2 files changed, 69 insertions, 0 deletions
diff --git a/NaturalTransformation/Instance/EmptyList.agda b/NaturalTransformation/Instance/EmptyList.agda
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+++ b/NaturalTransformation/Instance/EmptyList.agda
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+{-# OPTIONS --without-K --safe #-}
+
+open import Level using (Level)
+
+module NaturalTransformation.Instance.EmptyList {c ℓ : Level} where
+
+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 Categories.Functor.Construction.Constant using (const)
+open import Data.List using ([])
+open import Functor.Instance.List {c} {ℓ} using (List)
+open import Relation.Binary using (Setoid)
+
+module List = Functor List
+
+⊤⇒[] : NaturalTransformation (const SingletonSetoid) List
+⊤⇒[] = ntHelper record
+    { η = λ X → Const.function SingletonSetoid (List.₀ X) []
+    ; commute = λ {_} {B} f → Setoid.refl (List.₀ B)
+    }
diff --git a/NaturalTransformation/Instance/ListAppend.agda b/NaturalTransformation/Instance/ListAppend.agda
new file mode 100644
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+++ b/NaturalTransformation/Instance/ListAppend.agda
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+{-# OPTIONS --without-K --safe #-}
+
+open import Level using (Level; _⊔_)
+
+module NaturalTransformation.Instance.ListAppend {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 (_++_; map)
+open import Data.List.Properties using (map-++)
+open import Data.List.Relation.Binary.Pointwise using (++⁺)
+open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
+open import Data.Product using (_,_)
+open import Functor.Instance.List {c} {ℓ} using (List)
+open import Function using (Func; _⟶ₛ_)
+open import Relation.Binary using (Setoid)
+
+module List = Functor List
+
+open Cartesian (Setoids-Cartesian {c} {c ⊔ ℓ}) using (products)
+open BinaryProducts products using (-×-)
+open Func
+
+++ₛ : {X : Setoid c ℓ} → List.₀ X ×ₛ List.₀ X ⟶ₛ List.₀ X
+++ₛ .to (xs , ys) = xs ++ ys
+++ₛ .cong (≈xs , ≈ys) = ++⁺ ≈xs ≈ys
+
+map-++ₛ
+  : {A B : Setoid c ℓ}
+    (f : Func A B)
+    (xs ys : Data.List.List (Setoid.Carrier A))
+  → (open Setoid (List.₀ B))
+  → map (to f) xs ++ map (to f) ys ≈ map (to f) (xs ++ ys)
+map-++ₛ {_} {B} f xs ys = ListB.sym (ListB.reflexive (map-++ (to f) xs ys))
+  where
+    module ListB = Setoid (List.₀ B)
+
+++ : NaturalTransformation (-×- ∘F (List ※ List)) List
+++ = ntHelper record
+    { η = λ X → ++ₛ {X}
+    ; commute = λ { {A} {B} f {xs , ys} → map-++ₛ f xs ys }
+    }