From 3bf15830058dab0baca2b8518e4fe1c4a7363e45 Mon Sep 17 00:00:00 2001 From: Jacques Comeaux Date: Tue, 13 Jan 2026 17:15:31 -0600 Subject: Remove old monoidal functors --- Functor/Monoidal/Instance/Nat/Circ.agda | 87 ------ Functor/Monoidal/Instance/Nat/Preimage.agda | 164 ------------ Functor/Monoidal/Instance/Nat/Pull.agda | 166 ------------ Functor/Monoidal/Instance/Nat/Push.agda | 209 --------------- Functor/Monoidal/Instance/Nat/System.agda | 394 ---------------------------- 5 files changed, 1020 deletions(-) delete mode 100644 Functor/Monoidal/Instance/Nat/Circ.agda delete mode 100644 Functor/Monoidal/Instance/Nat/Preimage.agda delete mode 100644 Functor/Monoidal/Instance/Nat/Pull.agda delete mode 100644 Functor/Monoidal/Instance/Nat/Push.agda delete mode 100644 Functor/Monoidal/Instance/Nat/System.agda (limited to 'Functor/Monoidal/Instance') diff --git a/Functor/Monoidal/Instance/Nat/Circ.agda b/Functor/Monoidal/Instance/Nat/Circ.agda deleted file mode 100644 index 1b45a75..0000000 --- a/Functor/Monoidal/Instance/Nat/Circ.agda +++ /dev/null @@ -1,87 +0,0 @@ -{-# 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 deleted file mode 100644 index 844df79..0000000 --- a/Functor/Monoidal/Instance/Nat/Preimage.agda +++ /dev/null @@ -1,164 +0,0 @@ -{-# 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 deleted file mode 100644 index b267f97..0000000 --- a/Functor/Monoidal/Instance/Nat/Pull.agda +++ /dev/null @@ -1,166 +0,0 @@ -{-# 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 deleted file mode 100644 index 2e8c0cf..0000000 --- a/Functor/Monoidal/Instance/Nat/Push.agda +++ /dev/null @@ -1,209 +0,0 @@ -{-# 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 deleted file mode 100644 index 6659fb3..0000000 --- a/Functor/Monoidal/Instance/Nat/System.agda +++ /dev/null @@ -1,394 +0,0 @@ -{-# 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 -- cgit v1.2.3