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{-# OPTIONS --without-K --safe #-}
open import Data.Nat using (ℕ)
open import Level using (Level; _⊔_)
-- The endofunctor in setoids sending A to Vector A n for a fixed n
module Data.Vector.Endofunctor (n : ℕ) {c ℓ : Level} where
import Data.Vec as Vec
open import Categories.Category.Instance.Setoids using (Setoids)
open import Categories.Functor using (Functor)
open import Data.Vec.Properties using (map-id; map-∘)
open import Data.Vec.Relation.Binary.Pointwise.Inductive using (map⁺)
open import Data.Vector.Core as Core using (Vector; Vectorₛ; module ≊)
open import Function using (Func; _⟶ₛ_; _⟨$⟩_)
open import Function.Construct.Composition using () renaming (function to compose)
open import Function.Construct.Identity using () renaming (function to Id)
open import Relation.Binary using (Setoid)
open Func
opaque
unfolding Vector
map : {c ℓ : Level} {A B : Setoid c ℓ} → A ⟶ₛ B → Vectorₛ A n ⟶ₛ Vectorₛ B n
map f .to = Vec.map (to f)
map f .cong = map⁺ (cong f)
abstract opaque
unfolding map
identity
: {A : Setoid c ℓ}
{V : Vector A n}
(open Core A using (_≊_))
→ map (Id A) ⟨$⟩ V ≊ V
identity {A} {V} = ≊.reflexive A (map-id V)
homomorphism
: {X Y Z : Setoid c ℓ}
{f : X ⟶ₛ Y}
{g : Y ⟶ₛ Z}
{V : Vector X n}
(open Core Z using (_≊_))
→ map (compose f g) ⟨$⟩ V ≊ map g ⟨$⟩ (map f ⟨$⟩ V)
homomorphism {_} {_} {Z} {f} {g} {V} = ≊.reflexive Z (map-∘ (to g) (to f) V)
F-resp-≈
: {A B : Setoid c ℓ}
{f g : A ⟶ₛ B}
→ ({x : Setoid.Carrier A} → (B Setoid.≈ to f x) (to g x))
→ {V : Vector A n}
(open Core B using (_≊_))
→ map f ⟨$⟩ V ≊ map g ⟨$⟩ V
F-resp-≈ {A} {B} {_} {g} f≈g {V} = map⁺ (λ x≈y → B.trans f≈g (cong g x≈y)) (≊.refl A)
where
module B = Setoid B
-- only a true endofunctor when c ≤ ℓ
Vec : Functor (Setoids c ℓ) (Setoids c (c ⊔ ℓ))
Vec = record
{ F₀ = λ A → Vectorₛ A n
; F₁ = map
; identity = identity
; homomorphism = homomorphism
; F-resp-≈ = F-resp-≈
}
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