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{-# OPTIONS --without-K --safe #-}
{-# OPTIONS --hidden-argument-puns #-}
module Functor.Instance.Nat.System where
open import Level using (suc; 0ℓ)
open import Categories.Category.Instance.Nat using (Nat)
open import Categories.Category using (Category)
open import Categories.Category.Instance.Cats using (Cats)
open import Categories.NaturalTransformation.NaturalIsomorphism using (_≃_; niHelper)
open import Categories.Functor using (Functor; _∘F_) renaming (id to idF)
open import Data.Circuit.Value using (Monoid)
open import Data.Fin using (Fin)
open import Data.Nat using (ℕ)
open import Data.Product.Base using (_,_; _×_)
open import Data.Setoid using (∣_∣)
open import Data.System {suc 0ℓ} using (System; _≤_; Systemₛ; Systems; ≤-refl; ≤-trans; _≈_)
open import Data.System.Values Monoid using (module ≋; module Object; Values; ≋-isEquiv)
open import Relation.Binary using (Setoid)
open import Function using (Func; _⟶ₛ_; _⟨$⟩_; _∘_; id)
open import Function.Construct.Identity using () renaming (function to Id)
open import Function.Construct.Setoid using (_∙_)
open import Functor.Instance.Nat.Pull using (Pull)
open import Functor.Instance.Nat.Push using (Push)
open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using (Setoids-×)
open import Category.Construction.CMonoids Setoids-×.symmetric using (CMonoids)
open import Object.Monoid.Commutative Setoids-×.symmetric using (CommutativeMonoid; CommutativeMonoid⇒)
open CommutativeMonoid⇒ using (arr)
open Object using (Valuesₘ)
open Func
open Functor
open _≤_
private
variable A B C : ℕ
opaque
unfolding Valuesₘ ≋-isEquiv
map : (Fin A → Fin B) → System A → System B
map {A} {B} f X = let open System X in record
{ S = S
; fₛ = fₛ ∙ arr (Pull.₁ f)
; fₒ = arr (Push.₁ f) ∙ fₒ
}
opaque
unfolding map
open System
map-≤ : (f : Fin A → Fin B) {X Y : System A} → X ≤ Y → map f X ≤ map f Y
⇒S (map-≤ f x≤y) = ⇒S x≤y
≗-fₛ (map-≤ f x≤y) = ≗-fₛ x≤y ∘ to (arr (Pull.₁ f))
≗-fₒ (map-≤ f x≤y) = cong (arr (Push.₁ f)) ∘ ≗-fₒ x≤y
opaque
unfolding map-≤
map-≤-refl
: (f : Fin A → Fin B)
→ {X : System A}
→ map-≤ f (≤-refl {A} {X}) ≈ ≤-refl
map-≤-refl f {X} = Setoid.refl (S (map f X))
opaque
unfolding map-≤
map-≤-trans
: (f : Fin A → Fin B)
→ {X Y Z : System A}
→ {h : X ≤ Y}
→ {g : Y ≤ Z}
→ map-≤ f (≤-trans h g) ≈ ≤-trans (map-≤ f h) (map-≤ f g)
map-≤-trans f {Z = Z} = Setoid.refl (S (map f Z))
opaque
unfolding map-≤
map-≈
: (f : Fin A → Fin B)
→ {X Y : System A}
→ {g h : X ≤ Y}
→ h ≈ g
→ map-≤ f h ≈ map-≤ f g
map-≈ f h≈g = h≈g
Sys₁ : (Fin A → Fin B) → Functor (Systems A) (Systems B)
Sys₁ {A} {B} f = record
{ F₀ = map f
; F₁ = λ C≤D → map-≤ f C≤D
; identity = map-≤-refl f
; homomorphism = map-≤-trans f
; F-resp-≈ = map-≈ f
}
opaque
unfolding map
map-id-≤ : (X : System A) → map id X ≤ X
map-id-≤ X .⇒S = Id (S X)
map-id-≤ X .≗-fₛ i s = cong (fₛ X) Pull.identity
map-id-≤ X .≗-fₒ s = Push.identity
opaque
unfolding map
map-id-≥ : (X : System A) → X ≤ map id X
map-id-≥ X .⇒S = Id (S X)
map-id-≥ X .≗-fₛ i s = cong (fₛ X) (≋.sym Pull.identity)
map-id-≥ X .≗-fₒ s = ≋.sym Push.identity
opaque
unfolding map-≤ map-id-≤
map-id-comm
: {X Y : System A}
(f : X ≤ Y)
→ ≤-trans (map-≤ id f) (map-id-≤ Y) ≈ ≤-trans (map-id-≤ X) f
map-id-comm {Y} f = Setoid.refl (S Y)
opaque
unfolding map-id-≤ map-id-≥
map-id-isoˡ
: (X : System A)
→ ≤-trans (map-id-≤ X) (map-id-≥ X) ≈ ≤-refl
map-id-isoˡ X = Setoid.refl (S X)
map-id-isoʳ
: (X : System A)
→ ≤-trans (map-id-≥ X) (map-id-≤ X) ≈ ≤-refl
map-id-isoʳ X = Setoid.refl (S X)
Sys-identity : Sys₁ {A} id ≃ idF
Sys-identity = niHelper record
{ η = map-id-≤
; η⁻¹ = map-id-≥
; commute = map-id-comm
; iso = λ X → record
{ isoˡ = map-id-isoˡ X
; isoʳ = map-id-isoʳ X
}
}
opaque
unfolding map
map-∘-≤
: (f : Fin A → Fin B)
(g : Fin B → Fin C)
(X : System A)
→ map (g ∘ f) X ≤ map g (map f X)
map-∘-≤ f g X .⇒S = Id (S X)
map-∘-≤ f g X .≗-fₛ i s = cong (fₛ X) Pull.homomorphism
map-∘-≤ f g X .≗-fₒ s = Push.homomorphism
opaque
unfolding map
map-∘-≥
: (f : Fin A → Fin B)
(g : Fin B → Fin C)
(X : System A)
→ map g (map f X) ≤ map (g ∘ f) X
map-∘-≥ f g X .⇒S = Id (S X)
map-∘-≥ f g X .≗-fₛ i s = cong (fₛ X) (≋.sym Pull.homomorphism)
map-∘-≥ f g X .≗-fₒ s = ≋.sym Push.homomorphism
Sys-homo
: (f : Fin A → Fin B)
(g : Fin B → Fin C)
→ Sys₁ (g ∘ f) ≃ Sys₁ g ∘F Sys₁ f
Sys-homo {A} f g = niHelper record
{ η = map-∘-≤ f g
; η⁻¹ = map-∘-≥ f g
; commute = map-∘-comm f g
; iso = λ X → record
{ isoˡ = isoˡ X
; isoʳ = isoʳ X
}
}
where
opaque
unfolding map-∘-≤ map-≤
map-∘-comm
: (f : Fin A → Fin B)
(g : Fin B → Fin C)
→ {X Y : System A}
(X≤Y : X ≤ Y)
→ ≤-trans (map-≤ (g ∘ f) X≤Y) (map-∘-≤ f g Y)
≈ ≤-trans (map-∘-≤ f g X) (map-≤ g (map-≤ f X≤Y))
map-∘-comm f g {Y} X≤Y = Setoid.refl (S Y)
module _ (X : System A) where
opaque
unfolding map-∘-≤ map-∘-≥
isoˡ : ≤-trans (map-∘-≤ f g X) (map-∘-≥ f g X) ≈ ≤-refl
isoˡ = Setoid.refl (S X)
isoʳ : ≤-trans (map-∘-≥ f g X) (map-∘-≤ f g X) ≈ ≤-refl
isoʳ = Setoid.refl (S X)
module _ {f g : Fin A → Fin B} (f≗g : f ≗ g) (X : System A) where
opaque
unfolding map
map-cong-≤ : map f X ≤ map g X
map-cong-≤ .⇒S = Id (S X)
map-cong-≤ .≗-fₛ i s = cong (fₛ X) (Pull.F-resp-≈ f≗g)
map-cong-≤ .≗-fₒ s = Push.F-resp-≈ f≗g
map-cong-≥ : map g X ≤ map f X
map-cong-≥ .⇒S = Id (S X)
map-cong-≥ .≗-fₛ i s = cong (fₛ X) (Pull.F-resp-≈ (≡.sym ∘ f≗g))
map-cong-≥ .≗-fₒ s = Push.F-resp-≈ (≡.sym ∘ f≗g)
opaque
unfolding map-≤ map-cong-≤
map-cong-comm
: {f g : Fin A → Fin B}
(f≗g : f ≗ g)
{X Y : System A}
(h : X ≤ Y)
→ ≤-trans (map-≤ f h) (map-cong-≤ f≗g Y)
≈ ≤-trans (map-cong-≤ f≗g X) (map-≤ g h)
map-cong-comm f≗g {Y} h = Setoid.refl (S Y)
opaque
unfolding map-cong-≤
map-cong-isoˡ
: {f g : Fin A → Fin B}
(f≗g : f ≗ g)
(X : System A)
→ ≤-trans (map-cong-≤ f≗g X) (map-cong-≥ f≗g X) ≈ ≤-refl
map-cong-isoˡ f≗g X = Setoid.refl (S X)
map-cong-isoʳ
: {f g : Fin A → Fin B}
(f≗g : f ≗ g)
(X : System A)
→ ≤-trans (map-cong-≥ f≗g X) (map-cong-≤ f≗g X) ≈ ≤-refl
map-cong-isoʳ f≗g X = Setoid.refl (S X)
Sys-resp-≈ : {f g : Fin A → Fin B} → f ≗ g → Sys₁ f ≃ Sys₁ g
Sys-resp-≈ f≗g = niHelper record
{ η = map-cong-≤ f≗g
; η⁻¹ = map-cong-≥ f≗g
; commute = map-cong-comm f≗g
; iso = λ X → record
{ isoˡ = map-cong-isoˡ f≗g X
; isoʳ = map-cong-isoʳ f≗g X
}
}
Sys : Functor Nat (Cats (suc 0ℓ) (suc 0ℓ) 0ℓ)
Sys .F₀ = Systems
Sys .F₁ = Sys₁
Sys .identity = Sys-identity
Sys .homomorphism = Sys-homo _ _
Sys .F-resp-≈ = Sys-resp-≈
module Sys = Functor Sys
|
