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+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Instance.Nat.Push where
+
+open import Categories.Functor using (Functor)
+open import Categories.Category.Instance.Nat using (Nat)
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Data.Circuit.Merge using (merge; merge-cong₁; merge-cong₂; merge-⁅⁆; merge-preimage)
+open import Data.Fin.Base using (Fin)
+open import Data.Fin.Preimage using (preimage; preimage-cong₁)
+open import Data.Nat.Base using (ℕ)
+open import Data.Subset.Functional using (⁅_⁆)
+open import Function.Base using (id; _∘_)
+open import Function.Bundles using (Func; _⟶ₛ_)
+open import Function.Construct.Identity using () renaming (function to Id)
+open import Function.Construct.Setoid using (setoid; _∙_)
+open import Level using (0ℓ)
+open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
+open import Data.Circuit.Value using (Value)
+open import Data.System.Values Value using (Values)
+
+open Func
+open Functor
+
+private
+  variable A B C : ℕ
+
+_≈_ : {X Y : Setoid 0ℓ 0ℓ} → Rel (X ⟶ₛ Y) 0ℓ
+_≈_ {X} {Y} = Setoid._≈_ (setoid X Y)
+infixr 4 _≈_
+
+-- action of Push on objects is Values n (Vector Value n)
+
+-- action of Push on morphisms (covariant)
+Push₁ : (Fin A → Fin B) → Values A ⟶ₛ Values B
+to (Push₁ f) v = merge v ∘ preimage f ∘ ⁅_⁆
+cong (Push₁ f) x≗y = merge-cong₁ x≗y ∘ preimage f ∘ ⁅_⁆
+
+-- Push respects identity
+Push-identity : Push₁ id ≈ Id (Values A)
+Push-identity {_} {v} = merge-⁅⁆ v
+
+-- Push respects composition
+Push-homomorphism
+    : {f : Fin A → Fin B}
+      {g : Fin B → Fin C}
+    → Push₁ (g ∘ f) ≈ Push₁ g ∙ Push₁ f
+Push-homomorphism {f = f} {g} {v} = merge-preimage f v ∘ preimage g ∘ ⁅_⁆
+
+-- Push respects equality
+Push-resp-≈
+    : {f g : Fin A → Fin B}
+    → f ≗ g
+    → Push₁ f ≈ Push₁ g
+Push-resp-≈ f≗g {v} = merge-cong₂ v ∘ preimage-cong₁ f≗g ∘ ⁅_⁆
+
+-- the Push functor
+Push : Functor Nat (Setoids 0ℓ 0ℓ)
+F₀ Push = Values
+F₁ Push = Push₁
+identity Push = Push-identity
+homomorphism Push = Push-homomorphism
+F-resp-≈ Push = Push-resp-≈