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{-# OPTIONS --without-K --safe #-}
module Functor.Monoidal.Instance.Nat.Preimage where
open import Category.Monoidal.Instance.Nat using (Natop,+,0; Natop-Cartesian)
open import Categories.Category.Instance.Nat using (Nat-Cocartesian)
open import Categories.Category.Instance.SingletonSet using (SingletonSetoid)
open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian)
open import Categories.Category.BinaryProducts using (module BinaryProducts)
open import Categories.Category.Cartesian using (Cartesian)
open import Categories.Category.Cocartesian using (Cocartesian; BinaryCoproducts)
open import Categories.Category.Product using (_⁂_)
open import Categories.Functor using (_∘F_)
open import Data.Subset.Functional using (Subset)
open import Data.Nat.Base using (ℕ; _+_)
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
open import Data.Product.Base using (_,_; _×_; Σ)
open import Data.Vec.Functional using ([]; _++_)
open import Data.Vec.Functional.Properties using (++-cong)
open import Data.Vec.Functional using (Vector; [])
open import Function.Bundles using (Func; _⟶ₛ_)
open import Functor.Instance.Nat.Preimage using (Preimage; Subsetₛ)
open import Level using (0ℓ)
open Cartesian (Setoids-Cartesian {0ℓ} {0ℓ}) using (products)
open BinaryProducts products using (-×-)
open Cocartesian Nat-Cocartesian using (module Dual; _+₁_; +-assocʳ; +-swap; +₁∘+-swap)
open Dual.op-binaryProducts using () renaming (-×- to -+-; assocˡ∘⟨⟩ to []∘assocʳ; swap∘⟨⟩ to []∘swap)
open import Data.Fin.Base using (Fin; splitAt; join; _↑ˡ_; _↑ʳ_)
open import Data.Fin.Properties using (splitAt-join; splitAt-↑ˡ)
open import Data.Sum.Base using ([_,_]′; map; map₁; map₂; inj₁; inj₂)
open import Data.Sum.Properties using ([,]-map; [,]-cong; [-,]-cong; [,-]-cong; [,]-∘)
open import Data.Fin.Preimage using (preimage)
open import Function.Base using (_∘_; id)
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_; module ≡-Reasoning)
open import Data.Bool.Base using (Bool)
open Func
Preimage-ε : SingletonSetoid {0ℓ} {0ℓ} ⟶ₛ Subsetₛ 0
to Preimage-ε x = []
cong Preimage-ε x ()
++ₛ : {n m : ℕ} → Subsetₛ n ×ₛ Subsetₛ m ⟶ₛ Subsetₛ (n + m)
to ++ₛ (xs , ys) = xs ++ ys
cong ++ₛ (≗xs , ≗ys) = ++-cong _ _ ≗xs ≗ys
preimage-++
: {n n′ m m′ : ℕ}
(f : Fin n → Fin n′)
(g : Fin m → Fin m′)
{xs : Subset n′}
{ys : Subset m′}
→ preimage f xs ++ preimage g ys ≗ preimage (f +₁ g) (xs ++ ys)
preimage-++ {n} {n′} {m} {m′} f g {xs} {ys} e = begin
(xs ∘ f ++ ys ∘ g) e ≡⟨ [,]-map (splitAt n e) ⟨
[ xs , ys ]′ (map f g (splitAt n e)) ≡⟨ ≡.cong [ xs , ys ]′ (splitAt-join n′ m′ (map f g (splitAt n e))) ⟨
[ xs , ys ]′ (splitAt n′ (join n′ m′ (map f g (splitAt n e)))) ≡⟨ ≡.cong ([ xs , ys ]′ ∘ splitAt n′) ([,]-map (splitAt n e)) ⟩
[ xs , ys ]′ (splitAt n′ ((f +₁ g) e)) ∎
where
open ≡-Reasoning
⊗-homomorphism : NaturalTransformation (-×- ∘F (Preimage ⁂ Preimage)) (Preimage ∘F -+-)
⊗-homomorphism = ntHelper record
{ η = λ (n , m) → ++ₛ {n} {m}
; commute = λ { {n′ , m′} {n , m} (f , g) {xs , ys} e → preimage-++ f g e }
}
open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} using (Setoids-×)
open import Categories.Functor.Monoidal.Symmetric Natop,+,0 Setoids-× using (module Lax)
open Lax using (SymmetricMonoidalFunctor)
++-assoc
: {m n o : ℕ}
(X : Subset m)
(Y : Subset n)
(Z : Subset o)
→ ((X ++ Y) ++ Z) ∘ +-assocʳ {m} ≗ X ++ (Y ++ Z)
++-assoc {m} {n} {o} X Y Z i = begin
((X ++ Y) ++ Z) (+-assocʳ {m} i) ≡⟨⟩
[ [ X , Y ]′ ∘ splitAt m , Z ]′ (splitAt (m + n) (+-assocʳ {m} i)) ≡⟨ [,]-cong ([,]-cong (inv ∘ X) (inv ∘ Y) ∘ splitAt m) (inv ∘ Z) (splitAt (m + n) (+-assocʳ {m} i)) ⟨
[ [ b ∘ X′ , b ∘ Y′ ]′ ∘ splitAt m , b ∘ Z′ ]′ (splitAt _ (+-assocʳ {m} i)) ≡⟨ [-,]-cong ([,]-∘ b ∘ splitAt m) (splitAt (m + n) (+-assocʳ {m} i)) ⟨
[ b ∘ [ X′ , Y′ ]′ ∘ splitAt m , b ∘ Z′ ]′ (splitAt _ (+-assocʳ {m} i)) ≡⟨ [,]-∘ b (splitAt (m + n) (+-assocʳ {m} i)) ⟨
b ([ [ X′ , Y′ ]′ ∘ splitAt m , Z′ ]′ (splitAt _ (+-assocʳ {m} i))) ≡⟨ ≡.cong b ([]∘assocʳ {2} {m} i) ⟩
b ([ X′ , [ Y′ , Z′ ]′ ∘ splitAt n ]′ (splitAt m i)) ≡⟨ [,]-∘ b (splitAt m i) ⟩
[ b ∘ X′ , b ∘ [ Y′ , Z′ ]′ ∘ splitAt n ]′ (splitAt m i) ≡⟨ [,-]-cong ([,]-∘ b ∘ splitAt n) (splitAt m i) ⟩
[ b ∘ X′ , [ b ∘ Y′ , b ∘ Z′ ]′ ∘ splitAt n ]′ (splitAt m i) ≡⟨ [,]-cong (inv ∘ X) ([,]-cong (inv ∘ Y) (inv ∘ Z) ∘ splitAt n) (splitAt m i) ⟩
[ X , [ Y , Z ]′ ∘ splitAt n ]′ (splitAt m i) ≡⟨⟩
(X ++ (Y ++ Z)) i ∎
where
open Bool
open Fin
f : Bool → Fin 2
f false = zero
f true = suc zero
b : Fin 2 → Bool
b zero = false
b (suc zero) = true
inv : b ∘ f ≗ id
inv false = ≡.refl
inv true = ≡.refl
X′ : Fin m → Fin 2
X′ = f ∘ X
Y′ : Fin n → Fin 2
Y′ = f ∘ Y
Z′ : Fin o → Fin 2
Z′ = f ∘ Z
open ≡-Reasoning
Preimage-unitaryˡ
: {n : ℕ}
(X : Subset n)
→ (X ++ []) ∘ (_↑ˡ 0) ≗ X
Preimage-unitaryˡ {n} X i = begin
[ X , [] ]′ (splitAt _ (i ↑ˡ 0)) ≡⟨ ≡.cong ([ X , [] ]′) (splitAt-↑ˡ n i 0) ⟩
[ X , [] ]′ (inj₁ i) ≡⟨⟩
X i ∎
where
open ≡-Reasoning
++-swap
: {n m : ℕ}
(X : Subset n)
(Y : Subset m)
→ (X ++ Y) ∘ +-swap {n} ≗ Y ++ X
++-swap {n} {m} X Y i = begin
[ X , Y ]′ (splitAt n (+-swap {n} i)) ≡⟨ [,]-cong (inv ∘ X) (inv ∘ Y) (splitAt n (+-swap {n} i)) ⟨
[ b ∘ X′ , b ∘ Y′ ]′ (splitAt n (+-swap {n} i)) ≡⟨ [,]-∘ b (splitAt n (+-swap {n} i)) ⟨
b ([ X′ , Y′ ]′ (splitAt n (+-swap {n} i))) ≡⟨ ≡.cong b ([]∘swap {2} {n} i) ⟩
b ([ Y′ , X′ ]′ (splitAt m i)) ≡⟨ [,]-∘ b (splitAt m i) ⟩
[ b ∘ Y′ , b ∘ X′ ]′ (splitAt m i) ≡⟨ [,]-cong (inv ∘ Y) (inv ∘ X) (splitAt m i) ⟩
[ Y , X ]′ (splitAt m i) ∎
where
open Bool
open Fin
f : Bool → Fin 2
f false = zero
f true = suc zero
b : Fin 2 → Bool
b zero = false
b (suc zero) = true
inv : b ∘ f ≗ id
inv false = ≡.refl
inv true = ≡.refl
X′ : Fin n → Fin 2
X′ = f ∘ X
Y′ : Fin m → Fin 2
Y′ = f ∘ Y
open ≡-Reasoning
open SymmetricMonoidalFunctor
Preimage,++,[] : SymmetricMonoidalFunctor
Preimage,++,[] .F = Preimage
Preimage,++,[] .isBraidedMonoidal = record
{ isMonoidal = record
{ ε = Preimage-ε
; ⊗-homo = ⊗-homomorphism
; associativity = λ { {m} {n} {o} {(X , Y) , Z} i → ++-assoc X Y Z i }
; unitaryˡ = λ _ → ≡.refl
; unitaryʳ = λ { {n} {X , _} i → Preimage-unitaryˡ X i }
}
; braiding-compat = λ { {n} {m} {X , Y} i → ++-swap X Y i }
}
|
