2 files changed, 38 insertions, 28 deletions

diff --git a/NaturalTransformation/Instance/EmptyMultiset.agda b/NaturalTransformation/Instance/EmptyMultiset.agda
index 9c3a779..bfec451 100644
--- a/NaturalTransformation/Instance/EmptyMultiset.agda
+++ b/NaturalTransformation/Instance/EmptyMultiset.agda
@@ -1,6 +1,6 @@
 {-# OPTIONS --without-K --safe #-}
 
-open import Level using (Level)
+open import Level using (Level; _⊔_)
 
 module NaturalTransformation.Instance.EmptyMultiset {c ℓ : Level} where
 
@@ -8,16 +8,27 @@ import Function.Construct.Constant as Const
 
 open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
 open import Categories.Functor using (Functor)
-open import Categories.Category.Instance.SingletonSet using (SingletonSetoid)
+open import Data.Setoid.Unit {c} {c ⊔ ℓ} using (⊤ₛ)
 open import Categories.Functor.Construction.Constant using (const)
-open import Data.List using ([])
+open import Data.Opaque.Multiset using (Multisetₛ; []ₛ; mapₛ)
 open import Functor.Instance.Multiset {c} {ℓ} using (Multiset)
+open import Function.Construct.Constant using () renaming (function to Const)
 open import Relation.Binary using (Setoid)
+open import Data.Setoid using (_⇒ₛ_)
+open import Function using (Func; _⟶ₛ_)
+open import Function.Construct.Setoid using (_∙_)
 
-module Multiset = Functor Multiset
+opaque
+  unfolding mapₛ
+  map-[]ₛ
+      : {A B : Setoid c ℓ}
+      → (f : A ⟶ₛ B)
+      → (open Setoid (⊤ₛ ⇒ₛ Multisetₛ B))
+      → []ₛ ≈ mapₛ f ∙ []ₛ
+  map-[]ₛ {B = B} f = Setoid.refl (Multisetₛ B)
 
-⊤⇒[] : NaturalTransformation (const SingletonSetoid) Multiset
+⊤⇒[] : NaturalTransformation (const ⊤ₛ) Multiset
 ⊤⇒[] = ntHelper record
-    { η = λ X → Const.function SingletonSetoid (Multiset.₀ X) []
-    ; commute = λ {_} {B} f → Setoid.refl (Multiset.₀ B)
+    { η = λ X → []ₛ {Aₛ = X}
+    ; commute = map-[]ₛ
     }
diff --git a/NaturalTransformation/Instance/MultisetAppend.agda b/NaturalTransformation/Instance/MultisetAppend.agda
index b0e8bc4..f786124 100644
--- a/NaturalTransformation/Instance/MultisetAppend.agda
+++ b/NaturalTransformation/Instance/MultisetAppend.agda
@@ -4,43 +4,42 @@ open import Level using (Level; _⊔_)
 
 module NaturalTransformation.Instance.MultisetAppend {c ℓ : Level} where
 
-open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
-open import Categories.Category.Product using (_※_)
+import Data.Opaque.List as L
+
 open import Categories.Category.BinaryProducts using (module BinaryProducts)
-open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian)
 open import Categories.Category.Cartesian using (Cartesian)
+open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian)
+open import Categories.Category.Product using (_※_)
 open import Categories.Functor using (Functor; _∘F_)
-open import Data.List using (List; _++_; map)
+open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
 open import Data.List.Properties using (map-++)
 open import Data.List.Relation.Binary.Permutation.Setoid.Properties using (++⁺)
-open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
+open import Data.Opaque.Multiset using (Multisetₛ; mapₛ; ++ₛ)
 open import Data.Product using (_,_)
+open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
+open import Function using (Func; _⟶ₛ_; _⟨$⟩_)
 open import Functor.Instance.Multiset {c} {ℓ} using (Multiset)
-open import Function using (Func; _⟶ₛ_)
 open import Relation.Binary using (Setoid)
 
-module Multiset = Functor Multiset
-
 open Cartesian (Setoids-Cartesian {c} {c ⊔ ℓ}) using (products)
 open BinaryProducts products using (-×-)
 open Func
 
-++ₛ : {X : Setoid c ℓ} → Multiset.₀ X ×ₛ Multiset.₀ X ⟶ₛ Multiset.₀ X
-++ₛ .to (xs , ys) = xs ++ ys
-++ₛ {A} .cong (≈xs , ≈ys) = ++⁺ A ≈xs ≈ys
+opaque
+  unfolding ++ₛ mapₛ
 
-map-++ₛ
-  : {A B : Setoid c ℓ}
-    (f : Func A B)
-    (xs ys : List (Setoid.Carrier A))
-  → (open Setoid (Multiset.₀ B))
-  → map (to f) xs ++ map (to f) ys ≈ map (to f) (xs ++ ys)
-map-++ₛ {_} {B} f xs ys =  sym (reflexive (map-++ (to f) xs ys))
-  where
-    open Setoid (Multiset.₀ B)
+  map-++ₛ
+    : {A B : Setoid c ℓ}
+      (f : Func A B)
+      (xs ys : Setoid.Carrier (Multiset.₀ A))
+    → (open Setoid (Multiset.₀ B))
+    → ++ₛ ⟨$⟩ (mapₛ f ⟨$⟩ xs , mapₛ f ⟨$⟩ ys) ≈ mapₛ f ⟨$⟩ (++ₛ ⟨$⟩ (xs , ys))
+  map-++ₛ {A} {B} f xs ys = sym (reflexive (map-++ (to f) xs ys))
+    where
+      open Setoid (Multiset.₀ B)
 
 ++ : NaturalTransformation (-×- ∘F (Multiset ※ Multiset)) Multiset
 ++ = ntHelper record
-    { η = λ X → ++ₛ {X}
+    { η = λ X → ++ₛ
     ; commute = λ { {A} {B} f {xs , ys} → map-++ₛ f xs ys }
     }