Simplify System definition and add System functor

1 files changed, 94 insertions, 0 deletions

diff --git a/Functor/Instance/Nat/System.agda b/Functor/Instance/Nat/System.agda
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+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Instance.Nat.System {ℓ} where
+
+open import Categories.Category.Instance.Nat using (Nat)
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Functor.Core using (Functor)
+open import Data.Fin.Base using (Fin)
+open import Data.Nat.Base using (ℕ)
+open import Data.Product.Base using (_,_; _×_)
+open import Data.System using (System; ≤-System; Systemₛ; module ≋)
+open import Function.Bundles using (Func; _⟶ₛ_)
+open import Function.Base using (id; _∘_)
+open import Function.Construct.Setoid using (_∙_)
+open import Functor.Instance.Nat.Pull using (Pull₁; Pull-resp-≈)
+open import Functor.Instance.Nat.Push using (Push₁; Push-identity; Push-homomorphism; Push-resp-≈)
+open import Level using (suc)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
+
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
+import Function.Construct.Identity as Id
+
+open Func
+open ≤-System
+open Functor
+
+private
+  variable A B C : ℕ
+
+map : (Fin A → Fin B) → System {ℓ} A → System B
+map f X = record
+    { S = S
+    ; fₛ = fₛ ∙ Pull₁ f
+    ; fₒ = Push₁ f ∙ fₒ
+    }
+  where
+    open System X
+
+≤-cong : (f : Fin A → Fin B) {X Y : System A} → ≤-System Y X → ≤-System (map f Y) (map f X)
+⇒S (≤-cong f x≤y) = ⇒S x≤y
+≗-fₛ (≤-cong f x≤y) = ≗-fₛ x≤y ∘ to (Pull₁ f)
+≗-fₒ (≤-cong f x≤y) = cong (Push₁ f) ∘ ≗-fₒ x≤y
+
+System₁ : (Fin A → Fin B) → Systemₛ A ⟶ₛ Systemₛ B
+to (System₁ f) = map f
+cong (System₁ f) (x≤y , y≤x) = ≤-cong f x≤y , ≤-cong f y≤x
+
+id-x≤x : {X : System A} → ≤-System (map id X) X
+⇒S (id-x≤x) = Id.function _
+≗-fₛ (id-x≤x {_} {x}) i s = System.refl x
+≗-fₒ (id-x≤x {A} {x}) s = Push-identity
+
+x≤id-x : {x : System A} → ≤-System x (map id x)
+⇒S x≤id-x = Id.function _
+≗-fₛ (x≤id-x {A} {x}) i s = System.refl x
+≗-fₒ (x≤id-x {A} {x}) s = ≋.sym Push-identity
+
+
+System-homomorphism
+    : {f : Fin A → Fin B}
+      {g : Fin B → Fin C} 
+      {X : System A}
+    → ≤-System (map (g ∘ f) X) (map g (map f X)) × ≤-System (map g (map f X)) (map (g ∘ f) X)
+System-homomorphism {f = f} {g} {X} = left , right
+  where
+    open System X
+    left : ≤-System (map (g ∘ f) X) (map g (map f X))
+    left .⇒S = Id.function S
+    left .≗-fₛ i s = refl
+    left .≗-fₒ s = Push-homomorphism
+    right : ≤-System (map g (map f X)) (map (g ∘ f) X)
+    right .⇒S = Id.function S
+    right .≗-fₛ i s = refl
+    right .≗-fₒ s = ≋.sym Push-homomorphism
+
+System-resp-≈
+    : {f g : Fin A → Fin B}
+    → f ≗ g
+    → {X : System A}
+    → (≤-System (map f X) (map g X)) × (≤-System (map g X) (map f X))
+System-resp-≈ {A} {B} {f = f} {g} f≗g {X} = both f≗g , both (≡.sym ∘ f≗g)
+  where
+    open System X
+    both : {f g : Fin A → Fin B} → f ≗ g → ≤-System (map f X) (map g X)
+    both f≗g .⇒S = Id.function S
+    both f≗g .≗-fₛ i s = cong fₛ (Pull-resp-≈ f≗g {i})
+    both {f} {g} f≗g .≗-fₒ s = Push-resp-≈ f≗g
+
+Sys : Functor Nat (Setoids (suc ℓ) ℓ)
+Sys .F₀ = Systemₛ
+Sys .F₁ = System₁
+Sys .identity = id-x≤x , x≤id-x
+Sys .homomorphism {x = X} = System-homomorphism {X = X}
+Sys .F-resp-≈ = System-resp-≈