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+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Monoidal.Instance.Nat.Pull where
+
+open import Category.Monoidal.Instance.Nat using (Natop,+,0; Natop-Cartesian)
+open import Categories.Category.Instance.Nat using (Nat-Cocartesian)
+open import Categories.Category.Instance.SingletonSet using (SingletonSetoid)
+open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
+open import Categories.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.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_; _⟨$⟩_)
+open import Functor.Instance.Nat.Pull using (Pull; Pull₁)
+open import Level using (0ℓ; Level)
+
+open import Data.Fin.Permutation using (Permutation; _⟨$⟩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.Properties using (splitAt-join; splitAt-↑ˡ; splitAt-↑ʳ; join-splitAt)
+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 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 Func
+
+module _ where
+
+  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
+  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 _ })
+
+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 }
+        }
+    ; braiding-compat = λ { {n} {m} {X , Y} i → Pull-swap X Y i }
+    }