Add labeled hypergraph decoration functor

2 files changed, 757 insertions, 6 deletions

diff --git a/DecorationFunctor/Hypergraph.agda b/DecorationFunctor/Hypergraph.agda
index a4e08df..5cd83f3 100644
--- a/DecorationFunctor/Hypergraph.agda
+++ b/DecorationFunctor/Hypergraph.agda
@@ -26,7 +26,7 @@ open import Category.Instance.Nat.FinitelyCocomplete using (Nat-FinitelyCocomple
 open import Data.Empty using (⊥-elim)
 open import Data.Fin using (#_)
 open import Data.Fin.Base using (Fin; splitAt; join; zero; suc; _↑ˡ_; _↑ʳ_; Fin′; toℕ; cast)
-open import Data.Fin.Patterns using (0F; 1F)
+open import Data.Fin.Patterns using (0F; 1F; 2F)
 open import Data.Fin.Permutation using (lift₀)
 open import Data.Fin.Properties using (splitAt-join; join-splitAt; cast-is-id; cast-trans; toℕ-cast; subst-is-cast; splitAt-↑ˡ; splitAt-↑ʳ; splitAt⁻¹-↑ˡ; ↑ˡ-injective)
 open import Data.Nat using (ℕ; _+_)
@@ -629,9 +629,11 @@ and-gate = record
     and-graph : Hypergraph 3
     and-graph = record
         { h = 1
-        ; a = λ { 0F → 3 }
-        ; j = λ { 0F 0F → # 0
-                ; 0F 1F → # 1
-                ; 0F 2F → # 2
-                }
+        ; a = λ { 0F → 2 }
+        ; j = λ { 0F → edge-0-nodes }
         }
+      where
+        edge-0-nodes : Fin 3 → Fin 3
+        edge-0-nodes 0F = # 0
+        edge-0-nodes 1F = # 1
+        edge-0-nodes 2F = # 2
diff --git a/DecorationFunctor/Hypergraph/Labeled.agda b/DecorationFunctor/Hypergraph/Labeled.agda
new file mode 100644
index 0000000..d8f6e64
--- /dev/null
+++ b/DecorationFunctor/Hypergraph/Labeled.agda
@@ -0,0 +1,749 @@
+{-# OPTIONS --without-K --safe #-}
+
+module DecorationFunctor.Hypergraph.Labeled where
+
+import Categories.Morphism as Morphism
+
+open import Categories.Category.BinaryProducts using (module BinaryProducts)
+open import Categories.Category.Cartesian using (Cartesian)
+open import Categories.Category.Cocartesian using (Cocartesian; module BinaryCoproducts)
+open import Categories.Category.Core using (Category)
+open import Categories.Category.Instance.Nat using (Nat-Cocartesian)
+open import Categories.Category.Instance.Nat using (Nat)
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Category.Instance.SingletonSet using (SingletonSetoid)
+open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian)
+open import Categories.Category.Product using (_⁂_)
+open import Categories.Functor using () renaming (_∘F_ to _∘′_)
+open import Categories.Functor.Core using (Functor)
+open import Categories.Functor.Monoidal.Symmetric using (module Lax)
+open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
+
+open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory)
+open import Category.Instance.Setoids.SymmetricMonoidal using (Setoids-×)
+open import Category.Instance.Nat.FinitelyCocomplete using (Nat-FinitelyCocomplete)
+
+open import Data.Empty using (⊥-elim)
+open import Data.Fin using (#_)
+open import Data.Fin.Base using (Fin; splitAt; join; zero; suc; _↑ˡ_; _↑ʳ_; Fin′; cast)
+open import Data.Fin.Patterns using (0F; 1F)
+open import Data.Fin.Permutation using (lift₀)
+open import Data.Fin.Properties
+  using
+    ( splitAt-join ; join-splitAt
+    ; cast-is-id ; cast-trans ; subst-is-cast
+    ; splitAt-↑ˡ ; splitAt-↑ʳ
+    ; splitAt⁻¹-↑ˡ ; ↑ˡ-injective
+    )
+open import Data.Nat using (ℕ; _+_)
+open import Data.Product.Base using (_,_; Σ)
+open import Data.Product.Relation.Binary.Pointwise.NonDependent using (×-setoid)
+open import Data.Sum.Base using (_⊎_; map; inj₁; inj₂; swap; map₂) renaming ([_,_]′ to [_,_])
+open import Data.Sum.Properties using (map-map; [,]-map; [,]-∘; [-,]-cong; [,-]-cong; [,]-cong; map-cong; swap-involutive)
+open import Data.Unit using (tt)
+open import Data.Unit.Properties using () renaming (≡-setoid to ⊤-setoid)
+
+open import Function.Base using (_∘_; id; const; case_of_; case_returning_of_)
+open import Function.Bundles using (Func; Inverse; _↔_; mk↔)
+open import Function.Construct.Composition using (_↔-∘_)
+open import Function.Construct.Identity using (↔-id)
+open import Function.Construct.Symmetry using (↔-sym)
+
+open import Level using (0ℓ; lift)
+
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.PropositionalEquality using (_≗_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; erefl; refl; sym; trans; cong; cong₂; subst; cong-app)
+open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence; module ≡-Reasoning; dcong; dcong₂; subst-∘; subst-subst; sym-cong; subst-subst-sym; trans-cong; cong-∘; trans-reflʳ)
+open import Relation.Nullary.Negation.Core using (¬_)
+
+open Cartesian (Setoids-Cartesian {0ℓ} {0ℓ}) using (products)
+open Cocartesian Nat-Cocartesian using (coproducts)
+open FinitelyCocompleteCategory Nat-FinitelyCocomplete
+    using ()
+    renaming (symmetricMonoidalCategory to Nat-smc)
+open Morphism (Setoids 0ℓ 0ℓ) using (_≅_)
+open Lax using (SymmetricMonoidalFunctor)
+
+open BinaryProducts products using (-×-)
+open BinaryCoproducts coproducts using (-+-) renaming (+-assoc to Nat-+-assoc)
+
+
+data Gate : ℕ → Set where
+    ZERO  : Gate 1
+    ONE   : Gate 1
+    NOT   : Gate 2
+    AND   : Gate 3
+    OR    : Gate 3
+    XOR   : Gate 3
+    NAND  : Gate 3
+    NOR   : Gate 3
+    XNOR  : Gate 3
+
+cast-gate : {e e′ : ℕ} → .(e ≡ e′) → Gate e → Gate e′
+cast-gate {1} {1} eq g = g
+cast-gate {2} {2} eq g = g
+cast-gate {3} {3} eq g = g
+
+cast-gate-trans
+    : {m n o : ℕ}
+    → .(eq₁ : m ≡ n)
+      .(eq₂ : n ≡ o)
+      (g : Gate m)
+    → cast-gate eq₂ (cast-gate eq₁ g) ≡ cast-gate (trans eq₁ eq₂) g
+cast-gate-trans {1} {1} {1} eq₁ eq₂ g = refl
+cast-gate-trans {2} {2} {2} eq₁ eq₂ g = refl
+cast-gate-trans {3} {3} {3} eq₁ eq₂ g = refl
+
+cast-gate-is-id : {m : ℕ} .(eq : m ≡ m) (g : Gate m) → cast-gate eq g ≡ g
+cast-gate-is-id {1} eq g = refl
+cast-gate-is-id {2} eq g = refl
+cast-gate-is-id {3} eq g = refl
+
+subst-is-cast-gate : {m n : ℕ} (eq : m ≡ n) (g : Gate m) → subst Gate eq g ≡ cast-gate eq g
+subst-is-cast-gate refl g = sym (cast-gate-is-id refl g)
+
+record Hypergraph (v : ℕ) : Set where
+
+  field
+    h : ℕ
+    a : Fin h → ℕ
+
+  arity : Fin h → ℕ
+  arity = ℕ.suc ∘ a
+
+  field
+    j : (e : Fin h) → Fin (arity e) → Fin v
+    l : (e : Fin h) → Gate (arity e)
+
+record Hypergraph-same {n : ℕ} (H H′ : Hypergraph n) : Set where
+
+  open Hypergraph H public
+  open Hypergraph H′ renaming (h to h′; a to a′; arity to arity′; j to j′; l to l′) public
+
+  field
+    ↔h : Fin h ↔ Fin h′
+
+  open Inverse ↔h public
+
+  field
+    ≗a : a ≗ a′ ∘ to
+
+  ≗arity : arity ≗ arity′ ∘ to
+  ≗arity e = cong ℕ.suc (≗a e)
+
+  field
+    ≗j  : (e : Fin h)
+          (i : Fin (arity e))
+        → j e i
+        ≡ j′ (to e) (cast (≗arity e) i)
+    ≗l  : (e : Fin h)
+        → l e
+        ≡ cast-gate (sym (≗arity e)) (l′ (to e))
+
+private
+
+  variable
+    n n′ m m′ o : ℕ
+    H H′ H″ H₁ H₁′ : Hypergraph n
+    H₂ H₂′ : Hypergraph m
+    H₃ : Hypergraph o
+
+Hypergraph-same-refl : Hypergraph-same H H
+Hypergraph-same-refl {_} {H} = record
+    { ↔h = ↔-id (Fin h)
+    ; ≗a = cong a ∘ erefl
+    ; ≗j = λ e i → cong (j e) (sym (cast-is-id refl i))
+    ; ≗l = λ { e → sym (cast-gate-is-id refl (l e)) }
+    }
+  where
+    open Hypergraph H
+
+sym-sym : {A : Set} {x y : A} (p : x ≡ y) → sym (sym p) ≡ p
+sym-sym refl = refl
+
+Hypergraph-same-sym : Hypergraph-same H H′ → Hypergraph-same H′ H
+Hypergraph-same-sym {V} {H} {H′} ≡H = record
+    { ↔h = ↔-sym ↔h
+    ; ≗a = sym ∘ a∘from≗a′
+    ; ≗j = ≗j′
+    ; ≗l = ≗l′
+    }
+  where
+    open Hypergraph-same ≡H
+    open ≡-Reasoning
+    a∘from≗a′ : a ∘ from ≗ a′
+    a∘from≗a′ x = begin
+        (a ∘ from) x        ≡⟨ (≗a ∘ from) x ⟩
+        (a′ ∘ to ∘ from) x  ≡⟨ (cong a′ ∘ inverseˡ ∘ erefl ∘ from) x ⟩
+        a′ x                ∎
+    id≗to∘from : id ≗ to ∘ from
+    id≗to∘from e = sym (inverseˡ refl)
+    ≗arity′ : arity′ ≗ arity ∘ from
+    ≗arity′ e = cong ℕ.suc (sym (a∘from≗a′ e))
+    ≗arity- : arity′ ≗ arity′ ∘ to ∘ from
+    ≗arity- e = cong arity′ (id≗to∘from e)
+
+    ≗j′ : (e : Fin h′) (i : Fin (arity′ e)) → j′ e i ≡ j (from e) (cast (≗arity′ e) i)
+    ≗j′ e i = begin
+        j′ e i                                                          ≡⟨ dcong₂ j′ (id≗to∘from e) (subst-∘ (id≗to∘from e)) ⟩
+        j′ (to (from e)) (subst Fin (cong arity′ (id≗to∘from e)) i)     ≡⟨ cong (j′ (to (from e))) (subst-is-cast (cong arity′ (id≗to∘from e)) i) ⟩
+        j′ (to (from e)) (cast (cong arity′ (id≗to∘from e)) i)          ≡⟨⟩
+        j′ (to (from e)) (cast (trans (≗arity′ e) (≗arity (from e))) i) ≡⟨ cong (j′ (to (from e))) (cast-trans (≗arity′ e) (≗arity (from e)) i) ⟨
+        j′ (to (from e)) (cast (≗arity (from e)) (cast (≗arity′ e) i))  ≡⟨ ≗j (from e) (cast (≗arity′ e) i) ⟨
+        j (from e) (cast (≗arity′ e) i) ∎
+
+    ≗l′ : (e : Fin h′) → l′ e ≡ cast-gate (sym (cong ℕ.suc (sym (a∘from≗a′ e)))) (l (from e))
+    ≗l′ e = begin
+      l′ e                                                                              ≡⟨ dcong l′ (sym (id≗to∘from e)) ⟨
+      subst (Gate ∘ arity′) (sym (id≗to∘from e)) (l′ (to (from e)))                     ≡⟨ subst-∘ (sym (id≗to∘from e)) ⟩
+      subst Gate (cong arity′ (sym (id≗to∘from e))) (l′ (to (from e)))                  ≡⟨ subst-is-cast-gate (cong arity′ (sym (id≗to∘from e))) (l′ (to (from e))) ⟩
+      cast-gate _ (l′ (to (from e)))                                                    ≡⟨ cast-gate-trans _ (sym (cong ℕ.suc (sym (a∘from≗a′ e)))) (l′ (to (from e))) ⟨
+      cast-gate (sym (cong ℕ.suc (sym (a∘from≗a′ e)))) (cast-gate _ (l′ (to (from e)))) ≡⟨ cong (cast-gate (sym (cong ℕ.suc (sym (a∘from≗a′ e))))) (≗l (from e)) ⟨
+      cast-gate (sym (cong ℕ.suc (sym (a∘from≗a′ e)))) (l (from e))                     ∎
+
+Hypergraph-same-trans : Hypergraph-same H H′ → Hypergraph-same H′ H″ → Hypergraph-same H H″
+Hypergraph-same-trans ≡H₁ ≡H₂ = record
+    { ↔h = ↔h ≡H₂ ↔-∘ ↔h ≡H₁
+    ; ≗a = λ { x → trans (≗a ≡H₁ x) (≗a ≡H₂ (to (↔h ≡H₁) x)) }
+    ; ≗j = λ { e i → trans (≗j ≡H₁ e i) (≗j₂ e i) }
+    ; ≗l = λ { e → trans (≗l ≡H₁ e) (≗l₂ e) }
+    }
+  where
+    open Hypergraph-same
+    open Inverse
+    open ≡-Reasoning
+    ≗j₂ : (e : Fin (h ≡H₁))
+          (i : Fin (arity ≡H₁ e))
+        → j ≡H₂ (to (↔h ≡H₁) e) (cast (≗arity ≡H₁ e) i)
+        ≡ j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (trans (≗arity ≡H₁ e) (≗arity ≡H₂ (to (↔h ≡H₁) e))) i)
+    ≗j₂ e i = begin
+        j ≡H₂ (to (↔h ≡H₁) e) (cast (≗arity ≡H₁ e) i)
+            ≡⟨ ≗j ≡H₂ (to (↔h ≡H₁) e) (cast (≗arity ≡H₁ e) i) ⟩
+        j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (≗arity ≡H₂ (to (↔h ≡H₁) e)) (cast (≗arity ≡H₁ e) i))
+            ≡⟨ cong (j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e))) (cast-trans (≗arity ≡H₁ e) (≗arity ≡H₂ (to (↔h ≡H₁) e)) i) ⟩
+        j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (trans (≗arity ≡H₁ e) (≗arity ≡H₂ (to (↔h ≡H₁) e))) i) ∎
+    ≗l₂ : (e : Fin (h ≡H₁)) → cast-gate _ (l′ ≡H₁ (to ≡H₁ e)) ≡ cast-gate _ (l′ ≡H₂ (to ≡H₂ (to ≡H₁ e)))
+    ≗l₂ e = trans (cong (cast-gate _) (≗l ≡H₂ (to ≡H₁ e))) (cast-gate-trans _ (sym (≗arity ≡H₁ e)) (l′ ≡H₂ (to ≡H₂ (to ≡H₁ e))))
+
+Hypergraph-setoid : ℕ → Setoid 0ℓ 0ℓ
+Hypergraph-setoid p = record
+    { Carrier = Hypergraph p
+    ; _≈_ = Hypergraph-same
+    ; isEquivalence = record
+        { refl = Hypergraph-same-refl
+        ; sym = Hypergraph-same-sym
+        ; trans = Hypergraph-same-trans
+        }
+    }
+
+map-nodes : (Fin n → Fin m) → Hypergraph n → Hypergraph m
+map-nodes f H = record
+    { h = h
+    ; a = a
+    ; j = λ e i → f (j e i)
+    ; l = l
+    }
+  where
+    open Hypergraph H
+
+Hypergraph-same-cong
+    : (f : Fin n → Fin m)  
+    → Hypergraph-same H H′
+    → Hypergraph-same (map-nodes f H) (map-nodes f H′)
+Hypergraph-same-cong f ≡H = record
+    { ↔h = ↔h
+    ; ≗a = ≗a
+    ; ≗j = λ { e i → cong f (≗j e i) }
+    ; ≗l = ≗l
+    }
+  where
+    open Hypergraph-same ≡H
+
+Hypergraph-Func : (Fin n → Fin m) → Func (Hypergraph-setoid n) (Hypergraph-setoid m)
+Hypergraph-Func f = record
+    { to = map-nodes f
+    ; cong = Hypergraph-same-cong f
+    }
+
+F-resp-≈
+    : {f g : Fin n → Fin m}
+    → f ≗ g
+    → Hypergraph-same (map-nodes f H) (map-nodes g H)
+F-resp-≈ {g = g} f≗g = record
+    { ↔h = ↔h
+    ; ≗a = ≗a
+    ; ≗j = λ { e i → trans (f≗g (j e i)) (cong g (≗j e i)) }
+    ; ≗l = ≗l
+    }
+  where
+    open Hypergraph-same Hypergraph-same-refl
+
+homomorphism
+    : (f : Fin n → Fin m)
+    → (g : Fin m → Fin o)
+    → Hypergraph-same (map-nodes (g ∘ f) H) (map-nodes g (map-nodes f H))
+homomorphism {n} {m} {o} {H} f g = record
+    { ↔h = ↔h
+    ; ≗a = ≗a
+    ; ≗j = λ e i → cong (g ∘ f) (≗j e i)
+    ; ≗l = ≗l
+    }
+  where
+    open Hypergraph-same Hypergraph-same-refl
+
+F : Functor Nat (Setoids 0ℓ 0ℓ)
+F = record
+    { F₀ = Hypergraph-setoid
+    ; F₁ = Hypergraph-Func
+    ; identity = λ { {n} {H} → Hypergraph-same-refl {H = H} }
+    ; homomorphism = λ { {f = f} {g = g} → homomorphism f g }
+    ; F-resp-≈ = λ f≗g → F-resp-≈ f≗g
+    }
+
+-- monoidal structure
+
+empty-hypergraph : Hypergraph 0
+empty-hypergraph = record
+    { h = 0
+    ; a = λ ()
+    ; j = λ ()
+    ; l = λ ()
+    }
+
+ε : Func (SingletonSetoid {0ℓ} {0ℓ}) (Hypergraph-setoid 0)
+ε = record
+    { to = const empty-hypergraph
+    ; cong = const Hypergraph-same-refl
+    }
+
+module _ (H₁ : Hypergraph n) (H₂ : Hypergraph m) where
+  private
+    module H₁ = Hypergraph H₁
+    module H₂ = Hypergraph H₂
+  j+j : (e : Fin (H₁.h + H₂.h))
+      → Fin (ℕ.suc ([ H₁.a , H₂.a ] (splitAt H₁.h e)))
+      → Fin (n + m)
+  j+j e i with splitAt H₁.h e
+  ... | inj₁ e₁ = H₁.j e₁ i ↑ˡ m
+  ... | inj₂ e₂ = n ↑ʳ H₂.j e₂ i
+  l+l : (e : Fin (H₁.h + H₂.h)) → Gate (ℕ.suc ([ H₁.a , H₂.a ] (splitAt H₁.h e)))
+  l+l e with splitAt H₁.h e
+  ... | inj₁ e₁ = H₁.l e₁
+  ... | inj₂ e₂ = H₂.l e₂
+
+together : Hypergraph n → Hypergraph m → Hypergraph (n + m)
+together {n} {m} H₁ H₂ = record
+    { h = h H₁ + h H₂
+    ; a = [ a H₁ , a H₂ ] ∘ splitAt (h H₁)
+    ; j = j+j H₁ H₂
+    ; l = l+l H₁ H₂
+    }
+  where
+    open Hypergraph
+
++-resp-↔
+    : {n n′ m m′ : ℕ}
+    → Fin n ↔ Fin n′
+    → Fin m ↔ Fin m′
+    → Fin (n + m) ↔ Fin (n′ + m′)
++-resp-↔ {n} {n′} {m} {m′} ↔n ↔m = record
+    { to = join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n
+    ; from = join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′
+    ; to-cong = cong (join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n)
+    ; from-cong = cong (join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′)
+    ; inverse = (λ { refl → to∘from _ }) , λ { refl → from∘to _ }
+    }
+  where
+    module ↔n = Inverse ↔n
+    module ↔m = Inverse ↔m
+    open ≡-Reasoning
+    to∘from : join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n ∘ join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′ ≗ id
+    to∘from x = begin
+        (join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n ∘ join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′) x
+            ≡⟨ cong
+                (join n′ m′ ∘ map ↔n.to ↔m.to)
+                (splitAt-join n m (map ↔n.from ↔m.from (splitAt n′ x))) ⟩
+        (join n′ m′ ∘ map ↔n.to ↔m.to ∘ map ↔n.from ↔m.from ∘ splitAt n′) x
+            ≡⟨ cong (join n′ m′) (map-map (splitAt n′ x)) ⟩
+        (join n′ m′ ∘ map (↔n.to ∘ ↔n.from) (↔m.to ∘ ↔m.from) ∘ splitAt n′) x
+            ≡⟨ cong
+                (join n′ m′)
+                (map-cong
+                    (λ _ → ↔n.inverseˡ refl)
+                    (λ _ → ↔m.inverseˡ refl)
+                    (splitAt n′ x)) ⟩
+        (join n′ m′ ∘ map id id ∘ splitAt n′) x ≡⟨ [,]-map (splitAt n′ x) ⟩
+        (join n′ m′ ∘ splitAt n′) x ≡⟨ join-splitAt n′ m′ x ⟩
+        x ∎
+    from∘to : join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′ ∘ join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n ≗ id
+    from∘to x = begin
+        (join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′ ∘ join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n) x
+            ≡⟨ cong
+                (join n m ∘ map ↔n.from ↔m.from)
+                (splitAt-join n′ m′ (map ↔n.to ↔m.to (splitAt n x))) ⟩
+        (join n m ∘ map ↔n.from ↔m.from ∘ map ↔n.to ↔m.to ∘ splitAt n) x
+            ≡⟨ cong (join n m) (map-map (splitAt n x)) ⟩
+        (join n m ∘ map (↔n.from ∘ ↔n.to) (↔m.from ∘ ↔m.to) ∘ splitAt n) x
+            ≡⟨ cong
+                (join n m)
+                (map-cong
+                    (λ _ → ↔n.inverseʳ refl)
+                    (λ _ → ↔m.inverseʳ refl)
+                    (splitAt n x)) ⟩
+        (join n m ∘ map id id ∘ splitAt n) x ≡⟨ [,]-map (splitAt n x) ⟩
+        (join n m ∘ splitAt n) x ≡⟨ join-splitAt n m x ⟩
+        x ∎
+
+together-resp-same
+    : Hypergraph-same H₁ H₁′
+    → Hypergraph-same H₂ H₂′
+    → Hypergraph-same (together H₁ H₂) (together H₁′ H₂′)
+together-resp-same {n} {H₁} {H₁′} {m} {H₂} {H₂′} ≡H₁ ≡H₂ = record
+    { ↔h = +-resp-↔ ≡H₁.↔h ≡H₂.↔h
+    ; ≗a = ≗a
+    ; ≗j = ≗j
+    ; ≗l = ≗l
+    }
+  where
+    module ≡H₁ = Hypergraph-same ≡H₁
+    module ≡H₂ = Hypergraph-same ≡H₂
+    module H₁+H₂ = Hypergraph (together H₁ H₂)
+    module H₁+H₂′ = Hypergraph (together H₁′ H₂′)
+    open ≡-Reasoning
+    open Inverse
+    open Hypergraph
+    ≗a  : [ ≡H₁.a , ≡H₂.a ] ∘ splitAt ≡H₁.h
+        ≗ [ ≡H₁.a′ , ≡H₂.a′ ] ∘ splitAt ≡H₁.h′
+        ∘ join ≡H₁.h′ ≡H₂.h′ ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h
+    ≗a e = begin
+      [ ≡H₁.a , ≡H₂.a ] (splitAt ≡H₁.h e)                         ≡⟨ [,]-cong ≡H₁.≗a ≡H₂.≗a (splitAt ≡H₁.h e) ⟩
+      ([ ≡H₁.a′ ∘ ≡H₁.to , ≡H₂.a′ ∘ ≡H₂.to ] ∘ splitAt ≡H₁.h) e   ≡⟨ [,]-map (splitAt ≡H₁.h e) ⟨
+      ([ ≡H₁.a′ , ≡H₂.a′ ] ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h) e ≡⟨ (cong [ ≡H₁.a′ , ≡H₂.a′ ] ∘ splitAt-join ≡H₁.h′ ≡H₂.h′ ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h) e ⟨
+      ([ ≡H₁.a′ , ≡H₂.a′ ] ∘ splitAt ≡H₁.h′ ∘ join ≡H₁.h′ ≡H₂.h′ ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h) e ∎
+    ≗arity : H₁+H₂.arity ≗ H₁+H₂′.arity ∘ join ≡H₁.h′ ≡H₂.h′ ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h
+    ≗arity = cong ℕ.suc ∘ ≗a
+    ≗j  : (e : Fin H₁+H₂.h)
+          (i : Fin (H₁+H₂.arity e))
+        → H₁+H₂.j e i
+        ≡ H₁+H₂′.j (to (+-resp-↔ ≡H₁.↔h ≡H₂.↔h) e) (cast (≗arity e) i)
+    ≗j e i with splitAt ≡H₁.h e
+    ... | inj₁ e₁ rewrite splitAt-↑ˡ ≡H₁.h′ (≡H₁.to e₁) ≡H₂.h′ = cong (_↑ˡ m) (≡H₁.≗j e₁ i)
+    ... | inj₂ e₂ rewrite splitAt-↑ʳ ≡H₁.h′ ≡H₂.h′ (≡H₂.to e₂) = cong (n ↑ʳ_) (≡H₂.≗j e₂ i)
+    ≗l : (e : Fin H₁+H₂.h) → l+l H₁ H₂ e ≡ cast-gate (sym (≗arity e)) (l+l H₁′ H₂′ (to (+-resp-↔ ≡H₁.↔h ≡H₂.↔h) e))
+    ≗l e with splitAt ≡H₁.h e | .{≗arity e}
+    ... | inj₁ e₁ rewrite splitAt-↑ˡ ≡H₁.h′ (≡H₁.to e₁) ≡H₂.h′ = ≡H₁.≗l e₁
+    ... | inj₂ e₂ rewrite splitAt-↑ʳ ≡H₁.h′ ≡H₂.h′ (≡H₂.to e₂) = ≡H₂.≗l e₂
+
+commute
+    : (f : Fin n → Fin n′)
+    → (g : Fin m → Fin m′)
+    → Hypergraph-same
+        (together (map-nodes f H₁) (map-nodes g H₂))
+        (map-nodes  ([ (_↑ˡ m′) ∘ f , (n′ ↑ʳ_) ∘ g ] ∘ splitAt n) (together H₁ H₂))
+commute {n} {n′} {m} {m′} {H₁} {H₂} f g = record
+    { ↔h = ≡H₁+H₂.↔h
+    ; ≗a = ≡H₁+H₂.≗a
+    ; ≗j = ≗j
+    ; ≗l = ≗l
+    }
+  where
+    module H₁ = Hypergraph H₁
+    module H₂ = Hypergraph H₂
+    module H₁+H₂ = Hypergraph (together H₁ H₂)
+    module ≡H₁+H₂ = Hypergraph-same {H = together H₁ H₂} Hypergraph-same-refl
+    open Hypergraph
+    open ≡-Reasoning
+    ≗j  : (e : Fin (H₁.h + H₂.h))
+          (i : Fin ((ℕ.suc ∘ [ H₁.a , H₂.a ] ∘ splitAt H₁.h) e))
+        → j (together (map-nodes f H₁) (map-nodes g H₂)) e i
+        ≡ j (map-nodes ([ (_↑ˡ m′) ∘ f , (n′ ↑ʳ_) ∘ g ] ∘ splitAt n) (together H₁ H₂)) (≡H₁+H₂.to e) (cast refl i)
+    ≗j e i rewrite (cast-is-id refl i) with splitAt H₁.h e
+    ... | inj₁ e₁ rewrite splitAt-↑ˡ n (H₁.j e₁ i) m = refl
+    ... | inj₂ e₂ rewrite splitAt-↑ʳ n m (H₂.j e₂ i) = refl
+    ≗l  : (e : Fin (H₁.h + H₂.h))
+        → l+l (map-nodes f H₁) (map-nodes g H₂) e
+        ≡ cast-gate refl (l+l H₁ H₂ (≡H₁+H₂.to e))
+    ≗l e rewrite cast-gate-is-id refl (l+l H₁ H₂ (≡H₁+H₂.to e)) with splitAt H₁.h e
+    ... | inj₁ e₁ = refl
+    ... | inj₂ e₁ = refl
+
+⊗-homomorphism : NaturalTransformation (-×- ∘′ (F ⁂ F)) (F ∘′ -+-)
+⊗-homomorphism = record
+    { η = λ { (m , n) → η }
+    ; commute = λ { (f , g) {H₁ , H₂} → commute {H₁ = H₁} {H₂ = H₂} f g }
+    ; sym-commute = λ { (f , g) {H₁ , H₂} → Hypergraph-same-sym (commute {H₁ = H₁} {H₂ = H₂} f g) }
+    }
+  where
+    η : Func (×-setoid (Hypergraph-setoid n) (Hypergraph-setoid m)) (Hypergraph-setoid (n + m))
+    η = record
+        { to = λ { (H₁ , H₂) → together H₁ H₂ }
+        ; cong = λ { (≡H₁ , ≡H₂) → together-resp-same ≡H₁ ≡H₂ }
+        }
+
++-assoc-↔ : ∀ (x y z : ℕ) → Fin (x + y + z) ↔ Fin (x + (y + z))
++-assoc-↔ x y z = record
+    { to = to
+    ; from = from
+    ; to-cong = λ { refl → refl }
+    ; from-cong = λ { refl → refl }
+    ; inverse = (λ { refl → isoˡ _ }) , λ { refl → isoʳ _ }
+    }
+  where
+    module Nat = Morphism Nat
+    open Nat._≅_ (Nat-+-assoc {x} {y} {z})
+
+associativity
+    : {X Y Z : ℕ}
+    → {H₁ : Hypergraph X}
+    → {H₂ : Hypergraph Y}
+    → {H₃ : Hypergraph Z}
+    → Hypergraph-same
+        (map-nodes (Inverse.to (+-assoc-↔ X Y Z)) (together (together H₁ H₂) H₃))
+        (together H₁ (together H₂ H₃))
+associativity {X} {Y} {Z} {H₁} {H₂} {H₃} = record
+    { ↔h = ↔h
+    ; ≗a = ≗a
+    ; ≗j = ≗j
+    ; ≗l = ≗l
+    }
+  where
+    module H₁ = Hypergraph H₁
+    module H₂ = Hypergraph H₂
+    module H₃ = Hypergraph H₃
+    open ≡-Reasoning
+    open Hypergraph
+    ↔h : Fin (H₁.h + H₂.h + H₃.h) ↔ Fin (H₁.h + (H₂.h + H₃.h))
+    ↔h = +-assoc-↔ H₁.h H₂.h H₃.h
+    ≗a  : (x : Fin (H₁.h + H₂.h + H₃.h))
+        → [ [ H₁.a , H₂.a ] ∘ splitAt H₁.h , H₃.a ] (splitAt (H₁.h + H₂.h) x)
+        ≡ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] (splitAt H₁.h ([ [ _↑ˡ H₂.h + H₃.h , (H₁.h ↑ʳ_) ∘ (_↑ˡ H₃.h) ] ∘ splitAt H₁.h , (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] (splitAt (H₁.h + H₂.h) x)))
+    ≗a x = begin
+        ([ [ H₁.a , H₂.a ] ∘ splitAt H₁.h , H₃.a ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨⟩
+        ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ inj₁ ] ∘ splitAt H₁.h , H₃.a ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨ [-,]-cong ([,-]-cong (cong [ H₂.a , H₃.a ] ∘ (λ i → splitAt-↑ˡ H₂.h i H₃.h)) ∘ splitAt H₁.h) (splitAt (H₁.h + H₂.h) x) ⟨
+        ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ∘ (_↑ˡ H₃.h) ] ∘ splitAt H₁.h , H₃.a ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨ [-,]-cong ([,]-∘ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h) (splitAt (H₁.h + H₂.h) x) ⟨
+        ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , H₃.a ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨⟩
+        ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , [ H₂.a , H₃.a ] ∘ inj₂ ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨ [,-]-cong (cong [ H₂.a , H₃.a ] ∘ splitAt-↑ʳ H₂.h H₃.h) (splitAt (H₁.h + H₂.h) x) ⟨
+        ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨⟩
+        ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ inj₂ ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨ [,]-∘ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] (splitAt (H₁.h + H₂.h) x) ⟨
+        ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ [ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , inj₂ ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨ cong [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ([,-]-cong (splitAt-↑ʳ H₁.h (H₂.h + H₃.h) ∘ (H₂.h ↑ʳ_)) (splitAt (H₁.h + H₂.h) x)) ⟨
+        ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ [ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , splitAt H₁.h ∘ (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨ cong [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ([-,]-cong (splitAt-join H₁.h (H₂.h + H₃.h) ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h) (splitAt (H₁.h + H₂.h) x)) ⟨
+        ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ [ splitAt H₁.h ∘ join H₁.h (H₂.h + H₃.h) ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , splitAt H₁.h ∘ (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨ cong [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ([,]-∘ (splitAt H₁.h) (splitAt (H₁.h + H₂.h) x)) ⟨
+        ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h ∘ [ join H₁.h (H₂.h + H₃.h) ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨⟩
+        ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h ∘ [ [ _↑ˡ H₂.h + H₃.h , H₁.h ↑ʳ_ ] ∘ [ inj₁ , inj₂ ∘ (_↑ˡ H₃.h) ] ∘ splitAt H₁.h , (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x
+            ≡⟨ cong ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h) ([-,]-cong ([,]-∘ [ _↑ˡ H₂.h + H₃.h , H₁.h ↑ʳ_ ] ∘ splitAt H₁.h) (splitAt (H₁.h + H₂.h) x)) ⟩
+        ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h ∘ [ [ _↑ˡ H₂.h + H₃.h , (H₁.h ↑ʳ_) ∘ (_↑ˡ H₃.h) ] ∘ splitAt H₁.h , (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x ∎
+    ≗j  : (e : Fin (H₁.h + H₂.h + H₃.h))
+          (i : Fin (ℕ.suc ([ [ H₁.a , H₂.a ] ∘ splitAt H₁.h , H₃.a ] (splitAt (H₁.h + H₂.h) e))))
+        → Inverse.to (+-assoc-↔ X Y Z) (j+j (together H₁ H₂) H₃ e i)
+        ≡ j+j H₁ (together H₂ H₃) (Inverse.to ↔h e) (cast (cong ℕ.suc (≗a e)) i)
+    ≗j e i with splitAt (H₁.h + H₂.h) e
+    ≗j e i | inj₁ e₁₂ with splitAt H₁.h e₁₂
+    ≗j e i | inj₁ e₁₂ | inj₁ e₁
+        rewrite splitAt-↑ˡ H₁.h e₁ (H₂.h + H₃.h)
+        rewrite splitAt-↑ˡ (X + Y) (H₁.j e₁ i ↑ˡ Y) Z
+        rewrite splitAt-↑ˡ X (H₁.j e₁ i) Y = cong ((_↑ˡ Y + Z) ∘ H₁.j e₁) (sym (cast-is-id refl i))
+    ≗j e i | inj₁ e₁₂ | inj₂ e₂
+        rewrite splitAt-↑ʳ H₁.h H₂.h e₂
+        rewrite splitAt-↑ʳ H₁.h (H₂.h + H₃.h) (e₂ ↑ˡ H₃.h)
+        rewrite splitAt-↑ˡ H₂.h e₂ H₃.h
+        rewrite splitAt-↑ˡ (X + Y) (X ↑ʳ H₂.j e₂ i) Z
+        rewrite splitAt-↑ʳ X Y (H₂.j e₂ i) = cong ((X ↑ʳ_) ∘ (_↑ˡ Z) ∘ H₂.j e₂) (sym (cast-is-id refl i))
+    ≗j e i | inj₂ e₃
+        rewrite splitAt-↑ʳ H₁.h (H₂.h + H₃.h) (H₂.h ↑ʳ e₃)
+        rewrite splitAt-↑ʳ H₂.h H₃.h e₃
+        rewrite splitAt-↑ʳ (X + Y) Z (H₃.j e₃ i) = cong ((X ↑ʳ_) ∘ (Y ↑ʳ_) ∘ H₃.j e₃) (sym (cast-is-id refl i))
+    ≗l  : (e : Fin (H₁.h + H₂.h + H₃.h))
+        → l (map-nodes (Inverse.to (+-assoc-↔ X Y Z)) (together (together H₁ H₂) H₃)) e
+        ≡ cast-gate (sym (cong ℕ.suc (≗a e))) (l (together H₁ (together H₂ H₃)) (Inverse.to ↔h e))
+    ≗l e with splitAt (H₁.h + H₂.h) e
+    ≗l e | inj₁ e₁₂ with splitAt H₁.h e₁₂
+    ≗l e | inj₁ e₁₂ | inj₁ e₁
+        rewrite splitAt-↑ˡ H₁.h e₁ (H₂.h + H₃.h) = sym (cast-gate-is-id refl (H₁.l e₁))
+    ≗l e | inj₁ e₁₂ | inj₂ e₂
+        rewrite splitAt-↑ʳ H₁.h (H₂.h + H₃.h) (e₂ ↑ˡ H₃.h)
+        rewrite splitAt-↑ˡ H₂.h e₂ H₃.h = sym (cast-gate-is-id refl (H₂.l e₂))
+    ≗l e | inj₂ e₃
+        rewrite splitAt-↑ʳ H₁.h (H₂.h + H₃.h) (H₂.h ↑ʳ e₃)
+        rewrite splitAt-↑ʳ H₂.h H₃.h e₃ = sym (cast-gate-is-id refl (H₃.l e₃))
+
+n+0↔n : ∀ n → Fin (n + 0) ↔ Fin n
+n+0↔n n = record
+    { to = to
+    ; from = from
+    ; to-cong = λ { refl → refl }
+    ; from-cong = λ { refl → refl }
+    ; inverse = (λ { refl → to∘from _ }) , λ { refl → from∘to _ }
+    }
+  where
+    to : Fin (n + 0) → Fin n
+    to x with inj₁ x₁ ← splitAt n x = x₁
+    from : Fin n → Fin (n + 0)
+    from x = x ↑ˡ 0
+    from∘to : (x : Fin (n + 0)) → from (to x) ≡ x
+    from∘to x with inj₁ x₁ ← splitAt n x in eq = splitAt⁻¹-↑ˡ eq
+    to∘from : (x : Fin n) → to (from x) ≡ x
+    to∘from x rewrite splitAt-↑ˡ n x 0 = refl
+
+unitaryʳ : Hypergraph-same (map-nodes ([ (λ x → x) , (λ ()) ] ∘ splitAt n) (together H empty-hypergraph)) H
+unitaryʳ {n} {H} = record
+    { ↔h = h+0↔h
+    ; ≗a = ≗a
+    ; ≗j = ≗j
+    ; ≗l = ≗l
+    }
+  where
+    module H = Hypergraph H
+    module H+0 = Hypergraph (together H empty-hypergraph)
+    h+0↔h : Fin H+0.h ↔ Fin H.h
+    h+0↔h = n+0↔n H.h
+    ≗a : (e : Fin (H.h + 0)) → [ H.a , (λ ()) ] (splitAt H.h e) ≡ H.a (Inverse.to h+0↔h e)
+    ≗a e with inj₁ e₁ ← splitAt H.h e in eq = refl
+    ≗j  : (e : Fin (H.h + 0))
+          (i : Fin (ℕ.suc ([ H.a , (λ ()) ] (splitAt H.h e))))
+        → [ (λ x → x) , (λ ()) ] (splitAt n (j+j H empty-hypergraph e i))
+        ≡ H.j (Inverse.to h+0↔h e) (cast (cong ℕ.suc (≗a e)) i)
+    ≗j e i = ≗j-aux (splitAt H.h e) refl (j+j H empty-hypergraph e) refl (≗a e) i
+      where
+        ≗j-aux
+            : (w : Fin H.h ⊎ Fin 0)
+            → (eq₁ : splitAt H.h e ≡ w)
+            → (w₁ : Fin (ℕ.suc ([ H.a , (λ ()) ] w)) → Fin (n + 0))
+            → j+j H empty-hypergraph e ≡ subst (λ hole → Fin (ℕ.suc ([ H.a , (λ ()) ] hole)) → Fin (n + 0)) (sym eq₁) w₁
+            → (w₂ : [ H.a , (λ ()) ] w ≡ H.a (Inverse.to h+0↔h e))
+              (i : Fin (ℕ.suc ([ H.a , (λ ()) ] w)))
+            → [ (λ x → x) , (λ ()) ] (splitAt n (w₁ i))
+            ≡ H.j (Inverse.to h+0↔h e) (cast (cong ℕ.suc w₂) i)
+        ≗j-aux (inj₁ e₁) eq w₁ eq₁ w₂ i
+            with (inj₁ x) ← splitAt n (w₁ i) in eq₂
+            rewrite eq = trans
+                (↑ˡ-injective 0 x (H.j e₁ i) (trans (splitAt⁻¹-↑ˡ eq₂) (sym (cong-app eq₁ i))))
+                (cong (H.j e₁) (sym (cast-is-id refl i)))
+    ≗l  : (e : Fin (H.h + 0))
+        → l+l H empty-hypergraph e
+        ≡ cast-gate (sym (cong ℕ.suc (≗a e))) (H.l (Inverse.to h+0↔h e))
+    ≗l e with splitAt H.h e | {(≗a e)}
+    ... | inj₁ e₁ = sym (cast-gate-is-id refl (H.l e₁))
+
++-comm-↔ : ∀ (n m : ℕ) → Fin (n + m) ↔ Fin (m + n)
++-comm-↔ n m = record
+    { to = join m n ∘ swap ∘ splitAt n
+    ; from = join n m ∘ swap ∘ splitAt m
+    ; to-cong = λ { refl → refl }
+    ; from-cong = λ { refl → refl }
+    ; inverse = (λ { refl → to∘from _ }) , λ { refl → from∘to _ }
+    }
+  where
+    open ≡-Reasoning
+    to∘from : join m n ∘ swap ∘ splitAt n ∘ join n m ∘ swap ∘ splitAt m ≗ id
+    to∘from x = begin
+        (join m n ∘ swap ∘ splitAt n ∘ join n m ∘ swap ∘ splitAt m) x ≡⟨ (cong (join m n ∘ swap) ∘ splitAt-join n m ∘ swap ∘ splitAt m) x ⟩
+        (join m n ∘ swap ∘ swap ∘ splitAt m) x                        ≡⟨ (cong (join m n) ∘ swap-involutive ∘ splitAt m) x ⟩
+        (join m n ∘ splitAt m) x                                      ≡⟨ join-splitAt m n x ⟩
+        x                                                             ∎
+    from∘to : join n m ∘ swap ∘ splitAt m ∘ join m n ∘ swap ∘ splitAt n ≗ id
+    from∘to x = begin
+        (join n m ∘ swap ∘ splitAt m ∘ join m n ∘ swap ∘ splitAt n) x ≡⟨ (cong (join n m ∘ swap) ∘ splitAt-join m n ∘ swap ∘ splitAt n) x ⟩
+        (join n m ∘ swap ∘ swap ∘ splitAt n) x                        ≡⟨ (cong (join n m) ∘ swap-involutive ∘ splitAt n) x ⟩
+        (join n m ∘ splitAt n) x                                      ≡⟨ join-splitAt n m x ⟩
+        x                                                             ∎
+
+[,]∘swap : {A B C : Set} {f : A → C} {g : B → C} → [ f , g ] ∘ swap ≗ [ g , f ]
+[,]∘swap (inj₁ x) = refl
+[,]∘swap (inj₂ y) = refl
+
+braiding
+    : {n m : ℕ}
+      {H₁ : Hypergraph n}
+      {H₂ : Hypergraph m}
+    → Hypergraph-same (map-nodes ([ m ↑ʳ_ , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂)) (together H₂ H₁)
+braiding {n} {m} {H₁} {H₂} = record
+    { ↔h = +-comm-↔ H₁.h H₂.h
+    ; ≗a = ≗a
+    ; ≗j = ≗j
+    ; ≗l = ≗l
+    }
+  where
+    open ≡-Reasoning
+    module H₁ = Hypergraph H₁
+    module H₂ = Hypergraph H₂
+    ≗a  : (e : Fin (H₁.h + H₂.h))
+        → [ H₁.a , H₂.a ] (splitAt H₁.h e)
+        ≡ [ H₂.a , H₁.a ] (splitAt H₂.h (join H₂.h H₁.h (swap (splitAt H₁.h e))))
+    ≗a e = begin
+        [ H₁.a , H₂.a ] (splitAt H₁.h e)                                        ≡⟨ [,]∘swap (splitAt H₁.h e) ⟨
+        [ H₂.a , H₁.a ] (swap (splitAt H₁.h e))                                 ≡⟨ cong [ H₂.a , H₁.a ] (splitAt-join H₂.h H₁.h (swap (splitAt H₁.h e))) ⟨
+        [ H₂.a , H₁.a ] (splitAt H₂.h (join H₂.h H₁.h (swap (splitAt H₁.h e)))) ∎
+    ≗j  : (e : Fin (Hypergraph.h (map-nodes ([ m ↑ʳ_ , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂))))
+          (i : Fin (Hypergraph.arity (map-nodes ([ m ↑ʳ_ , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂)) e))
+        → Hypergraph.j (map-nodes ([ _↑ʳ_ m , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂)) e i
+        ≡ Hypergraph.j (together H₂ H₁) (Inverse.to (+-comm-↔ H₁.h H₂.h) e) (cast (cong ℕ.suc (≗a e)) i)
+    ≗j e i with splitAt H₁.h e
+    ≗j e i | inj₁ e₁
+        rewrite splitAt-↑ˡ n (H₁.j e₁ i) m
+        rewrite splitAt-↑ʳ H₂.h H₁.h e₁ = cong ((m ↑ʳ_) ∘ H₁.j e₁) (sym (cast-is-id refl i))
+    ≗j e i | inj₂ e₂
+        rewrite splitAt-↑ʳ n m (H₂.j e₂ i)
+        rewrite splitAt-↑ˡ H₂.h e₂ H₁.h = cong ((_↑ˡ n) ∘ H₂.j e₂) (sym (cast-is-id refl i))
+    ≗l  : (e : Fin (H₁.h + H₂.h))
+        → l+l H₁ H₂ e
+        ≡ cast-gate (sym (cong ℕ.suc (≗a e))) (l+l H₂ H₁ (Inverse.to (+-comm-↔ H₁.h H₂.h) e))
+    ≗l e with splitAt H₁.h e | .{≗a e}
+    ≗l e | inj₁ e₁ rewrite splitAt-↑ʳ H₂.h H₁.h e₁ = sym (cast-gate-is-id refl (H₁.l e₁))
+    ≗l e | inj₂ e₂ rewrite splitAt-↑ˡ H₂.h e₂ H₁.h = sym (cast-gate-is-id refl (H₂.l e₂))
+
+hypergraph : SymmetricMonoidalFunctor Nat-smc (Setoids-× {0ℓ})
+hypergraph = record
+    { F = F
+    ; isBraidedMonoidal = record
+        { isMonoidal = record
+            { ε = ε
+            ; ⊗-homo = ntHelper record
+                { η = λ { (m , n) → η }
+                ; commute = λ { (f , g) {H₁ , H₂} → commute {H₁ = H₁} {H₂ = H₂} f g }
+                }
+            ; associativity = λ { {X} {Y} {Z} {(H₁ , H₂) , H₃} → associativity {X} {Y} {Z} {H₁} {H₂} {H₃} }
+            ; unitaryˡ = Hypergraph-same-refl
+            ; unitaryʳ = unitaryʳ
+            }
+        ; braiding-compat = λ { {X} {Y} {H₁ , H₂} → braiding {X} {Y} {H₁} {H₂} }
+        }
+    }
+  where
+    η : Func (×-setoid (Hypergraph-setoid n) (Hypergraph-setoid m)) (Hypergraph-setoid (n + m))
+    η = record
+        { to = λ { (H₁ , H₂) → together H₁ H₂ }
+        ; cong = λ { (≡H₁ , ≡H₂) → together-resp-same ≡H₁ ≡H₂ }
+        }
+
+module F = SymmetricMonoidalFunctor hypergraph
+
+and-gate : Func (SingletonSetoid {0ℓ} {0ℓ}) (F.₀ 3)
+and-gate = record
+    { to = λ { (lift tt) → and-graph }
+    ; cong = λ { (lift tt) → Hypergraph-same-refl }
+    }
+  where
+    and-graph : Hypergraph 3
+    and-graph = record
+        { h = 1
+        ; a = λ { 0F → 2 }
+        ; j = λ { 0F → edge-0-nodes }
+        ; l = λ { 0F → AND }
+        }
+      where
+        edge-0-nodes : Fin 3 → Fin 3
+        edge-0-nodes 0F = # 0
+        edge-0-nodes 1F = # 1
+        edge-0-nodes 2F = # 2