From 07a14947f6fee3219d575a15938bf33764cce791 Mon Sep 17 00:00:00 2001
From: Jacques Comeaux <jacquesrcomeaux@protonmail.com>
Date: Wed, 5 Nov 2025 03:14:18 -0600
Subject: Add free commutaive monoid functor

---
 NaturalTransformation/Instance/MultisetAppend.agda | 46 ++++++++++++++++++++++
 1 file changed, 46 insertions(+)
 create mode 100644 NaturalTransformation/Instance/MultisetAppend.agda

(limited to 'NaturalTransformation/Instance/MultisetAppend.agda')

diff --git a/NaturalTransformation/Instance/MultisetAppend.agda b/NaturalTransformation/Instance/MultisetAppend.agda
new file mode 100644
index 0000000..b0e8bc4
--- /dev/null
+++ b/NaturalTransformation/Instance/MultisetAppend.agda
@@ -0,0 +1,46 @@
+{-# OPTIONS --without-K --safe #-}
+
+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 (_※_)
+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.Functor using (Functor; _∘F_)
+open import Data.List using (List; _++_; map)
+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.Product using (_,_)
+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
+
+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)
+
+++ : NaturalTransformation (-×- ∘F (Multiset ※ Multiset)) Multiset
+++ = ntHelper record
+    { η = λ X → ++ₛ {X}
+    ; commute = λ { {A} {B} f {xs , ys} → map-++ₛ f xs ys }
+    }
-- 
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