot-Algebra : ∀ {EA : Elgot-Algebra} {X : Obj} → Elgot-Algebra
|
||||
Exponential-Elgot-Algebra {EA} {X} = record
|
||||
{ A = A ^ X
|
||||
; algebra = record
|
||||
{ _# = λ {Z} f → λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)
|
||||
; #-Fixpoint = λ {X} {f} → begin
|
||||
λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ≈⟨ λ-cong #ᵃ-Fixpoint ⟩
|
||||
λg ([ id , ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ ] ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) ≈⟨ λ-cong ((pullˡ []∘+₁) ○ ([]-cong₂ identityˡ identityʳ) ⟩∘⟨refl) ⟩
|
||||
λg ([ eval′ , ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ ] ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) ≈˘⟨ λ-unique′ (begin
|
||||
eval′ ∘ (([ id , λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ] ∘ f) ⁂ id) ≈˘⟨ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²) ⟩
|
||||
eval′ ∘ ([ id , λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ] ⁂ id) ∘ (f ⁂ id) ≈⟨ refl⟩∘⟨ (⟨⟩-unique (begin
|
||||
π₁ ∘ [ id , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ⁂ id ] ∘ distributeʳ⁻¹ ≈⟨ pullˡ (∘[] ○ []-cong₂ id-comm π₁∘⁂) ⟩
|
||||
[ id ∘ π₁ , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ∘ π₁ ] ∘ distributeʳ⁻¹ ≈˘⟨ pullˡ []∘+₁ ⟩
|
||||
[ id , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ] ∘ (π₁ +₁ π₁) ∘ distributeʳ⁻¹ ≈⟨ refl⟩∘⟨ distributeʳ⁻¹-π₁ ⟩
|
||||
[ id , λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ] ∘ π₁ ∎)
|
||||
(begin
|
||||
π₂ ∘ [ id , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ⁂ id ] ∘ distributeʳ⁻¹ ≈⟨ pullˡ (∘[] ○ []-cong₂ identityʳ (π₂∘⁂ ○ identityˡ)) ⟩
|
||||
[ π₂ , π₂ ] ∘ distributeʳ⁻¹ ≈⟨ distributeʳ⁻¹-π₂ ○ (sym identityˡ) ⟩
|
||||
id ∘ π₂ ∎)
|
||||
⟩∘⟨refl) ⟩
|
||||
eval′ ∘ ([ id , (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ⁂ id ] ∘ distributeʳ⁻¹) ∘ (f ⁂ id) ≈⟨ pullˡ (pullˡ ∘[]) ⟩
|
||||
([ eval′ ∘ id , eval′ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) ⁂ id) ] ∘ distributeʳ⁻¹) ∘ (f ⁂ id) ≈⟨ assoc ○ ([]-cong₂ identityʳ β′) ⟩∘⟨refl ⟩
|
||||
[ eval′ , ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ ] ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ∎) ⟩
|
||||
[ id , λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ] ∘ f ∎
|
||||
; #-Uniformity = #-Uniformity
|
||||
; #-Folding = #-Folding
|
||||
; #-resp-≈ = λ {Z} {f} {g} f≈g → λ-cong (#ᵃ-resp-≈ (refl⟩∘⟨ refl⟩∘⟨ ⁂-cong₂ f≈g refl))
|
||||
}
|
||||
}
|
||||
where
|
||||
open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
|
||||
#-Uniformity : ∀ {D E} {f : D ⇒ A ^ X + D} {g : E ⇒ A ^ X + E} {h : D ⇒ E} → (id +₁ h) ∘ f ≈ g ∘ h → λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ≈ λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᵃ) ∘ h
|
||||
#-Uniformity {D} {E} {f} {g} {h} eq = sym (λ-unique′ (begin
|
||||
eval′ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᵃ) ∘ h) ⁂ id) ≈˘⟨ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identityˡ) ⟩
|
||||
eval′ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᵃ)) ⁂ id) ∘ (h ⁂ id) ≈⟨ pullˡ β′ ⟩
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᵃ ∘ (h ⁂ id) ≈˘⟨ #ᵃ-Uniformity by-uni ⟩
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ ∎))
|
||||
where
|
||||
by-uni : (id +₁ (h ⁂ id)) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ≈ ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) ∘ (h ⁂ id)
|
||||
by-uni = begin
|
||||
(id +₁ (h ⁂ id)) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ identityʳ) ⟩
|
||||
(eval′ +₁ (h ⁂ id)) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ≈˘⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityʳ identityˡ) ⟩
|
||||
(eval′ +₁ id) ∘ (id +₁ (h ⁂ id)) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) ≈⟨ refl⟩∘⟨ pullˡ ((+₁-cong₂ (sym (⟨⟩-unique id-comm id-comm)) refl) ⟩∘⟨refl ○ distributeʳ⁻¹-natural id id h) ⟩
|
||||
(eval′ +₁ id) ∘ (distributeʳ⁻¹ ∘ ((id +₁ h) ⁂ id)) ∘ (f ⁂ id) ≈⟨ refl⟩∘⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ eq identity²) ⟩
|
||||
(eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ∘ h ⁂ id) ≈˘⟨ pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ refl identity²)) ⟩
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) ∘ (h ⁂ id) ∎
|
||||
#-Folding : ∀ {D E} {f : D ⇒ A ^ X + D} {h : E ⇒ D + E} → λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) +₁ h) ⁂ id)) #ᵃ) ≈ λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) #ᵃ)
|
||||
#-Folding {D} {E} {f} {h} = λ-cong (begin
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) +₁ h) ⁂ id)) #ᵃ ≈⟨ #ᵃ-resp-≈ (refl⟩∘⟨ sym (distributeʳ⁻¹-natural id (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ)) h)) ⟩
|
||||
((eval′ +₁ id) ∘ ((λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ) ⁂ id) +₁ (h ⁂ id)) ∘ distributeʳ⁻¹) #ᵃ ≈⟨ #ᵃ-resp-≈ (pullˡ (+₁∘+₁ ○ +₁-cong₂ β′ identityˡ)) ⟩
|
||||
((((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ h ⁂ id) ∘ distributeʳ⁻¹) #ᵃ ≈⟨ #ᵃ-Uniformity by-uni₁ ⟩
|
||||
(((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ distributeʳ⁻¹ ∘ (h ⁂ id)) #ᵃ ∘ distributeʳ⁻¹ ≈⟨ #ᵃ-Folding ⟩∘⟨refl ⟩
|
||||
[ ((id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) , (i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id)) ] #ᵃ ∘ distributeʳ⁻¹ ≈˘⟨ #ᵃ-Uniformity by-uni₂ ⟩
|
||||
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) #ᵃ ∎)
|
||||
where
|
||||
by-uni₁ : (id +₁ distributeʳ⁻¹) ∘ (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ h ⁂ id) ∘ distributeʳ⁻¹ ≈ ((((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ distributeʳ⁻¹ ∘ (h ⁂ id))) ∘ distributeʳ⁻¹
|
||||
by-uni₁ = begin
|
||||
(id +₁ distributeʳ⁻¹) ∘ (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ h ⁂ id) ∘ distributeʳ⁻¹ ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ refl) ⟩
|
||||
((((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id)) #ᵃ +₁ distributeʳ⁻¹ ∘ (h ⁂ id))) ∘ distributeʳ⁻¹ ∎
|
||||
by-uni₂ : (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id) ≈ [ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ∘ distributeʳ⁻¹
|
||||
by-uni₂ = Iso⇒Epi (IsIso.iso isIsoʳ) ((id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) ([ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ∘ distributeʳ⁻¹) (begin
|
||||
((id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) ∘ distributeʳ ≈⟨ ∘[] ⟩
|
||||
[ (((id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) ∘ (i₁ ⁂ id)) , (((id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ([ (id +₁ i₁) ∘ f , i₂ ∘ h ] ⁂ id)) ∘ (i₂ ⁂ id)) ] ≈⟨ []-cong₂ (pullʳ (pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ inject₁ identity²)))) (pullʳ (pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ inject₂ identity²)))) ⟩
|
||||
[ (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((id +₁ i₁) ∘ f) ⁂ id) , (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((i₂ ∘ h) ⁂ id) ] ≈⟨ []-cong₂ (refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ sym (⁂∘⁂ ○ ⁂-cong₂ refl identity²)) (refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ sym (⁂∘⁂ ○ ⁂-cong₂ refl identity²)) ⟩
|
||||
[ (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((id +₁ i₁)) ⁂ id) ∘ (f ⁂ id) , (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (i₂ ⁂ id) ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (refl⟩∘⟨ refl⟩∘⟨ pullˡ (sym (distributeʳ⁻¹-natural id id i₁))) (refl⟩∘⟨ refl⟩∘⟨ pullˡ distributeʳ⁻¹-i₂) ⟩
|
||||
[ (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ ((id ⁂ id +₁ i₁ ⁂ id) ∘ distributeʳ⁻¹) ∘ (f ⁂ id) , (id +₁ distributeʳ⁻¹) ∘ (eval′ +₁ id) ∘ i₂ ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (pullˡ +₁∘+₁) (pullˡ +₁∘+₁) ⟩
|
||||
[ (id ∘ eval′ +₁ distributeʳ⁻¹ ∘ id) ∘ ((id ⁂ id +₁ i₁ ⁂ id) ∘ distributeʳ⁻¹) ∘ (f ⁂ id) , (id ∘ eval′ +₁ distributeʳ⁻¹ ∘ id) ∘ i₂ ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (pullˡ (pullˡ +₁∘+₁)) (pullˡ +₁∘i₂) ⟩
|
||||
[ ((((id ∘ eval′) ∘ (id ⁂ id)) +₁ ((distributeʳ⁻¹ ∘ id) ∘ (i₁ ⁂ id))) ∘ distributeʳ⁻¹) ∘ (f ⁂ id) , (i₂ ∘ (distributeʳ⁻¹ ∘ id)) ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (((+₁-cong₂ (identityˡ ⟩∘⟨refl) (identityʳ ⟩∘⟨refl ○ distributeʳ⁻¹-i₁)) ⟩∘⟨refl) ⟩∘⟨refl) (pullʳ (pullʳ identityˡ)) ⟩
|
||||
[ (((eval′ ∘ (id ⁂ id)) +₁ i₁) ∘ distributeʳ⁻¹) ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ≈⟨ []-cong₂ (assoc ○ (+₁-cong₂ (elimʳ (⟨⟩-unique id-comm id-comm)) refl) ⟩∘⟨refl) refl ⟩
|
||||
[ (eval′ +₁ i₁) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ≈˘⟨ []-cong₂ (pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ identityʳ)) refl ⟩
|
||||
[ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoʳ) ⟩
|
||||
([ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ∘ distributeʳ⁻¹) ∘ distributeʳ ∎)
|
||||
```
|
||||
154
agda/src/Category/Construction/ElgotAlgebras/Products.lagda.md
Normal file
154
agda/src/Category/Construction/ElgotAlgebras/Products.lagda.md
Normal file
|
|
@ -0,0 +1,154 @@
|
|||
<!--
|
||||
```agda
|
||||
open import Level
|
||||
open import Categories.Category.Cocartesian using (Cocartesian)
|
||||
open import Categories.Category.Cartesian using (Cartesian)
|
||||
open import Categories.Category.CartesianClosed using (CartesianClosed)
|
||||
open import Categories.Category.BinaryProducts using (BinaryProducts)
|
||||
open import Categories.Object.Terminal using (Terminal)
|
||||
open import Categories.Object.Product using (Product)
|
||||
open import Categories.Category.Core
|
||||
open import Category.Ambient using (Ambient)
|
||||
|
||||
import Categories.Morphism.Reasoning as MR
|
||||
```
|
||||
-->
|
||||
|
||||
# The category of elgot algebras on C is cartesian if C is cartesian
|
||||
|
||||
```agda
|
||||
module Category.Construction.ElgotAlgebras.Products {o ℓ e} {C : Category o ℓ e} (cocartesian : Cocartesian C) (cartesian : Cartesian C) where
|
||||
open Category C
|
||||
open Equiv
|
||||
open HomReasoning
|
||||
open MR C
|
||||
open import Category.Construction.ElgotAlgebras cocartesian
|
||||
open Cartesian cartesian
|
||||
open Cocartesian cocartesian
|
||||
open import Algebra.Elgot cocartesian
|
||||
open BinaryProducts products renaming (unique to ⟨⟩-unique; unique′ to ⟨⟩-unique′)
|
||||
|
||||
-- if the carrier contains a terminal, so does elgot-algebras
|
||||
Terminal-Elgot-Algebras : Terminal Elgot-Algebras
|
||||
Terminal-Elgot-Algebras = record
|
||||
{ ⊤ = record
|
||||
{ A = ⊤
|
||||
; algebra = record
|
||||
{ _# = λ x → !
|
||||
; #-Fixpoint = λ {_ f} → !-unique ([ id , ! ] ∘ f)
|
||||
; #-Uniformity = λ {_ _ _ _ h} _ → !-unique (! ∘ h)
|
||||
; #-Folding = refl
|
||||
; #-resp-≈ = λ _ → refl
|
||||
}
|
||||
}
|
||||
; ⊤-is-terminal = record
|
||||
{ ! = λ {A} → record { h = ! ; preserves = λ {X} {f} → sym (!-unique (! ∘ (A Elgot-Algebra.#) f)) }
|
||||
; !-unique = λ {A} f → !-unique (Elgot-Algebra-Morphism.h f)
|
||||
}
|
||||
}
|
||||
where
|
||||
open Terminal terminal
|
||||
|
||||
-- the product of the carrier of two elgot algebras is again an elgot algebra
|
||||
Product-Elgot-Algebra : ∀ {EA EB : Elgot-Algebra} → Elgot-Algebra
|
||||
Product-Elgot-Algebra {EA} {EB} = record
|
||||
{ A = A × B
|
||||
; algebra = record
|
||||
{ _# = λ {X : Obj} (h : X ⇒ A×B + X) → ⟨ ((π₁ +₁ id) ∘ h)#ᵃ , ((π₂ +₁ id) ∘ h)#ᵇ ⟩
|
||||
; #-Fixpoint = λ {X} {f} → begin
|
||||
⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ≈⟨ ⟨⟩-cong₂ #ᵃ-Fixpoint #ᵇ-Fixpoint ⟩
|
||||
⟨ [ id , ((π₁ +₁ id) ∘ f)#ᵃ ] ∘ ((π₁ +₁ id) ∘ f) , [ id , ((π₂ +₁ id) ∘ f)#ᵇ ] ∘ ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ ⟨⟩-cong₂ (pullˡ []∘+₁) (pullˡ []∘+₁) ⟩
|
||||
⟨ [ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] ∘ f , [ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ∘ f ⟩ ≈˘⟨ ⟨⟩∘ ⟩
|
||||
(⟨ [ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] , [ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ⟩ ∘ f) ≈⟨ ∘-resp-≈ˡ (⟨⟩-unique′
|
||||
(begin
|
||||
π₁ ∘ ⟨ [ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] , [ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ⟩ ≈⟨ project₁ ⟩
|
||||
[ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] ≈⟨ []-cong₂ id-comm-sym (trans identityʳ (sym project₁)) ⟩
|
||||
[ π₁ ∘ id , π₁ ∘ ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ≈˘⟨ ∘[] ⟩
|
||||
π₁ ∘ [ id , ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ∎)
|
||||
(begin
|
||||
π₂ ∘ ⟨ [ id ∘ π₁ , ((π₁ +₁ id) ∘ f)#ᵃ ∘ id ] , [ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ⟩ ≈⟨ project₂ ⟩
|
||||
[ id ∘ π₂ , ((π₂ +₁ id) ∘ f)#ᵇ ∘ id ] ≈⟨ []-cong₂ id-comm-sym (trans identityʳ (sym project₂)) ⟩
|
||||
[ π₂ ∘ id , π₂ ∘ ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ≈˘⟨ ∘[] ⟩
|
||||
π₂ ∘ [ id , ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ∎)
|
||||
)⟩
|
||||
([ id , ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ] ∘ f) ∎
|
||||
; #-Uniformity = λ {X Y f g h} uni → ⟨⟩-unique′
|
||||
(begin
|
||||
π₁ ∘ ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ≈⟨ project₁ ⟩
|
||||
(((π₁ +₁ id) ∘ f)#ᵃ) ≈⟨ #ᵃ-Uniformity
|
||||
(begin
|
||||
-- TODO factor these out or adjust +₁ swap... maybe call it +₁-id-comm
|
||||
(id +₁ h) ∘ (π₁ +₁ id) ∘ f ≈⟨ pullˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm)) ⟩
|
||||
(π₁ ∘ id +₁ id ∘ h) ∘ f ≈˘⟨ pullˡ +₁∘+₁ ⟩
|
||||
(π₁ +₁ id) ∘ (id +₁ h) ∘ f ≈⟨ pushʳ uni ⟩
|
||||
((π₁ +₁ id) ∘ g) ∘ h ∎)⟩
|
||||
(((π₁ +₁ id) ∘ g)#ᵃ ∘ h) ≈˘⟨ pullˡ project₁ ⟩
|
||||
π₁ ∘ ⟨ ((π₁ +₁ id) ∘ g)#ᵃ , ((π₂ +₁ id) ∘ g)#ᵇ ⟩ ∘ h ∎)
|
||||
(begin
|
||||
π₂ ∘ ⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ ≈⟨ project₂ ⟩
|
||||
((π₂ +₁ id) ∘ f)#ᵇ ≈⟨ #ᵇ-Uniformity
|
||||
(begin
|
||||
(id +₁ h) ∘ (π₂ +₁ id) ∘ f ≈⟨ pullˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm))⟩
|
||||
((π₂ ∘ id +₁ id ∘ h) ∘ f) ≈˘⟨ pullˡ +₁∘+₁ ⟩
|
||||
(π₂ +₁ id) ∘ ((id +₁ h)) ∘ f ≈⟨ pushʳ uni ⟩
|
||||
((π₂ +₁ id) ∘ g) ∘ h ∎)⟩
|
||||
((π₂ +₁ id) ∘ g)#ᵇ ∘ h ≈˘⟨ pullˡ project₂ ⟩
|
||||
π₂ ∘ ⟨ ((π₁ +₁ id) ∘ g)#ᵃ , ((π₂ +₁ id) ∘ g)#ᵇ ⟩ ∘ h ∎)
|
||||
; #-Folding = λ {X} {Y} {f} {h} → ⟨⟩-cong₂ (foldingˡ {X} {Y}) (foldingʳ {X} {Y})
|
||||
; #-resp-≈ = λ fg → ⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ʳ fg)) (#ᵇ-resp-≈ (∘-resp-≈ʳ fg))
|
||||
}
|
||||
}
|
||||
where
|
||||
open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
|
||||
open Elgot-Algebra EB using () renaming (A to B; _# to _#ᵇ; #-Fixpoint to #ᵇ-Fixpoint; #-Uniformity to #ᵇ-Uniformity; #-Folding to #ᵇ-Folding; #-resp-≈ to #ᵇ-resp-≈)
|
||||
+₁-id-swap : ∀ {X Y C} {f : X ⇒ (A × B) + X} {h : Y ⇒ X + Y} (π : A × B ⇒ C) → [ (id +₁ i₁) ∘ ((π +₁ id) ∘ f) , i₂ ∘ h ] ≈ (π +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ]
|
||||
+₁-id-swap {X} {Y} {C} {f} {h} π = begin ([ (id +₁ i₁) ∘ ((π +₁ id) ∘ f) , i₂ ∘ h ] ) ≈⟨ ([]-congʳ sym-assoc) ⟩
|
||||
([ ((id +₁ i₁) ∘ (π +₁ id)) ∘ f , i₂ ∘ h ] ) ≈⟨ []-cong₂ (∘-resp-≈ˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm))) (∘-resp-≈ˡ (sym identityʳ)) ⟩
|
||||
(([ (π ∘ id +₁ id ∘ i₁) ∘ f , (i₂ ∘ id) ∘ h ])) ≈˘⟨ []-cong₂ (pullˡ +₁∘+₁) (pullˡ +₁∘i₂) ⟩
|
||||
(([ (π +₁ id) ∘ (id +₁ i₁) ∘ f , (π +₁ id) ∘ i₂ ∘ h ])) ≈˘⟨ ∘[] ⟩
|
||||
((π +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ]) ∎
|
||||
foldingˡ : ∀ {X} {Y} {f} {h} → (((π₁ +₁ id) ∘ (⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ) ≈ ((π₁ +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ
|
||||
foldingˡ {X} {Y} {f} {h} = begin
|
||||
((π₁ +₁ id) ∘ (⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ ≈⟨ #ᵃ-resp-≈ (trans +₁∘+₁ (+₁-cong₂ project₁ identityˡ)) ⟩
|
||||
((((π₁ +₁ id) ∘ f)#ᵃ +₁ h)#ᵃ) ≈⟨ #ᵃ-Folding ⟩
|
||||
([ (id +₁ i₁) ∘ ((π₁ +₁ id) ∘ f) , i₂ ∘ h ] #ᵃ) ≈⟨ #ᵃ-resp-≈ (+₁-id-swap π₁)⟩
|
||||
((π₁ +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ ∎
|
||||
foldingʳ : ∀ {X} {Y} {f} {h} → ((π₂ +₁ id) ∘ (⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈ ((π₂ +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ
|
||||
foldingʳ {X} {Y} {f} {h} = begin
|
||||
((π₂ +₁ id) ∘ (⟨ ((π₁ +₁ id) ∘ f)#ᵃ , ((π₂ +₁ id) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈⟨ #ᵇ-resp-≈ (trans +₁∘+₁ (+₁-cong₂ project₂ identityˡ)) ⟩
|
||||
((((π₂ +₁ id) ∘ f)#ᵇ +₁ h)#ᵇ) ≈⟨ #ᵇ-Folding ⟩
|
||||
[ (id +₁ i₁) ∘ ((π₂ +₁ id) ∘ f) , i₂ ∘ h ] #ᵇ ≈⟨ #ᵇ-resp-≈ (+₁-id-swap π₂) ⟩
|
||||
((π₂ +₁ id) ∘ [ (id +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ ∎
|
||||
|
||||
Product-Elgot-Algebras : ∀ (EA EB : Elgot-Algebra) → Product Elgot-Algebras EA EB
|
||||
Product-Elgot-Algebras EA EB = record
|
||||
{ A×B = Product-Elgot-Algebra {EA} {EB}
|
||||
; π₁ = record { h = π₁ ; preserves = λ {X} {f} → project₁ }
|
||||
; π₂ = record { h = π₂ ; preserves = λ {X} {f} → project₂ }
|
||||
; ⟨_,_⟩ = λ {E} f g → let
|
||||
open Elgot-Algebra-Morphism f renaming (h to f′; preserves to preservesᶠ)
|
||||
open Elgot-Algebra-Morphism g renaming (h to g′; preserves to preservesᵍ)
|
||||
open Elgot-Algebra E renaming (_# to _#ᵉ) in record { h = ⟨ f′ , g′ ⟩ ; preserves = λ {X} {h} →
|
||||
begin
|
||||
⟨ f′ , g′ ⟩ ∘ (h #ᵉ) ≈⟨ ⟨⟩∘ ⟩
|
||||
⟨ f′ ∘ (h #ᵉ) , g′ ∘ (h #ᵉ) ⟩ ≈⟨ ⟨⟩-cong₂ preservesᶠ preservesᵍ ⟩
|
||||
⟨ ((f′ +₁ id) ∘ h) #ᵃ , ((g′ +₁ id) ∘ h) #ᵇ ⟩ ≈˘⟨ ⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ project₁ identity²))) (#ᵇ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ project₂ identity²))) ⟩
|
||||
⟨ ((π₁ ∘ ⟨ f′ , g′ ⟩ +₁ id ∘ id) ∘ h) #ᵃ , ((π₂ ∘ ⟨ f′ , g′ ⟩ +₁ id ∘ id) ∘ h) #ᵇ ⟩ ≈˘⟨ ⟨⟩-cong₂ (#ᵃ-resp-≈ (pullˡ +₁∘+₁)) (#ᵇ-resp-≈ (pullˡ +₁∘+₁)) ⟩
|
||||
⟨ ((π₁ +₁ id) ∘ (⟨ f′ , g′ ⟩ +₁ id) ∘ h) #ᵃ , ((π₂ +₁ id) ∘ (⟨ f′ , g′ ⟩ +₁ id) ∘ h) #ᵇ ⟩ ∎ }
|
||||
; project₁ = project₁
|
||||
; project₂ = project₂
|
||||
; unique = ⟨⟩-unique
|
||||
}
|
||||
where
|
||||
open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
|
||||
open Elgot-Algebra EB using () renaming (A to B; _# to _#ᵇ; #-Fixpoint to #ᵇ-Fixpoint; #-Uniformity to #ᵇ-Uniformity; #-Folding to #ᵇ-Folding; #-resp-≈ to #ᵇ-resp-≈)
|
||||
open Elgot-Algebra (Product-Elgot-Algebra {EA} {EB}) using () renaming (_# to _#ᵖ)
|
||||
|
||||
|
||||
-- if the carrier is cartesian, so is the category of algebras
|
||||
Cartesian-Elgot-Algebras : Cartesian Elgot-Algebras
|
||||
Cartesian-Elgot-Algebras = record
|
||||
{ terminal = Terminal-Elgot-Algebras
|
||||
; products = record { product = λ {EA EB} → Product-Elgot-Algebras EA EB }
|
||||
}
|
||||
```
|
||||
|
|
@ -1,7 +1,5 @@
|
|||
<!--
|
||||
```agda
|
||||
{-# OPTIONS --no-lossy-unification #-}
|
||||
|
||||
open import Level
|
||||
open import Categories.Category
|
||||
open import Categories.Monad
|
||||
|
|
|
|||
|
|
@ -1,6 +1,5 @@
|
|||
<!--
|
||||
```agda
|
||||
{-# OPTIONS --allow-unsolved-metas #-}
|
||||
open import Level
|
||||
|
||||
open import Categories.Functor
|
||||
|
|
|
|||
|
|
@ -1,6 +1,5 @@
|
|||
<!--
|
||||
```agda
|
||||
{-# OPTIONS --no-lossy-unification #-}
|
||||
open import Level
|
||||
|
||||
open import Data.Product using (_,_; proj₁; proj₂)
|
||||
|
|
|
|||
|
|
@ -17,8 +17,7 @@ open import Categories.Monad.Construction.Kleisli
|
|||
```agda
|
||||
module Monad.Instance.K {o ℓ e} (ambient : Ambient o ℓ e) where
|
||||
open Ambient ambient
|
||||
open import Category.Construction.ElgotAlgebras {C = C}
|
||||
open Cat cocartesian using (Elgot-Algebras)
|
||||
open import Category.Construction.ElgotAlgebras cocartesian
|
||||
open import Algebra.Elgot cocartesian using (Elgot-Algebra)
|
||||
open import Algebra.Elgot.Free cocartesian using (FreeElgotAlgebra; elgotForgetfulF)
|
||||
open import Algebra.Elgot.Stable distributive using (IsStableFreeElgotAlgebra)
|
||||
|
|
|
|||
|
|
@ -1,7 +1,5 @@
|
|||
<!--
|
||||
```agda
|
||||
{-# OPTIONS --no-lossy-unification #-}
|
||||
|
||||
open import Level
|
||||
open import Category.Ambient using (Ambient)
|
||||
open import Categories.Monad.Commutative
|
||||
|
|
@ -20,8 +18,7 @@ open MIK ambient
|
|||
open MonadK MK
|
||||
open import Monad.Instance.K.Strong ambient MK
|
||||
open import Monad.Instance.K.EquationalLifting ambient MK
|
||||
open import Category.Construction.ElgotAlgebras {C = C}
|
||||
open Cat cocartesian
|
||||
open import Category.Construction.ElgotAlgebras cocartesian
|
||||
open import Algebra.Elgot cocartesian
|
||||
open import Algebra.Elgot.Free cocartesian using (FreeElgotAlgebra; elgotForgetfulF)
|
||||
open import Algebra.Elgot.Stable distributive using (IsStableFreeElgotAlgebra; IsStableFreeElgotAlgebraˡ; isStable⇒isStableˡ)
|
||||
|
|
|
|||
|
|
@ -18,8 +18,7 @@ open Ambient ambient
|
|||
open MIK ambient
|
||||
open MonadK MK
|
||||
open import Monad.Instance.K.Strong ambient MK
|
||||
open import Category.Construction.ElgotAlgebras {C = C}
|
||||
open Cat cocartesian
|
||||
open import Category.Construction.ElgotAlgebras cocartesian
|
||||
open import Algebra.Elgot cocartesian
|
||||
open import Algebra.Elgot.Stable distributive using (IsStableFreeElgotAlgebra)
|
||||
|
||||
|
|
|
|||
|
|
@ -24,8 +24,7 @@ open import Monad.Instance.K ambient
|
|||
open import Monad.Instance.K.Commutative ambient MK
|
||||
open import Monad.Instance.K.Strong ambient MK
|
||||
open import Category.Construction.PreElgotMonads ambient
|
||||
open import Category.Construction.ElgotAlgebras {C = C}
|
||||
open Cat cocartesian
|
||||
open import Category.Construction.ElgotAlgebras cocartesian
|
||||
|
||||
open Equiv
|
||||
open HomReasoning
|
||||
|
|
|
|||
|
|
@ -22,8 +22,7 @@ import Monad.Instance.K as MIK
|
|||
```agda
|
||||
module Monad.Instance.K.Strong {o ℓ e} (ambient : Ambient o ℓ e) (MK : MIK.MonadK ambient) where
|
||||
open Ambient ambient
|
||||
open import Category.Construction.ElgotAlgebras {C = C}
|
||||
open Cat cocartesian
|
||||
open import Category.Construction.ElgotAlgebras cocartesian
|
||||
open import Algebra.Elgot cocartesian
|
||||
open import Algebra.Elgot.Free cocartesian using (FreeElgotAlgebra; elgotForgetfulF)
|
||||
open import Algebra.Elgot.Stable distributive using (IsStableFreeElgotAlgebra)
|
||||
|
|
|
|||
|
|
@ -30,8 +30,7 @@ open import Monad.Instance.K.Commutative ambient MK
|
|||
open import Monad.Instance.K.Strong ambient MK
|
||||
open import Monad.Instance.K.PreElgot ambient MK
|
||||
open import Category.Construction.StrongPreElgotMonads ambient
|
||||
open import Category.Construction.ElgotAlgebras {C = C}
|
||||
open Cat cocartesian
|
||||
open import Category.Construction.ElgotAlgebras cocartesian
|
||||
|
||||
open Equiv
|
||||
open HomReasoning
|
||||
|
|
|
|||
|
|
@ -1,5 +1,5 @@
|
|||
```agda
|
||||
{-# OPTIONS --allow-unsolved-metas --guardedness #-}
|
||||
{-# OPTIONS --guardedness #-}
|
||||
|
||||
open import Level renaming (zero to ℓ-zero; suc to ℓ-suc)
|
||||
open import Function.Bundles using (_⟨$⟩_) renaming (Func to _⟶_)
|
||||
|
|
@ -66,13 +66,6 @@ module Bisimilarity (A : Setoid c (c ⊔ ℓ)) where
|
|||
|
||||
open _∼′_ public
|
||||
|
||||
record _∼′′_ (x y : Delay′ ∣ A ∣) : Set (c ⊔ ℓ) where
|
||||
coinductive
|
||||
|
||||
field
|
||||
force∼′′ : force x ∼ force y
|
||||
open _∼′′_ public
|
||||
|
||||
-- strong bisimilarity of now and later leads to a clash
|
||||
now∼later : ∀ {ℓ'}{Z : Set ℓ'}{x : ∣ A ∣}{y : Delay′ ∣ A ∣} → now x ∼ later y → Z
|
||||
now∼later ()
|
||||
|
|
@ -179,7 +172,7 @@ module Bisimilarity (A : Setoid c (c ⊔ ℓ)) where
|
|||
|
||||
open _≲′_ public
|
||||
|
||||
open Bisimilarity renaming (_≈_ to [_][_≈_]; _≈′_ to [_][_≈′_]; _∼_ to [_][_∼_]; _∼′_ to [_][_∼′_]; _∼′′_ to [_][_∼′′_]; _↓_ to [_][_↓_]; _≲_ to [_][_≲_]; _≲′_ to [_][_≲′_])
|
||||
open Bisimilarity renaming (_≈_ to [_][_≈_]; _≈′_ to [_][_≈′_]; _∼_ to [_][_∼_]; _∼′_ to [_][_∼′_]; _↓_ to [_][_↓_]; _≲_ to [_][_≲_]; _≲′_ to [_][_≲′_])
|
||||
|
||||
|
||||
module DelayMonad where
|
||||
|
|
@ -187,8 +180,6 @@ module DelayMonad where
|
|||
Delayₛ A = record { Carrier = Delay ∣ A ∣ ; _≈_ = [_][_≈_] A ; isEquivalence = record { refl = ≈-refl A ; sym = ≈-sym A ; trans = ≈-trans A } }
|
||||
Delayₛ∼ : Setoid c (c ⊔ ℓ) → Setoid c (c ⊔ ℓ)
|
||||
Delayₛ∼ A = record { Carrier = Delay ∣ A ∣ ; _≈_ = [_][_∼_] A ; isEquivalence = record { refl = ∼-refl A ; sym = ∼-sym A ; trans = ∼-trans A } }
|
||||
Delayₛ∼′ : Setoid c (c ⊔ ℓ) → Setoid c (c ⊔ ℓ)
|
||||
Delayₛ∼′ A = record { Carrier = Delay′ ∣ A ∣ ; _≈_ = [_][_∼′′_] A ; isEquivalence = {! !} }
|
||||
<_> = _⟨$⟩_
|
||||
open _⟶_ using (cong)
|
||||
|
||||
|
|
|
|||
|
|
@ -1,6 +1,6 @@
|
|||
<!--
|
||||
```agda
|
||||
{-# OPTIONS --allow-unsolved-metas --guardedness #-}
|
||||
{-# OPTIONS --guardedness #-}
|
||||
open import Level renaming (zero to ℓ-zero; suc to ℓ-suc)
|
||||
open import Relation.Binary
|
||||
open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)
|
||||
|
|
@ -36,8 +36,7 @@ module Monad.Instance.Setoids.K {ℓ : Level} where
|
|||
open import Algebra.Elgot cocartesian
|
||||
open import Algebra.Elgot.Free cocartesian
|
||||
open import Algebra.Elgot.Stable distributive
|
||||
open import Category.Construction.ElgotAlgebras {C = Setoids ℓ ℓ}
|
||||
open Cat cocartesian
|
||||
open import Category.Construction.ElgotAlgebras {C = Setoids ℓ ℓ} cocartesian
|
||||
open import Monad.PreElgot (setoidAmbient {ℓ})
|
||||
open Bisimilarity renaming (_≈_ to [_][_≈_]; _≈′_ to [_][_≈′_]; _∼_ to [_][_∼_]; _∼′_ to [_][_∼′_]; _↓_ to [_][_↓_])
|
||||
open DelayMonad
|
||||
|
|
@ -388,8 +387,8 @@ module Monad.Instance.Setoids.K {ℓ : Level} where
|
|||
iter-id {later x} = later≈ (iter-id′ {force x})
|
||||
|
||||
open CartesianClosed (Setoids-CCC ℓ) renaming (exp to setoid-exp)
|
||||
open import Algebra.Elgot.Properties distributive setoid-exp using (free-isStableˡ)
|
||||
open import Algebra.Elgot.Properties distributive setoid-exp using (free-isStable)
|
||||
|
||||
delayK : MonadK
|
||||
delayK = record { freealgebras = freeAlgebra ; stable = λ X → isStableˡ⇒isStable (freeAlgebra X) (free-isStableˡ (freeAlgebra X)) }
|
||||
delayK = record { freealgebras = freeAlgebra ; stable = λ X → free-isStable (freeAlgebra X) }
|
||||
```
|
||||
Loading…
Reference in a new issue