{-# 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-≈