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-rw-r--r--Functor/Monoidal/Instance/Nat/Circ.agda87
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-{-# OPTIONS --without-K --safe #-}
-
-open import Level using (Level; _⊔_; 0ℓ; suc)
-
-module Functor.Monoidal.Instance.Nat.Circ where
-
-import Categories.Object.Monoid as MonoidObject
-import Data.Permutation.Sort as ↭-Sort
-import Function.Reasoning as →-Reasoning
-
-open import Category.Instance.Setoids.SymmetricMonoidal {suc 0ℓ} {suc 0ℓ} using (Setoids-×)
-import Categories.Category.Monoidal.Reasoning as ⊗-Reasoning
-open import Category.Monoidal.Instance.Nat using (Nat,+,0)
-open import Categories.Category.Construction.Monoids using (Monoids)
-open import Categories.Category.Instance.Nat using (Nat; Nat-Cocartesian)
-open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory)
-open import Data.Setoid.Unit using (⊤ₛ)
-open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian)
-open import Categories.Category.Cartesian using (Cartesian)
-open Cartesian (Setoids-Cartesian {suc 0ℓ} {suc 0ℓ}) using (products)
-open import Categories.Category.BinaryProducts using (module BinaryProducts)
-open import Categories.Functor using (_∘F_)
-open BinaryProducts products using (-×-)
-open import Categories.Category.Product using (_⁂_)
-open import Categories.Category.Cocartesian using (Cocartesian)
-open import Categories.Category.Instance.Nat using (Nat-Cocartesian)
-open import Categories.Functor.Monoidal.Symmetric using (module Lax)
-open import Categories.Functor using (Functor)
-open import Categories.Category.Cocartesian.Bundle using (CocartesianCategory)
-open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
-open import Data.Circuit using (Circuit; Circuitₛ; mkCircuit; mkCircuitₛ; _≈_; mk≈; map)
-open import Data.Circuit.Gate using (Gates)
-open import Data.Nat using (ℕ; _+_)
-open import Data.Product using (_,_)
-open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
-open import Function using (_⟶ₛ_; Func; _⟨$⟩_; _∘_; id)
-open import Functor.Instance.Nat.Circ {suc 0ℓ} using (Circ; module Multiset∘Edge)
-open import Functor.Instance.Nat.Edge {suc 0ℓ} using (Edge)
-open import Function.Construct.Setoid using (_∙_)
-
-module Setoids-× = SymmetricMonoidalCategory Setoids-×
-
-open import Functor.Instance.FreeCMonoid {suc 0ℓ} {suc 0ℓ} using (FreeCMonoid)
-
-Nat-Cocartesian-Category : CocartesianCategory 0ℓ 0ℓ 0ℓ
-Nat-Cocartesian-Category = record { cocartesian = Nat-Cocartesian }
-
-open import Functor.Monoidal.Construction.MultisetOf
- {𝒞 = Nat-Cocartesian-Category} (Edge Gates) FreeCMonoid using (MultisetOf,++,[])
-
-open Lax using (SymmetricMonoidalFunctor)
-
-module MultisetOf,++,[] = SymmetricMonoidalFunctor MultisetOf,++,[]
-
-open SymmetricMonoidalFunctor
-
-ε⇒ : ⊤ₛ ⟶ₛ Circuitₛ 0
-ε⇒ = mkCircuitₛ ∙ MultisetOf,++,[].ε
-
-open Cocartesian Nat-Cocartesian using (-+-)
-
-open Func
-
-η : {n m : ℕ} → Circuitₛ n ×ₛ Circuitₛ m ⟶ₛ Circuitₛ (n + m)
-η {n} {m} .to (mkCircuit X , mkCircuit Y) = mkCircuit (MultisetOf,++,[].⊗-homo.η (n , m) ⟨$⟩ (X , Y))
-η {n} {m} .cong (mk≈ x , mk≈ y) = mk≈ (cong (MultisetOf,++,[].⊗-homo.η (n , m)) (x , y))
-
-⊗-homomorphism : NaturalTransformation (-×- ∘F (Circ ⁂ Circ)) (Circ ∘F -+-)
-⊗-homomorphism = ntHelper record
- { η = λ (n , m) → η {n} {m}
- ; commute = λ { (f , g) {mkCircuit X , mkCircuit Y} → mk≈ (MultisetOf,++,[].⊗-homo.commute (f , g) {X , Y}) }
- }
-
-Circ,⊗,ε : SymmetricMonoidalFunctor Nat,+,0 Setoids-×
-Circ,⊗,ε .F = Circ
-Circ,⊗,ε .isBraidedMonoidal = record
- { isMonoidal = record
- { ε = ε⇒
- ; ⊗-homo = ⊗-homomorphism
- ; associativity = λ { {n} {m} {o} {(mkCircuit x , mkCircuit y) , mkCircuit z} →
- mk≈ (MultisetOf,++,[].associativity {n} {m} {o} {(x , y) , z}) }
- ; unitaryˡ = λ { {n} {_ , mkCircuit x} → mk≈ (MultisetOf,++,[].unitaryˡ {n} {_ , x}) }
- ; unitaryʳ = λ { {n} {mkCircuit x , _} → mk≈ (MultisetOf,++,[].unitaryʳ {n} {x , _}) }
- }
- ; braiding-compat = λ { {n} {m} {mkCircuit x , mkCircuit y} →
- mk≈ (MultisetOf,++,[].braiding-compat {n} {m} {x , y}) }
- }