diff options
Diffstat (limited to 'Functor/Monoidal/Instance')
| -rw-r--r-- | Functor/Monoidal/Instance/Nat/Circ.agda | 87 | ||||
| -rw-r--r-- | Functor/Monoidal/Instance/Nat/Preimage.agda | 164 | ||||
| -rw-r--r-- | Functor/Monoidal/Instance/Nat/Pull.agda | 166 | ||||
| -rw-r--r-- | Functor/Monoidal/Instance/Nat/Push.agda | 209 | ||||
| -rw-r--r-- | Functor/Monoidal/Instance/Nat/System.agda | 394 |
5 files changed, 1020 insertions, 0 deletions
diff --git a/Functor/Monoidal/Instance/Nat/Circ.agda b/Functor/Monoidal/Instance/Nat/Circ.agda new file mode 100644 index 0000000..1b45a75 --- /dev/null +++ b/Functor/Monoidal/Instance/Nat/Circ.agda @@ -0,0 +1,87 @@ +{-# 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}) } + } diff --git a/Functor/Monoidal/Instance/Nat/Preimage.agda b/Functor/Monoidal/Instance/Nat/Preimage.agda new file mode 100644 index 0000000..844df79 --- /dev/null +++ b/Functor/Monoidal/Instance/Nat/Preimage.agda @@ -0,0 +1,164 @@ +{-# 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 Data.Setoid.Unit using (⊤ₛ) +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 : ⊤ₛ {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} {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 } + } diff --git a/Functor/Monoidal/Instance/Nat/Pull.agda b/Functor/Monoidal/Instance/Nat/Pull.agda new file mode 100644 index 0000000..b267f97 --- /dev/null +++ b/Functor/Monoidal/Instance/Nat/Pull.agda @@ -0,0 +1,166 @@ +{-# OPTIONS --without-K --safe #-} + +module Functor.Monoidal.Instance.Nat.Pull where + +import Categories.Morphism as Morphism + +open import Level using (0ℓ; Level) + +open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using (Setoids-×) +open import Category.Monoidal.Instance.Nat using (Natop,+,0; Natop-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.Instance.Nat using (Nat) +open import Categories.Category.Instance.Nat using (Nat-Cocartesian) +open import Data.Setoid.Unit using (⊤ₛ) +open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory) +open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian) +open import Categories.Category.Product using (_⁂_) +open import Categories.Functor using (_∘F_) +open import Categories.Functor.Monoidal.Symmetric Natop,+,0 Setoids-× using (module Strong) +open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper) +open import Categories.NaturalTransformation.NaturalIsomorphism using (NaturalIsomorphism; niHelper) +open import Data.Circuit.Value using (Monoid) +open import Data.Vector using (++-assoc) +open import Data.Fin.Base using (Fin; splitAt; join) +open import Data.Fin.Permutation using (Permutation; _⟨$⟩ʳ_; _⟨$⟩ˡ_) +open import Data.Fin.Preimage using (preimage) +open import Data.Fin.Properties using (splitAt-join; splitAt-↑ˡ; splitAt-↑ʳ; join-splitAt) +open import Data.Nat.Base using (ℕ; _+_) +open import Data.Product.Base using (_,_; _×_; Σ; proj₁; proj₂) +open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_) +open import Data.Setoid using (∣_∣) +open import Data.Subset.Functional using (Subset) +open import Data.Sum.Base using ([_,_]′; map; map₁; map₂; inj₁; inj₂) +open import Data.Sum.Properties using ([,]-map; [,]-cong; [-,]-cong; [,-]-cong; [,]-∘) +open import Data.System.Values Monoid using (Values; <ε>; []-unique; _++_; ++ₛ; splitₛ; _≋_; []) +open import Data.Unit.Polymorphic using (tt) +open import Function using (Func; _⟶ₛ_; _⟨$⟩_; _∘_) +open import Function.Construct.Constant using () renaming (function to Const) +open import Functor.Instance.Nat.Pull using (Pull; Pull-defs) +open import Relation.Binary using (Setoid) +open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_; module ≡-Reasoning) + +open Cartesian (Setoids-Cartesian {0ℓ} {0ℓ}) using (products) + +open BinaryProducts products using (-×-) +open Cocartesian Nat-Cocartesian using (module Dual; _+₁_; +-assocʳ; +-comm; +-swap; +₁∘+-swap; i₁; i₂) +open Dual.op-binaryProducts using () renaming (-×- to -+-; assocˡ∘⟨⟩ to []∘assocʳ; swap∘⟨⟩ to []∘swap) +open Func +open Morphism (Setoids-×.U) using (_≅_; module Iso) +open Strong using (SymmetricMonoidalFunctor) +open ≡-Reasoning + +private + + open _≅_ + open Iso + + Pull-ε : ⊤ₛ ≅ Values 0 + from Pull-ε = Const ⊤ₛ (Values 0) [] + to Pull-ε = Const (Values 0) ⊤ₛ tt + isoˡ (iso Pull-ε) = tt + isoʳ (iso Pull-ε) {x} = []-unique [] x + + opaque + unfolding _++_ + unfolding Pull-defs + Pull-++ + : {n n′ m m′ : ℕ} + (f : Fin n → Fin n′) + (g : Fin m → Fin m′) + {xs : ∣ Values n′ ∣} + {ys : ∣ Values m′ ∣} + → (Pull.₁ f ⟨$⟩ xs) ++ (Pull.₁ g ⟨$⟩ ys) ≋ Pull.₁ (f +₁ g) ⟨$⟩ (xs ++ ys) + Pull-++ {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) (join n′ m′ (map f g (splitAt n e))) ≡⟨ ≡.cong (xs ++ ys) ([,]-map (splitAt n e)) ⟩ + (xs ++ ys) ((f +₁ g) e) ∎ + + module _ {n m : ℕ} where + + opaque + unfolding splitₛ + + open import Function.Construct.Setoid using (setoid) + open module ⇒ₛ {A} {B} = Setoid (setoid {0ℓ} {0ℓ} {0ℓ} {0ℓ} A B) using (_≈_) + open import Function.Construct.Setoid using (_∙_) + open import Function.Construct.Identity using () renaming (function to Id) + + split∘++ : splitₛ ∙ ++ₛ ≈ Id (Values n ×ₛ Values m) + split∘++ {xs , ys} .proj₁ i = ≡.cong [ xs , ys ]′ (splitAt-↑ˡ n i m) + split∘++ {xs , ys} .proj₂ i = ≡.cong [ xs , ys ]′ (splitAt-↑ʳ n m i) + + ++∘split : ++ₛ {n} ∙ splitₛ ≈ Id (Values (n + m)) + ++∘split {x} i = ≡.trans (≡.sym ([,]-∘ x (splitAt n i))) (≡.cong x (join-splitAt n m i)) + + ⊗-homomorphism : NaturalIsomorphism (-×- ∘F (Pull ⁂ Pull)) (Pull ∘F -+-) + ⊗-homomorphism = niHelper record + { η = λ (n , m) → ++ₛ {n} {m} + ; η⁻¹ = λ (n , m) → splitₛ {n} {m} + ; commute = λ { {n , m} {n′ , m′} (f , g) {xs , ys} → Pull-++ f g } + ; iso = λ (n , m) → record + { isoˡ = split∘++ + ; isoʳ = ++∘split + } + } + + module _ {n m : ℕ} where + + opaque + unfolding Pull-++ + + Pull-i₁ + : (X : ∣ Values n ∣) + (Y : ∣ Values m ∣) + → Pull.₁ i₁ ⟨$⟩ (X ++ Y) ≋ X + Pull-i₁ X Y i = ≡.cong [ X , Y ]′ (splitAt-↑ˡ n i m) + + Pull-i₂ + : (X : ∣ Values n ∣) + (Y : ∣ Values m ∣) + → Pull.₁ i₂ ⟨$⟩ (X ++ Y) ≋ Y + Pull-i₂ X Y i = ≡.cong [ X , Y ]′ (splitAt-↑ʳ n m i) + + opaque + unfolding Pull-++ + + Push-assoc + : {m n o : ℕ} + (X : ∣ Values m ∣) + (Y : ∣ Values n ∣) + (Z : ∣ Values o ∣) + → Pull.₁ (+-assocʳ {m} {n} {o}) ⟨$⟩ ((X ++ Y) ++ Z) ≋ X ++ (Y ++ Z) + Push-assoc {m} {n} {o} X Y Z i = ++-assoc X Y Z i + + Pull-swap + : {n m : ℕ} + (X : ∣ Values n ∣) + (Y : ∣ Values m ∣) + → Pull.₁ (+-swap {n}) ⟨$⟩ (X ++ Y) ≋ Y ++ X + Pull-swap {n} {m} X Y i = begin + ((X ++ Y) ∘ +-swap {n}) i ≡⟨ [,]-∘ (X ++ Y) (splitAt m i) ⟩ + [ (X ++ Y) ∘ i₂ , (X ++ Y) ∘ i₁ ]′ (splitAt m i) ≡⟨ [-,]-cong (Pull-i₂ X Y) (splitAt m i) ⟩ + [ Y , (X ++ Y) ∘ i₁ ]′ (splitAt m i) ≡⟨ [,-]-cong (Pull-i₁ X Y) (splitAt m i) ⟩ + [ Y , X ]′ (splitAt m i) ≡⟨⟩ + (Y ++ X) i ∎ + +open SymmetricMonoidalFunctor + +Pull,++,[] : SymmetricMonoidalFunctor +Pull,++,[] .F = Pull +Pull,++,[] .isBraidedMonoidal = record + { isStrongMonoidal = record + { ε = Pull-ε + ; ⊗-homo = ⊗-homomorphism + ; associativity = λ { {_} {_} {_} {(X , Y) , Z} → Push-assoc X Y Z } + ; unitaryˡ = λ { {n} {_ , X} → Pull-i₂ {0} {n} [] X } + ; unitaryʳ = λ { {n} {X , _} → Pull-i₁ {n} {0} X [] } + } + ; braiding-compat = λ { {n} {m} {X , Y} → Pull-swap X Y } + } + +module Pull,++,[] = SymmetricMonoidalFunctor Pull,++,[] diff --git a/Functor/Monoidal/Instance/Nat/Push.agda b/Functor/Monoidal/Instance/Nat/Push.agda new file mode 100644 index 0000000..2e8c0cf --- /dev/null +++ b/Functor/Monoidal/Instance/Nat/Push.agda @@ -0,0 +1,209 @@ +{-# OPTIONS --without-K --safe #-} + +module Functor.Monoidal.Instance.Nat.Push where + +open import Categories.Category.Instance.Nat using (Nat) +open import Data.Bool.Base using (Bool; false) +open import Data.Subset.Functional using (Subset; ⁅_⁆; ⊥) +open import Function.Base using (_∘_; case_of_; _$_; id) +open import Function.Bundles using (Func; _⟶ₛ_; _⟨$⟩_) +open import Level using (0ℓ; Level) +open import Relation.Binary using (Rel; Setoid) +open import Functor.Instance.Nat.Push using (Push; Push-defs) +open import Data.Setoid.Unit using (⊤ₛ) +open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper) +open import Data.Vec.Functional as Vec using (Vector) +open import Data.Vector using (++-assoc; ++-↑ˡ; ++-↑ʳ) +-- open import Data.Vec.Functional.Properties using (++-cong) +open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian) +open import Function.Construct.Constant using () renaming (function to Const) +open import Categories.Category.BinaryProducts using (module BinaryProducts) +open import Categories.Category.Cartesian using (Cartesian) +open Cartesian (Setoids-Cartesian {0ℓ} {0ℓ}) using (products) +open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory) +open import Categories.Category.Instance.Nat using (Nat-Cocartesian) +open import Categories.Category.Cocartesian using (Cocartesian) +open import Categories.Category.Product using (_⁂_) +open import Categories.Functor using () renaming (_∘F_ to _∘′_) +open Cocartesian Nat-Cocartesian using (module Dual; i₁; i₂; -+-; _+₁_; +-assoc; +-assocʳ; +-assocˡ; +-comm; +-swap; +₁∘+-swap) +open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_) +open import Data.Nat using (ℕ; _+_) +open import Data.Fin using (Fin) +open import Data.Product.Base using (_,_; _×_; Σ) +open import Data.Fin.Preimage using (preimage; preimage-⊥; preimage-cong₂) +open import Functor.Monoidal.Instance.Nat.Preimage using (preimage-++) +open import Data.Sum.Base using ([_,_]; [_,_]′; inj₁; inj₂) +open import Data.Sum.Properties using ([,]-cong; [,-]-cong; [-,]-cong; [,]-∘; [,]-map) +open import Data.Circuit.Merge using (merge-with; merge; merge-⊥; merge-[]; ⁅⁆-++; merge-++; merge-cong₁; merge-cong₂; merge-suc; _when_; join-merge; merge-preimage-ρ; merge-⁅⁆) +open import Data.Circuit.Value using (Value; join; join-comm; join-assoc; Monoid) +open import Data.Fin.Base using (splitAt; _↑ˡ_; _↑ʳ_) renaming (join to joinAt) +open import Data.Fin.Properties using (splitAt-↑ˡ; splitAt-↑ʳ; splitAt⁻¹-↑ˡ; splitAt⁻¹-↑ʳ; ↑ˡ-injective; ↑ʳ-injective; _≟_) +open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≢_; _≗_; module ≡-Reasoning) +open BinaryProducts products using (-×-) +open Value using (U) +open Bool using (false) + +open import Function.Bundles using (Equivalence) +open import Category.Monoidal.Instance.Nat using (Nat,+,0) +open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using (Setoids-×) +open import Categories.Functor.Monoidal.Symmetric Nat,+,0 Setoids-× using (module Lax) +open Lax using (SymmetricMonoidalFunctor) +open import Categories.Morphism Nat using (_≅_) +open import Function.Bundles using (Inverse) +open import Data.Fin.Permutation using (Permutation; _⟨$⟩ʳ_; _⟨$⟩ˡ_) +open Dual.op-binaryProducts using () renaming (assocˡ∘⟨⟩ to []∘assocʳ; swap∘⟨⟩ to []∘swap) +open import Relation.Nullary.Decidable using (does; does-⇔; dec-false) +open import Data.Setoid using (∣_∣) + +open ℕ + +open import Data.System.Values Monoid using (Values; <ε>; ++ₛ; _++_; head; tail; _≋_) + +open Func +open ≡-Reasoning + +private + + Push-ε : ⊤ₛ {0ℓ} {0ℓ} ⟶ₛ Values 0 + Push-ε = Const ⊤ₛ (Values 0) <ε> + + opaque + + unfolding _++_ + + unfolding Push-defs + Push-++ + : {n n′ m m′ : ℕ } + → (f : Fin n → Fin n′) + → (g : Fin m → Fin m′) + → (xs : ∣ Values n ∣) + → (ys : ∣ Values m ∣) + → (Push.₁ f ⟨$⟩ xs) ++ (Push.₁ g ⟨$⟩ ys) + ≋ Push.₁ (f +₁ g) ⟨$⟩ (xs ++ ys) + Push-++ {n} {n′} {m} {m′} f g xs ys i = begin + ((merge xs ∘ preimage f ∘ ⁅_⁆) ++ (merge ys ∘ preimage g ∘ ⁅_⁆)) i + ≡⟨ [,]-cong left right (splitAt n′ i) ⟩ + [ (λ x → merge (xs ++ ys) _) , (λ x → merge (xs ++ ys) _) ]′ (splitAt n′ i) + ≡⟨ [,]-∘ (merge (xs ++ ys) ∘ (preimage (f +₁ g))) (splitAt n′ i) ⟨ + merge (xs ++ ys) (preimage (f +₁ g) ((⁅⁆++⊥ Vec.++ ⊥++⁅⁆) i)) ≡⟨ merge-cong₂ (xs ++ ys) (preimage-cong₂ (f +₁ g) (⁅⁆-++ {n′} i)) ⟩ + merge (xs ++ ys) (preimage (f +₁ g) ⁅ i ⁆) ∎ + where + ⁅⁆++⊥ : Vector (Subset (n′ + m′)) n′ + ⁅⁆++⊥ x = ⁅ x ⁆ Vec.++ ⊥ + ⊥++⁅⁆ : Vector (Subset (n′ + m′)) m′ + ⊥++⁅⁆ x = ⊥ Vec.++ ⁅ x ⁆ + left : (x : Fin n′) → merge xs (preimage f ⁅ x ⁆) ≡ merge (xs ++ ys) (preimage (f +₁ g) (⁅ x ⁆ Vec.++ ⊥)) + left x = begin + merge xs (preimage f ⁅ x ⁆) ≡⟨ join-comm U (merge xs (preimage f ⁅ x ⁆)) ⟩ + join (merge xs (preimage f ⁅ x ⁆)) U ≡⟨ ≡.cong (join (merge _ _)) (merge-⊥ ys) ⟨ + join (merge xs (preimage f ⁅ x ⁆)) (merge ys ⊥) ≡⟨ ≡.cong (join (merge _ _)) (merge-cong₂ ys (preimage-⊥ g)) ⟨ + join (merge xs (preimage f ⁅ x ⁆)) (merge ys (preimage g ⊥)) ≡⟨ merge-++ xs ys (preimage f ⁅ x ⁆) (preimage g ⊥) ⟨ + merge (xs ++ ys) ((preimage f ⁅ x ⁆) Vec.++ (preimage g ⊥)) ≡⟨ merge-cong₂ (xs ++ ys) (preimage-++ f g) ⟩ + merge (xs ++ ys) (preimage (f +₁ g) (⁅ x ⁆ Vec.++ ⊥)) ∎ + right : (x : Fin m′) → merge ys (preimage g ⁅ x ⁆) ≡ merge (xs ++ ys) (preimage (f +₁ g) (⊥ Vec.++ ⁅ x ⁆)) + right x = begin + merge ys (preimage g ⁅ x ⁆) ≡⟨⟩ + join U (merge ys (preimage g ⁅ x ⁆)) ≡⟨ ≡.cong (λ h → join h (merge _ _)) (merge-⊥ xs) ⟨ + join (merge xs ⊥) (merge ys (preimage g ⁅ x ⁆)) ≡⟨ ≡.cong (λ h → join h (merge _ _)) (merge-cong₂ xs (preimage-⊥ f)) ⟨ + join (merge xs (preimage f ⊥)) (merge ys (preimage g ⁅ x ⁆)) ≡⟨ merge-++ xs ys (preimage f ⊥) (preimage g ⁅ x ⁆) ⟨ + merge (xs ++ ys) ((preimage f ⊥) Vec.++ (preimage g ⁅ x ⁆)) ≡⟨ merge-cong₂ (xs ++ ys) (preimage-++ f g) ⟩ + merge (xs ++ ys) (preimage (f +₁ g) (⊥ Vec.++ ⁅ x ⁆)) ∎ + + ⊗-homomorphism : NaturalTransformation (-×- ∘′ (Push ⁂ Push)) (Push ∘′ -+-) + ⊗-homomorphism = ntHelper record + { η = λ (n , m) → ++ₛ {n} {m} + ; commute = λ { (f , g) {xs , ys} → Push-++ f g xs ys } + } + + opaque + + unfolding Push-defs + unfolding _++_ + + Push-assoc + : {m n o : ℕ} + (X : ∣ Values m ∣) + (Y : ∣ Values n ∣) + (Z : ∣ Values o ∣) + → (Push.₁ (+-assocˡ {m} {n} {o}) ⟨$⟩ ((X ++ Y) ++ Z)) ≋ X ++ Y ++ Z + Push-assoc {m} {n} {o} X Y Z i = begin + merge ((X ++ Y) ++ Z) (preimage (+-assocˡ {m}) ⁅ i ⁆) ≡⟨ merge-preimage-ρ ↔-mno ((X ++ Y) ++ Z) ⁅ i ⁆ ⟩ + merge (((X ++ Y) ++ Z) ∘ (+-assocʳ {m})) (⁅ i ⁆) ≡⟨⟩ + merge (((X ++ Y) ++ Z) ∘ (+-assocʳ {m})) (preimage id ⁅ i ⁆) ≡⟨ merge-cong₁ (++-assoc X Y Z) (preimage id ⁅ i ⁆) ⟩ + merge (X ++ (Y ++ Z)) (preimage id ⁅ i ⁆) ≡⟨ Push.identity i ⟩ + (X ++ (Y ++ Z)) i ∎ + where + open Inverse + module +-assoc = _≅_ (+-assoc {m} {n} {o}) + ↔-mno : Permutation ((m + n) + o) (m + (n + o)) + ↔-mno .to = +-assocˡ {m} + ↔-mno .from = +-assocʳ {m} + ↔-mno .to-cong ≡.refl = ≡.refl + ↔-mno .from-cong ≡.refl = ≡.refl + ↔-mno .inverse = (λ { ≡.refl → +-assoc.isoˡ _ }) , λ { ≡.refl → +-assoc.isoʳ _ } + + Push-unitaryˡ + : {n : ℕ} + (X : ∣ Values n ∣) + → Push.₁ id ⟨$⟩ (<ε> ++ X) ≋ X + Push-unitaryˡ = merge-⁅⁆ + + preimage-++′ + : {n m o : ℕ} + (f : Vector (Fin o) n) + (g : Vector (Fin o) m) + (S : Subset o) + → preimage (f Vec.++ g) S ≗ preimage f S Vec.++ preimage g S + preimage-++′ {n} f g S = [,]-∘ S ∘ splitAt n + + Push-unitaryʳ + : {n : ℕ} + (X : ∣ Values n ∣) + → Push.₁ (id Vec.++ (λ())) ⟨$⟩ (X ++ (<ε> {0})) ≋ X + Push-unitaryʳ {n} X i = begin + merge (X ++ <ε>) (preimage (id Vec.++ (λ ())) ⁅ i ⁆) ≡⟨ merge-cong₂ (X Vec.++ <ε>) (preimage-++′ id (λ ()) ⁅ i ⁆) ⟩ + merge (X ++ <ε>) (⁅ i ⁆ Vec.++ preimage (λ ()) ⁅ i ⁆) ≡⟨ merge-++ X <ε> ⁅ i ⁆ (preimage (λ ()) ⁅ i ⁆) ⟩ + join (merge X ⁅ i ⁆) (merge <ε> (preimage (λ ()) ⁅ i ⁆)) ≡⟨ ≡.cong (join (merge X ⁅ i ⁆)) (merge-[] <ε> (preimage (λ ()) ⁅ i ⁆)) ⟩ + join (merge X ⁅ i ⁆) U ≡⟨ join-comm (merge X ⁅ i ⁆) U ⟩ + merge X ⁅ i ⁆ ≡⟨ merge-⁅⁆ X i ⟩ + X i ∎ + + Push-swap + : {n m : ℕ} + (X : ∣ Values n ∣) + (Y : ∣ Values m ∣) + → Push.₁ (+-swap {m}) ⟨$⟩ (X ++ Y) ≋ (Y ++ X) + Push-swap {n} {m} X Y i = begin + merge (X ++ Y) (preimage (+-swap {m}) ⁅ i ⁆) ≡⟨ merge-preimage-ρ n+m↔m+n (X ++ Y) ⁅ i ⁆ ⟩ + merge ((X ++ Y) ∘ +-swap {n}) ⁅ i ⁆ ≡⟨ merge-⁅⁆ ((X ++ Y) ∘ (+-swap {n})) i ⟩ + ((X ++ Y) ∘ +-swap {n}) i ≡⟨ [,]-∘ (X ++ Y) (splitAt m i) ⟩ + [ (X ++ Y) ∘ i₂ , (X ++ Y) ∘ i₁ ]′ (splitAt m i) ≡⟨ [-,]-cong (++-↑ʳ X Y) (splitAt m i) ⟩ + [ Y , (X ++ Y) ∘ i₁ ]′ (splitAt m i) ≡⟨ [,-]-cong (++-↑ˡ X Y) (splitAt m i) ⟩ + [ Y , X ]′ (splitAt m i) ≡⟨⟩ + (Y ++ X) i ∎ + where + open ≡-Reasoning + open Inverse + module +-swap = _≅_ (+-comm {m} {n}) + n+m↔m+n : Permutation (n + m) (m + n) + n+m↔m+n .to = +-swap.to + n+m↔m+n .from = +-swap.from + n+m↔m+n .to-cong ≡.refl = ≡.refl + n+m↔m+n .from-cong ≡.refl = ≡.refl + n+m↔m+n .inverse = (λ { ≡.refl → +-swap.isoˡ _ }) , (λ { ≡.refl → +-swap.isoʳ _ }) + +open SymmetricMonoidalFunctor +Push,++,[] : SymmetricMonoidalFunctor +Push,++,[] .F = Push +Push,++,[] .isBraidedMonoidal = record + { isMonoidal = record + { ε = Push-ε + ; ⊗-homo = ⊗-homomorphism + ; associativity = λ { {n} {m} {o} {(X , Y) , Z} → Push-assoc X Y Z } + ; unitaryˡ = λ { {n} {_ , X} → Push-unitaryˡ X } + ; unitaryʳ = λ { {n} {X , _} → Push-unitaryʳ X } + } + ; braiding-compat = λ { {n} {m} {X , Y} → Push-swap X Y } + } + +module Push,++,[] = SymmetricMonoidalFunctor Push,++,[] diff --git a/Functor/Monoidal/Instance/Nat/System.agda b/Functor/Monoidal/Instance/Nat/System.agda new file mode 100644 index 0000000..6659fb3 --- /dev/null +++ b/Functor/Monoidal/Instance/Nat/System.agda @@ -0,0 +1,394 @@ +{-# OPTIONS --without-K --safe #-} + +module Functor.Monoidal.Instance.Nat.System where + +import Categories.Category.Monoidal.Utilities as ⊗-Util +import Data.Circuit.Value as Value +import Data.Vec.Functional as Vec +import Relation.Binary.PropositionalEquality as ≡ + +open import Level using (0ℓ; suc; Level) + +open import Category.Monoidal.Instance.Nat using (Nat,+,0; Natop,+,0) +open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory; BraidedMonoidalCategory) +open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using () renaming (Setoids-× to 0ℓ-Setoids-×) +open import Category.Instance.Setoids.SymmetricMonoidal {suc 0ℓ} {suc 0ℓ} using (Setoids-×) + +module Natop,+,0 = SymmetricMonoidalCategory Natop,+,0 renaming (braidedMonoidalCategory to B) +module 0ℓ-Setoids-× = SymmetricMonoidalCategory 0ℓ-Setoids-× renaming (braidedMonoidalCategory to B) + +open import Functor.Monoidal.Instance.Nat.Pull using (Pull,++,[]) +open import Categories.Functor.Monoidal.Braided Natop,+,0.B 0ℓ-Setoids-×.B using (module Strong) + +Pull,++,[]B : Strong.BraidedMonoidalFunctor +Pull,++,[]B = record { isBraidedMonoidal = Pull,++,[].isBraidedMonoidal } +module Pull,++,[]B = Strong.BraidedMonoidalFunctor (record { isBraidedMonoidal = Pull,++,[].isBraidedMonoidal }) + +open import Categories.Category.BinaryProducts using (module BinaryProducts) +open import Categories.Category.Cartesian using (Cartesian) +open import Categories.Category.Cocartesian using (Cocartesian) +open import Categories.Category.Instance.Nat using (Nat; Nat-Cocartesian; Natop) +open import Categories.Category.Instance.Setoids using (Setoids) +open import Data.Setoid.Unit using (⊤ₛ) +open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian) +open import Categories.Category.Product using (Product) +open import Categories.Category.Product using (_⁂_) +open import Categories.Functor using (Functor) +open import Categories.Functor using (_∘F_) +open import Categories.Functor.Monoidal.Symmetric Nat,+,0 Setoids-× using (module Lax) +open import Categories.NaturalTransformation.Core using (NaturalTransformation; ntHelper) +open import Data.Circuit.Value using (Monoid) +open import Data.Fin using (Fin) +open import Data.Nat using (ℕ; _+_) +open import Data.Product using (_,_; dmap; _×_) renaming (map to ×-map) +open import Data.Product.Function.NonDependent.Setoid using (_×-function_; proj₁ₛ; proj₂ₛ; <_,_>ₛ; swapₛ) +open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_) +open import Data.Setoid using (_⇒ₛ_; ∣_∣) +open import Data.System {suc 0ℓ} using (Systemₛ; System; discrete; _≤_) +open import Data.System.Values Monoid using (++ₛ; splitₛ; Values; ++-cong; _++_; []) +open import Data.System.Values Value.Monoid using (_≋_; module ≋) +open import Data.Unit.Polymorphic using (⊤; tt) +open import Function using (Func; _⟶ₛ_; _⟨$⟩_; _∘_; id; case_of_) +open import Function.Construct.Constant using () renaming (function to Const) +open import Function.Construct.Identity using () renaming (function to Id) +open import Function.Construct.Setoid using (_∙_; setoid) +open import Functor.Instance.Nat.Pull using (Pull) +open import Functor.Instance.Nat.Push using (Push) +open import Functor.Instance.Nat.System using (Sys; Sys-defs) +open import Functor.Monoidal.Braided.Strong.Properties Pull,++,[]B using (braiding-compat-inv) +open import Functor.Monoidal.Instance.Nat.Push using (Push,++,[]) +open import Functor.Monoidal.Strong.Properties Pull,++,[].monoidalFunctor using (associativity-inv) +open import Functor.Monoidal.Strong.Properties Pull,++,[].monoidalFunctor using (unitaryʳ-inv; unitaryˡ-inv; module Shorthands) +open import Relation.Binary using (Setoid) +open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_) + +open module ⇒ₛ {A} {B} = Setoid (setoid {0ℓ} {0ℓ} {0ℓ} {0ℓ} A B) using (_≈_) + +open Cartesian (Setoids-Cartesian {suc 0ℓ} {suc 0ℓ}) using (products) + +open BinaryProducts products using (-×-) +open Cocartesian Nat-Cocartesian using (module Dual; i₁; i₂; -+-; _+₁_; +-assocʳ; +-assocˡ; +-comm; +-swap; +₁∘+-swap; +₁∘i₁; +₁∘i₂) +open Dual.op-binaryProducts using () renaming (×-assoc to +-assoc) +open SymmetricMonoidalCategory using () renaming (braidedMonoidalCategory to B) + +open Func + +Sys-ε : ⊤ₛ {suc 0ℓ} {suc 0ℓ} ⟶ₛ Systemₛ 0 +Sys-ε = Const ⊤ₛ (Systemₛ 0) (discrete 0) + +private + + variable + n m o : ℕ + c₁ c₂ c₃ c₄ c₅ c₆ : Level + ℓ₁ ℓ₂ ℓ₃ ℓ₄ ℓ₅ ℓ₆ : Level + +_×-⇒_ + : {A : Setoid c₁ ℓ₁} + {B : Setoid c₂ ℓ₂} + {C : Setoid c₃ ℓ₃} + {D : Setoid c₄ ℓ₄} + {E : Setoid c₅ ℓ₅} + {F : Setoid c₆ ℓ₆} + → A ⟶ₛ B ⇒ₛ C + → D ⟶ₛ E ⇒ₛ F + → A ×ₛ D ⟶ₛ B ×ₛ E ⇒ₛ C ×ₛ F +_×-⇒_ f g .to (x , y) = to f x ×-function to g y +_×-⇒_ f g .cong (x , y) = cong f x , cong g y + +⊗ : System n × System m → System (n + m) +⊗ {n} {m} (S₁ , S₂) = record + { S = S₁.S ×ₛ S₂.S + ; fₛ = S₁.fₛ ×-⇒ S₂.fₛ ∙ splitₛ + ; fₒ = ++ₛ ∙ S₁.fₒ ×-function S₂.fₒ + } + where + module S₁ = System S₁ + module S₂ = System S₂ + +opaque + + _~_ : {A B : Setoid 0ℓ 0ℓ} → Func A B → Func A B → Set + _~_ = _≈_ + infix 4 _~_ + + sym-~ + : {A B : Setoid 0ℓ 0ℓ} + {x y : Func A B} + → x ~ y + → y ~ x + sym-~ {A} {B} {x} {y} = 0ℓ-Setoids-×.Equiv.sym {A} {B} {x} {y} + +⊗ₛ + : {n m : ℕ} + → Systemₛ n ×ₛ Systemₛ m ⟶ₛ Systemₛ (n + m) +⊗ₛ .to = ⊗ +⊗ₛ {n} {m} .cong {a , b} {c , d} ((a≤c , c≤a) , (b≤d , d≤b)) = left , right + where + module a = System a + module b = System b + module c = System c + module d = System d + module a≤c = _≤_ a≤c + module b≤d = _≤_ b≤d + module c≤a = _≤_ c≤a + module d≤b = _≤_ d≤b + + open _≤_ + left : ⊗ₛ ⟨$⟩ (a , b) ≤ ⊗ₛ ⟨$⟩ (c , d) + left .⇒S = a≤c.⇒S ×-function b≤d.⇒S + left .≗-fₛ i with (i₁ , i₂) ← splitₛ ⟨$⟩ i = dmap (a≤c.≗-fₛ i₁) (b≤d.≗-fₛ i₂) + left .≗-fₒ = cong ++ₛ ∘ dmap a≤c.≗-fₒ b≤d.≗-fₒ + + right : ⊗ₛ ⟨$⟩ (c , d) ≤ ⊗ₛ ⟨$⟩ (a , b) + right .⇒S = c≤a.⇒S ×-function d≤b.⇒S + right .≗-fₛ i with (i₁ , i₂) ← splitₛ ⟨$⟩ i = dmap (c≤a.≗-fₛ i₁) (d≤b.≗-fₛ i₂) + right .≗-fₒ = cong ++ₛ ∘ dmap c≤a.≗-fₒ d≤b.≗-fₒ + +opaque + + unfolding Sys-defs + + System-⊗-≤ + : {n n′ m m′ : ℕ} + (X : System n) + (Y : System m) + (f : Fin n → Fin n′) + (g : Fin m → Fin m′) + → ⊗ (Sys.₁ f ⟨$⟩ X , Sys.₁ g ⟨$⟩ Y) ≤ Sys.₁ (f +₁ g) ⟨$⟩ ⊗ (X , Y) + System-⊗-≤ {n} {n′} {m} {m′} X Y f g = record + { ⇒S = Id (X.S ×ₛ Y.S) + ; ≗-fₛ = λ i s → cong (X.fₛ ×-⇒ Y.fₛ) (Pull,++,[].⊗-homo.⇐.sym-commute (f , g) {i}) {s} + ; ≗-fₒ = λ (s₁ , s₂) → Push,++,[].⊗-homo.commute (f , g) {X.fₒ′ s₁ , Y.fₒ′ s₂} + } + where + module X = System X + module Y = System Y + + System-⊗-≥ + : {n n′ m m′ : ℕ} + (X : System n) + (Y : System m) + (f : Fin n → Fin n′) + (g : Fin m → Fin m′) + → Sys.₁ (f +₁ g) ⟨$⟩ (⊗ (X , Y)) ≤ ⊗ (Sys.₁ f ⟨$⟩ X , Sys.₁ g ⟨$⟩ Y) + System-⊗-≥ {n} {n′} {m} {m′} X Y f g = record + { ⇒S = Id (X.S ×ₛ Y.S) + ; ≗-fₛ = λ i s → cong (X.fₛ ×-⇒ Y.fₛ) (Pull,++,[].⊗-homo.⇐.commute (f , g) {i}) {s} + ; ≗-fₒ = λ (s₁ , s₂) → Push,++,[].⊗-homo.sym-commute (f , g) {X.fₒ′ s₁ , Y.fₒ′ s₂} + } + where + module X = System X + module Y = System Y + +⊗-homomorphism : NaturalTransformation (-×- ∘F (Sys ⁂ Sys)) (Sys ∘F -+-) +⊗-homomorphism = ntHelper record + { η = λ (n , m) → ⊗ₛ {n} {m} + ; commute = λ { (f , g) {X , Y} → System-⊗-≤ X Y f g , System-⊗-≥ X Y f g } + } + +opaque + + unfolding Sys-defs + + ⊗-assoc-≤ + : (X : System n) + (Y : System m) + (Z : System o) + → Sys.₁ (+-assocˡ {n}) ⟨$⟩ (⊗ (⊗ (X , Y) , Z)) ≤ ⊗ (X , ⊗ (Y , Z)) + ⊗-assoc-≤ {n} {m} {o} X Y Z = record + { ⇒S = assocˡ + ; ≗-fₛ = λ i ((s₁ , s₂) , s₃) → cong (X.fₛ ×-⇒ (Y.fₛ ×-⇒ Z.fₛ) ∙ assocˡ) (associativity-inv {x = i}) {s₁ , s₂ , s₃} + ; ≗-fₒ = λ ((s₁ , s₂) , s₃) → Push,++,[].associativity {x = (X.fₒ′ s₁ , Y.fₒ′ s₂) , Z.fₒ′ s₃} + } + where + open Cartesian (Setoids-Cartesian {0ℓ} {0ℓ}) using () renaming (products to 0ℓ-products) + open BinaryProducts 0ℓ-products using (assocˡ) + + module X = System X + module Y = System Y + module Z = System Z + + ⊗-assoc-≥ + : (X : System n) + (Y : System m) + (Z : System o) + → ⊗ (X , ⊗ (Y , Z)) ≤ Sys.₁ (+-assocˡ {n}) ⟨$⟩ (⊗ (⊗ (X , Y) , Z)) + ⊗-assoc-≥ {n} {m} {o} X Y Z = record + { ⇒S = ×-assocʳ + ; ≗-fₛ = λ i (s₁ , s₂ , s₃) → cong ((X.fₛ ×-⇒ Y.fₛ) ×-⇒ Z.fₛ) (sym-split-assoc {i}) {(s₁ , s₂) , s₃} + ; ≗-fₒ = λ (s₁ , s₂ , s₃) → sym-++-assoc {(X.fₒ′ s₁ , Y.fₒ′ s₂) , Z.fₒ′ s₃} + } + where + open Cartesian (Setoids-Cartesian {0ℓ} {0ℓ}) using () renaming (products to prod) + open BinaryProducts prod using () renaming (assocʳ to ×-assocʳ; assocˡ to ×-assocˡ) + + +-assocℓ : Fin ((n + m) + o) → Fin (n + (m + o)) + +-assocℓ = +-assocˡ {n} {m} {o} + + opaque + + unfolding _~_ + + associativity-inv-~ : splitₛ ×-function Id (Values o) ∙ splitₛ ∙ Pull.₁ +-assocℓ ~ ×-assocʳ ∙ Id (Values n) ×-function splitₛ ∙ splitₛ + associativity-inv-~ {i} = associativity-inv {n} {m} {o} {i} + + associativity-~ : Push.₁ (+-assocˡ {n} {m} {o}) ∙ ++ₛ ∙ ++ₛ ×-function Id (Values o) ~ ++ₛ ∙ Id (Values n) ×-function ++ₛ ∙ ×-assocˡ + associativity-~ {i} = Push,++,[].associativity {n} {m} {o} {i} + + sym-split-assoc-~ : ×-assocʳ ∙ Id (Values n) ×-function splitₛ ∙ splitₛ ~ splitₛ ×-function Id (Values o) ∙ splitₛ ∙ Pull.₁ +-assocℓ + sym-split-assoc-~ = sym-~ associativity-inv-~ + + sym-++-assoc-~ : ++ₛ ∙ Id (Values n) ×-function ++ₛ ∙ ×-assocˡ ~ Push.₁ (+-assocˡ {n} {m} {o}) ∙ ++ₛ ∙ ++ₛ ×-function Id (Values o) + sym-++-assoc-~ = sym-~ associativity-~ + + opaque + + unfolding _~_ + + sym-split-assoc : ×-assocʳ ∙ Id (Values n) ×-function splitₛ ∙ splitₛ ≈ splitₛ ×-function Id (Values o) ∙ splitₛ ∙ Pull.₁ +-assocℓ + sym-split-assoc {i} = sym-split-assoc-~ {i} + + sym-++-assoc : ++ₛ ∙ Id (Values n) ×-function ++ₛ ∙ ×-assocˡ ≈ Push.₁ (+-assocˡ {n} {m} {o}) ∙ ++ₛ ∙ ++ₛ ×-function Id (Values o) + sym-++-assoc {i} = sym-++-assoc-~ + + module X = System X + module Y = System Y + module Z = System Z + + Sys-unitaryˡ-≤ : (X : System n) → Sys.₁ id ⟨$⟩ (⊗ (discrete 0 , X)) ≤ X + Sys-unitaryˡ-≤ {n} X = record + { ⇒S = proj₂ₛ + ; ≗-fₛ = λ i (_ , s) → cong (X.fₛ ∙ proj₂ₛ {A = ⊤ₛ {0ℓ}}) (unitaryˡ-inv {n} {i}) + ; ≗-fₒ = λ (_ , s) → Push,++,[].unitaryˡ {n} {tt , X.fₒ′ s} + } + where + module X = System X + + Sys-unitaryˡ-≥ : (X : System n) → X ≤ Sys.₁ id ⟨$⟩ (⊗ (discrete 0 , X)) + Sys-unitaryˡ-≥ {n} X = record + { ⇒S = < Const X.S ⊤ₛ tt , Id X.S >ₛ + ; ≗-fₛ = λ i s → cong (disc.fₛ ×-⇒ X.fₛ ∙ ε⇒ ×-function Id (Values n)) (sym-split-unitaryˡ {i}) + ; ≗-fₒ = λ s → sym-++-unitaryˡ {_ , X.fₒ′ s} + } + where + module X = System X + open SymmetricMonoidalCategory 0ℓ-Setoids-× using (module Equiv) + open ⊗-Util.Shorthands 0ℓ-Setoids-×.monoidal using (λ⇐) + open Shorthands using (ε⇐; ε⇒) + module disc = System (discrete 0) + sym-split-unitaryˡ + : λ⇐ ≈ ε⇐ ×-function Id (Values n) ∙ splitₛ ∙ Pull.₁ ((λ ()) Vec.++ id) + sym-split-unitaryˡ = + 0ℓ-Setoids-×.Equiv.sym + {Values n} + {⊤ₛ ×ₛ Values n} + {ε⇐ ×-function Id (Values n) ∙ splitₛ ∙ Pull.₁ ((λ ()) Vec.++ id)} + {λ⇐} + (unitaryˡ-inv {n}) + sym-++-unitaryˡ : proj₂ₛ {A = ⊤ₛ {0ℓ} {0ℓ}} ≈ Push.₁ ((λ ()) Vec.++ id) ∙ ++ₛ ∙ Push,++,[].ε ×-function Id (Values n) + sym-++-unitaryˡ = + 0ℓ-Setoids-×.Equiv.sym + {⊤ₛ ×ₛ Values n} + {Values n} + {Push.₁ ((λ ()) Vec.++ id) ∙ ++ₛ ∙ Push,++,[].ε ×-function Id (Values n)} + {proj₂ₛ} + (Push,++,[].unitaryˡ {n}) + + + Sys-unitaryʳ-≤ : (X : System n) → Sys.₁ (id Vec.++ (λ ())) ⟨$⟩ (⊗ {n} {0} (X , discrete 0)) ≤ X + Sys-unitaryʳ-≤ {n} X = record + { ⇒S = proj₁ₛ + ; ≗-fₛ = λ i (s , _) → cong (X.fₛ ∙ proj₁ₛ {B = ⊤ₛ {0ℓ}}) (unitaryʳ-inv {n} {i}) + ; ≗-fₒ = λ (s , _) → Push,++,[].unitaryʳ {n} {X.fₒ′ s , tt} + } + where + module X = System X + + Sys-unitaryʳ-≥ : (X : System n) → X ≤ Sys.₁ (id Vec.++ (λ ())) ⟨$⟩ (⊗ {n} {0} (X , discrete 0)) + Sys-unitaryʳ-≥ {n} X = record + { ⇒S = < Id X.S , Const X.S ⊤ₛ tt >ₛ + ; ≗-fₛ = λ i s → cong (X.fₛ ×-⇒ disc.fₛ ∙ Id (Values n) ×-function ε⇒) sym-split-unitaryʳ {s , tt} + ; ≗-fₒ = λ s → sym-++-unitaryʳ {X.fₒ′ s , tt} + } + where + module X = System X + module disc = System (discrete 0) + open ⊗-Util.Shorthands 0ℓ-Setoids-×.monoidal using (ρ⇐) + open Shorthands using (ε⇐; ε⇒) + sym-split-unitaryʳ + : ρ⇐ ≈ Id (Values n) ×-function ε⇐ ∙ splitₛ ∙ Pull.₁ (id Vec.++ (λ ())) + sym-split-unitaryʳ = + 0ℓ-Setoids-×.Equiv.sym + {Values n} + {Values n ×ₛ ⊤ₛ} + {Id (Values n) ×-function ε⇐ ∙ splitₛ ∙ Pull.₁ (id Vec.++ (λ ()))} + {ρ⇐} + (unitaryʳ-inv {n}) + sym-++-unitaryʳ : proj₁ₛ {B = ⊤ₛ {0ℓ}} ≈ Push.₁ (id Vec.++ (λ ())) ∙ ++ₛ ∙ Id (Values n) ×-function Push,++,[].ε + sym-++-unitaryʳ = + 0ℓ-Setoids-×.Equiv.sym + {Values n ×ₛ ⊤ₛ} + {Values n} + {Push.₁ (id Vec.++ (λ ())) ∙ ++ₛ ∙ Id (Values n) ×-function Push,++,[].ε} + {proj₁ₛ} + (Push,++,[].unitaryʳ {n}) + + Sys-braiding-compat-≤ + : (X : System n) + (Y : System m) + → Sys.₁ (+-swap {m} {n}) ⟨$⟩ (⊗ (X , Y)) ≤ ⊗ (Y , X) + Sys-braiding-compat-≤ {n} {m} X Y = record + { ⇒S = swapₛ + ; ≗-fₛ = λ i (s₁ , s₂) → cong (Y.fₛ ×-⇒ X.fₛ ∙ swapₛ) (braiding-compat-inv {m} {n} {i}) {s₂ , s₁} + ; ≗-fₒ = λ (s₁ , s₂) → Push,++,[].braiding-compat {n} {m} {X.fₒ′ s₁ , Y.fₒ′ s₂} + } + where + module X = System X + module Y = System Y + + Sys-braiding-compat-≥ + : (X : System n) + (Y : System m) + → ⊗ (Y , X) ≤ Sys.₁ (+-swap {m} {n}) ⟨$⟩ ⊗ (X , Y) + Sys-braiding-compat-≥ {n} {m} X Y = record + { ⇒S = swapₛ + ; ≗-fₛ = λ i (s₂ , s₁) → cong (X.fₛ ×-⇒ Y.fₛ) (sym-braiding-compat-inv {i}) + ; ≗-fₒ = λ (s₂ , s₁) → sym-braiding-compat-++ {X.fₒ′ s₁ , Y.fₒ′ s₂} + } + where + module X = System X + module Y = System Y + sym-braiding-compat-inv : swapₛ ∙ splitₛ {m} ≈ splitₛ ∙ Pull.₁ (+-swap {m} {n}) + sym-braiding-compat-inv {i} = + 0ℓ-Setoids-×.Equiv.sym + {Values (m + n)} + {Values n ×ₛ Values m} + {splitₛ ∙ Pull.₁ (+-swap {m} {n})} + {swapₛ ∙ splitₛ {m}} + (λ {j} → braiding-compat-inv {m} {n} {j}) {i} + sym-braiding-compat-++ : ++ₛ {m} ∙ swapₛ ≈ Push.₁ (+-swap {m} {n}) ∙ ++ₛ + sym-braiding-compat-++ {i} = + 0ℓ-Setoids-×.Equiv.sym + {Values n ×ₛ Values m} + {Values (m + n)} + {Push.₁ (+-swap {m} {n}) ∙ ++ₛ} + {++ₛ {m} ∙ swapₛ} + (Push,++,[].braiding-compat {n} {m}) + +open Lax.SymmetricMonoidalFunctor + +Sys,⊗,ε : Lax.SymmetricMonoidalFunctor +Sys,⊗,ε .F = Sys +Sys,⊗,ε .isBraidedMonoidal = record + { isMonoidal = record + { ε = Sys-ε + ; ⊗-homo = ⊗-homomorphism + ; associativity = λ { {n} {m} {o} {(X , Y), Z} → ⊗-assoc-≤ X Y Z , ⊗-assoc-≥ X Y Z } + ; unitaryˡ = λ { {n} {_ , X} → Sys-unitaryˡ-≤ X , Sys-unitaryˡ-≥ X } + ; unitaryʳ = λ { {n} {X , _} → Sys-unitaryʳ-≤ X , Sys-unitaryʳ-≥ X } + } + ; braiding-compat = λ { {n} {m} {X , Y} → Sys-braiding-compat-≤ X Y , Sys-braiding-compat-≥ X Y } + } + +module Sys,⊗,ε = Lax.SymmetricMonoidalFunctor Sys,⊗,ε |