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{-# OPTIONS --without-K --safe #-}
open import Level using (Level; _⊔_; suc)
module Adjoint.Instance.List {ℓ : Level} where
import Data.List as L
import Data.List.Relation.Binary.Pointwise as PW
open import Categories.Category.Monoidal.Bundle using (MonoidalCategory; SymmetricMonoidalCategory)
open import Category.Instance.Setoids.SymmetricMonoidal {ℓ} {ℓ} using (Setoids-×)
module S = SymmetricMonoidalCategory Setoids-×
open import Categories.Adjoint using (_⊣_)
open import Categories.Category.Construction.Monoids using (Monoids)
open import Categories.Functor using (Functor; id; _∘F_)
open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
open import Categories.Object.Monoid S.monoidal using (Monoid; IsMonoid; Monoid⇒)
open import Data.Monoid using (toMonoid; toMonoid⇒)
open import Data.Opaque.List using ([-]ₛ; Listₛ; mapₛ; foldₛ; ++ₛ-homo; []ₛ-homo; fold-mapₛ; fold)
open import Data.Product using (_,_; uncurry)
open import Data.Setoid using (∣_∣)
open import Function using (_⟶ₛ_; _⟨$⟩_)
open import Functor.Forgetful.Instance.Monoid {suc ℓ} {ℓ} {ℓ} using () renaming (Forget to Forget′)
open import Functor.Free.Instance.Monoid {ℓ} {ℓ} using (Listₘ; mapₘ; ListMonoid) renaming (Free to Free′)
open import Relation.Binary using (Setoid)
open Monoid
open Monoid⇒
open import Categories.Category using (Category)
Mon[S] : Category (suc ℓ) ℓ ℓ
Mon[S] = Monoids S.monoidal
Free : Functor S.U Mon[S]
Free = Free′
Forget : Functor Mon[S] S.U
Forget = Forget′ S.monoidal
opaque
unfolding [-]ₛ mapₘ
map-[-]ₛ
: {X Y : Setoid ℓ ℓ}
(f : X ⟶ₛ Y)
{x : ∣ X ∣}
→ (open Setoid (Listₛ Y))
→ [-]ₛ ⟨$⟩ (f ⟨$⟩ x)
≈ arr (mapₘ f) ⟨$⟩ ([-]ₛ ⟨$⟩ x)
map-[-]ₛ {X} {Y} f {x} = Setoid.refl (Listₛ Y)
unit : NaturalTransformation id (Forget ∘F Free)
unit = ntHelper record
{ η = λ X → [-]ₛ {ℓ} {ℓ} {X}
; commute = map-[-]ₛ
}
opaque
unfolding toMonoid ListMonoid
foldₘ : (X : Monoid) → Monoid⇒ (Listₘ (Carrier X)) X
foldₘ X .arr = foldₛ (toMonoid X)
foldₘ X .preserves-μ {xs , ys} = ++ₛ-homo (toMonoid X) xs ys
foldₘ X .preserves-η {_} = []ₛ-homo (toMonoid X)
opaque
unfolding foldₘ toMonoid⇒ mapₘ
fold-mapₘ
: {X Y : Monoid}
(f : Monoid⇒ X Y)
{x : ∣ Listₛ (Carrier X) ∣}
→ (open Setoid (Carrier Y))
→ arr (foldₘ Y) ⟨$⟩ (arr (mapₘ (arr f)) ⟨$⟩ x)
≈ arr f ⟨$⟩ (arr (foldₘ X) ⟨$⟩ x)
fold-mapₘ {X} {Y} f = uncurry (fold-mapₛ (toMonoid X) (toMonoid Y)) (toMonoid⇒ X Y f)
counit : NaturalTransformation (Free ∘F Forget) id
counit = ntHelper record
{ η = foldₘ
; commute = fold-mapₘ
}
opaque
unfolding mapₘ foldₘ fold
zig : (Aₛ : Setoid ℓ ℓ)
{xs : ∣ Listₛ Aₛ ∣}
→ (open Setoid (Listₛ Aₛ))
→ arr (foldₘ (Listₘ Aₛ)) ⟨$⟩ (arr (mapₘ [-]ₛ) ⟨$⟩ xs) ≈ xs
zig Aₛ {xs = L.[]} = Setoid.refl (Listₛ Aₛ)
zig Aₛ {xs = x L.∷ xs} = Setoid.refl Aₛ PW.∷ zig Aₛ {xs = xs}
opaque
unfolding foldₘ fold
zag : (M : Monoid)
{x : ∣ Carrier M ∣}
→ (open Setoid (Carrier M))
→ arr (foldₘ M) ⟨$⟩ ([-]ₛ ⟨$⟩ x) ≈ x
zag M {x} = Setoid.sym (Carrier M) (identityʳ M {x , _})
List⊣ : Free ⊣ Forget
List⊣ = record
{ unit = unit
; counit = counit
; zig = λ {X} → zig X
; zag = λ {M} → zag M
}
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