Clean up System functors

1 files changed, 133 insertions, 157 deletions

diff --git a/Functor/Monoidal/Instance/Nat/Pull.agda b/Functor/Monoidal/Instance/Nat/Pull.agda
index c2b6958..1d3d338 100644
--- a/Functor/Monoidal/Instance/Nat/Pull.agda
+++ b/Functor/Monoidal/Instance/Nat/Pull.agda
@@ -2,191 +2,167 @@
 
 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.Instance.Nat using (Nat-Cocartesian)
-open import Categories.Category.Instance.SingletonSet using (SingletonSetoid)
-open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
-open import Categories.NaturalTransformation.NaturalIsomorphism using (NaturalIsomorphism; niHelper)
-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.Instance.Nat using (Nat)
+open import Categories.Category.Instance.Nat using (Nat-Cocartesian)
+open import Categories.Category.Instance.SingletonSet using () renaming (SingletonSetoid to ⊤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 Data.Subset.Functional using (Subset)
-open import Data.Nat.Base using (ℕ; _+_)
-open import Relation.Binary using (Setoid)
-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_; _⟨$⟩_)
-open import Functor.Instance.Nat.Pull using (Pull; Pull₁)
-open import Level using (0ℓ; Level)
-
+open import Categories.Functor.Monoidal.Symmetric Natop,+,0 Setoids-× using (module Strong)
+open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
+open import Categories.NaturalTransformation.NaturalIsomorphism using (NaturalIsomorphism; niHelper)
+open import Data.Circuit.Value using (Monoid)
+open import Data.Vector using (++-assoc)
+open import Data.Fin.Base using (Fin; splitAt; join)
 open import Data.Fin.Permutation using (Permutation; _⟨$⟩ʳ_; _⟨$⟩ˡ_)
-open Cartesian (Setoids-Cartesian {0ℓ} {0ℓ}) using (products)
-open BinaryProducts products using (-×-)
-open Cocartesian Nat-Cocartesian using (module Dual; _+₁_; +-assocʳ; +-comm; +-swap; +₁∘+-swap; i₁; i₂)
-open Dual.op-binaryProducts using () renaming (-×- to -+-; assocˡ∘⟨⟩ to []∘assocʳ; swap∘⟨⟩ to []∘swap)
-
-open import Data.Fin.Base using (Fin; splitAt; join; _↑ˡ_; _↑ʳ_)
+open import Data.Fin.Preimage using (preimage)
 open import Data.Fin.Properties using (splitAt-join; splitAt-↑ˡ; splitAt-↑ʳ; join-splitAt)
+open import Data.Nat.Base using (ℕ; _+_)
+open import Data.Product.Base using (_,_; _×_; Σ; proj₁; proj₂)
+open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
+open import Data.Setoid using (∣_∣)
+open import Data.Subset.Functional using (Subset)
 open import Data.Sum.Base using ([_,_]′; map; map₁; map₂; inj₁; inj₂)
 open import Data.Sum.Properties using ([,]-map; [,]-cong; [-,]-cong; [,-]-cong; [,]-∘)
-open import Data.Fin.Preimage using (preimage)
-open import Function.Base using (_∘_; id)
+open import Data.System.Values Monoid using (Values; <ε>; []-unique; _++_; ++ₛ; splitₛ; _≋_; [])
+open import Data.Unit.Polymorphic using (tt)
+open import Function using (Func; _⟶ₛ_; _⟨$⟩_; _∘_)
+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 import Data.Bool.Base using (Bool)
-open import Data.Setoid using (∣_∣)
-open import Data.Circuit.Value using (Value)
-open import Data.System.Values Value using (Values)
-
-open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using (Setoids-×)
-open import Categories.Functor.Monoidal.Symmetric Natop,+,0 Setoids-× using (module Strong)
-open Strong using (SymmetricMonoidalFunctor)
-open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory)
 
 module Setoids-× = SymmetricMonoidalCategory Setoids-×
-import Function.Construct.Constant as Const
 
+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
 
-module _ where
+private
 
-  open import Categories.Morphism (Setoids-×.U) using (_≅_; module Iso)
-  open import Data.Unit.Polymorphic using (tt)
   open _≅_
   open Iso
 
-  Pull-ε : SingletonSetoid ≅ Values 0
-  from Pull-ε = Const.function SingletonSetoid (Values 0) []
-  to Pull-ε = Const.function (Values 0) SingletonSetoid tt
+  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) ()
-
-++ₛ : {n m : ℕ} → Values n ×ₛ Values m ⟶ₛ Values (n + m)
-to ++ₛ (xs , ys) = xs ++ ys
-cong ++ₛ (≗xs , ≗ys) = ++-cong _ _ ≗xs ≗ys
-
-splitₛ : {n m : ℕ} → Values (n + m) ⟶ₛ Values n ×ₛ Values m
-to (splitₛ {n} {m}) v = v ∘ (_↑ˡ m) , v ∘ (n ↑ʳ_)
-cong (splitₛ {n} {m}) v₁≋v₂ = v₁≋v₂ ∘ (_↑ˡ m) , v₁≋v₂ ∘ (n ↑ʳ_)
-
-Pull-++
-    : {n n′ m m′ : ℕ}
-      (f : Fin n → Fin n′)
-      (g : Fin m → Fin m′)
-      {xs : ∣ Values n′ ∣}
-      {ys : ∣ Values m′ ∣}
-    → (Pull₁ f ⟨$⟩ xs) ++ (Pull₁ g ⟨$⟩ ys) ≗ Pull₁ (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 ]′ (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 : 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 (f , g) → Pull-++ f g
-    ; iso = λ (n , m) → record
-        { isoˡ = λ { {x , y} → (λ i → ≡.cong [ x , y ] (splitAt-↑ˡ n i m)) , (λ i → ≡.cong [ x , y ] (splitAt-↑ʳ n m i)) }
-        ; isoʳ = λ { {x} i → ≡.trans (≡.sym ([,]-∘ x (splitAt n i))) (≡.cong x (join-splitAt n m i)) }
-        }
-    }
-  where
-    open import Data.Sum.Base using ([_,_])
-    open import Data.Product.Base using (proj₁; proj₂)
-
-++-↑ˡ
-    : {n m : ℕ}
-      (X : ∣ Values n ∣)
-      (Y : ∣ Values m ∣)
-    → (X ++ Y) ∘ i₁ ≗ X
-++-↑ˡ {n} {m} X Y i = ≡.cong [ X , Y ]′ (splitAt-↑ˡ n i m)
-
-++-↑ʳ
-    : {n m : ℕ}
-      (X : ∣ Values n ∣)
-      (Y : ∣ Values m ∣)
-    → (X ++ Y) ∘ i₂ ≗ Y
-++-↑ʳ {n} {m} X Y i = ≡.cong [ X , Y ]′ (splitAt-↑ʳ n m i)
-
--- TODO move to Data.Vector
-++-assoc
-    : {m n o : ℕ}
-      (X : ∣ Values m ∣)
-      (Y : ∣ Values n ∣)
-      (Z : ∣ Values 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) ++ Z) ([ i₁ ∘ i₁ , _ ]′ (splitAt m i))                    ≡⟨ [,]-∘ ((X ++ Y) ++ Z) (splitAt m i) ⟩
-    [ ((X ++ Y) ++ Z) ∘ i₁ ∘ i₁ , _ ]′ (splitAt m i)                    ≡⟨ [-,]-cong (++-↑ˡ (X ++ Y) Z ∘ i₁) (splitAt m i) ⟩
-    [ (X ++ Y) ∘ i₁ , _ ]′ (splitAt m i)                                ≡⟨ [-,]-cong (++-↑ˡ X Y) (splitAt m i) ⟩
-    [ X , ((X ++ Y) ++ Z) ∘ [ _ , _ ]′ ∘ splitAt n ]′ (splitAt m i)     ≡⟨ [,-]-cong ([,]-∘ ((X ++ Y) ++ Z) ∘ splitAt n) (splitAt m i) ⟩
-    [ X , [ (_ ++ Z) ∘ i₁ ∘ i₂ {m} , _ ]′ ∘ splitAt n ]′ (splitAt m i)  ≡⟨ [,-]-cong ([-,]-cong (++-↑ˡ (X ++ Y) Z ∘ i₂) ∘ splitAt n) (splitAt m i) ⟩
-    [ X , [ (X ++ Y) ∘ i₂ , _ ]′ ∘ splitAt n ]′ (splitAt m i)           ≡⟨ [,-]-cong ([-,]-cong (++-↑ʳ X Y) ∘ splitAt n) (splitAt m i) ⟩
-    [ X , [ Y , ((X ++ Y) ++ Z) ∘ i₂ ]′ ∘ splitAt n ]′ (splitAt m i)    ≡⟨ [,-]-cong ([,-]-cong (++-↑ʳ (X ++ Y) Z) ∘ splitAt n) (splitAt m i) ⟩
-    [ X , [ Y , Z ]′ ∘ splitAt n ]′ (splitAt m i)                       ≡⟨⟩
-    (X ++ (Y ++ Z)) i                                                   ∎
-  where
-    open Bool
-    open Fin
-    open ≡-Reasoning
-
--- TODO also Data.Vector
-Pull-unitaryˡ
-    : {n : ℕ}
-      (X : ∣ Values n ∣)
-    → (X ++ []) ∘ i₁ ≗ X
-Pull-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
-
-open import Function.Bundles using (Inverse)
-open import Categories.Category.Instance.Nat using (Nat)
-open import Categories.Morphism Nat using (_≅_)
-Pull-swap
-    : {n m : ℕ}
-      (X : ∣ Values n ∣)
-      (Y : ∣ Values m ∣)
-    → (X ++ Y) ∘ (+-swap {n}) ≗ 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 (++-↑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 _ })
+  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 { {m} {n} {o} {(X , Y) , Z} i → ++-assoc X Y Z i  }
-        ; unitaryˡ = λ _ → ≡.refl
-        ; unitaryʳ = λ { {n} {X , _} i → Pull-unitaryˡ X i }
+        ; 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} i → Pull-swap X Y i }
+    ; braiding-compat = λ { {n} {m} {X , Y} → Pull-swap X Y }
     }
+
+module Pull,++,[] = SymmetricMonoidalFunctor Pull,++,[]