5 files changed, 289 insertions, 1 deletions
diff --git a/Functor/Instance/Decorate.agda b/Functor/Instance/Decorate.agda
index e87f20c..30cec87 100644
--- a/Functor/Instance/Decorate.agda
+++ b/Functor/Instance/Decorate.agda
@@ -40,7 +40,7 @@ module DecoratedCospans = Category DecoratedCospans
 module mc𝒞 = CocartesianMonoidal 𝒞.U 𝒞.cocartesian
 
 -- For every cospan there exists a free decorated cospan
--- i.e. the original cospan with the empty decoration
+-- i.e. the original cospan with the discrete decoration
 
 private
   variable
diff --git a/Functor/Instance/Nat/Preimage.agda b/Functor/Instance/Nat/Preimage.agda
new file mode 100644
index 0000000..7da00f4
--- /dev/null
+++ b/Functor/Instance/Nat/Preimage.agda
@@ -0,0 +1,65 @@
+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Instance.Nat.Preimage where
+
+open import Categories.Category.Instance.Nat using (Natop)
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Functor using (Functor)
+open import Data.Fin.Base using (Fin)
+open import Data.Bool.Base using (Bool)
+open import Data.Nat.Base using (ℕ)
+open import Data.Subset.Functional using (Subset)
+open import Data.Vec.Functional.Relation.Binary.Equality.Setoid using (≋-setoid)
+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 Functor
+open Func
+
+_≈_ : {X Y : Setoid 0ℓ 0ℓ} → Rel (X ⟶ₛ Y) 0ℓ
+_≈_ {X} {Y} = Setoid._≈_ (setoid X Y)
+infixr 4 _≈_
+
+private
+  variable A B C : ℕ
+
+-- action on objects (Subset n)
+Subsetₛ : ℕ → Setoid 0ℓ 0ℓ
+Subsetₛ = ≋-setoid (≡.setoid Bool)
+
+-- action of Preimage on morphisms (contravariant)
+Preimage₁ : (Fin A → Fin B) → Subsetₛ B ⟶ₛ Subsetₛ A
+to (Preimage₁ f) i = i ∘ f
+cong (Preimage₁ f) x≗y = x≗y ∘ f
+
+-- Preimage respects identity
+Preimage-identity : Preimage₁ id ≈ Id (Subsetₛ A)
+Preimage-identity {A} = Setoid.refl (Subsetₛ A)
+
+-- Preimage flips composition
+Preimage-homomorphism
+    : {A B C : ℕ}
+      (f : Fin A → Fin B)
+      (g : Fin B → Fin C)
+    → Preimage₁ (g ∘ f) ≈ Preimage₁ f ∙ Preimage₁ g
+Preimage-homomorphism {A} _ _ = Setoid.refl (Subsetₛ A)
+
+-- Preimage respects equality
+Preimage-resp-≈
+    : {f g : Fin A → Fin B}
+    → f ≗ g
+    → Preimage₁ f ≈ Preimage₁ g
+Preimage-resp-≈ f≗g {v} = ≡.cong v ∘ f≗g
+
+-- the Preimage functor
+Preimage : Functor Natop (Setoids 0ℓ 0ℓ)
+F₀ Preimage = Subsetₛ
+F₁ Preimage = Preimage₁
+identity Preimage = Preimage-identity
+homomorphism Preimage {f = f} {g} {v} = Preimage-homomorphism g f {v}
+F-resp-≈ Preimage = Preimage-resp-≈
diff --git a/Functor/Instance/Nat/Pull.agda b/Functor/Instance/Nat/Pull.agda
new file mode 100644
index 0000000..5b74399
--- /dev/null
+++ b/Functor/Instance/Nat/Pull.agda
@@ -0,0 +1,63 @@
+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Instance.Nat.Pull where
+
+open import Categories.Category.Instance.Nat using (Natop)
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Functor using (Functor)
+open import Data.Fin.Base using (Fin)
+open import Data.Nat.Base using (ℕ)
+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 Functor
+open Func
+
+_≈_ : {X Y : Setoid 0ℓ 0ℓ} → Rel (X ⟶ₛ Y) 0ℓ
+_≈_ {X} {Y} = Setoid._≈_ (setoid X Y)
+infixr 4 _≈_
+
+private
+  variable A B C : ℕ
+
+
+-- action on objects is Values n (Vector Value n)
+
+-- action of Pull on morphisms (contravariant)
+Pull₁ : (Fin A → Fin B) → Values B ⟶ₛ Values A
+to (Pull₁ f) i = i ∘ f
+cong (Pull₁ f) x≗y = x≗y ∘ f
+
+-- Pull respects identity
+Pull-identity : Pull₁ id ≈ Id (Values A)
+Pull-identity {A} = Setoid.refl (Values A)
+
+-- Pull flips composition
+Pull-homomorphism
+    : {A B C : ℕ}
+      (f : Fin A → Fin B)
+      (g : Fin B → Fin C)
+    → Pull₁ (g ∘ f) ≈ Pull₁ f ∙ Pull₁ g
+Pull-homomorphism {A} _ _ = Setoid.refl (Values A)
+
+-- Pull respects equality
+Pull-resp-≈
+    : {f g : Fin A → Fin B}
+    → f ≗ g
+    → Pull₁ f ≈ Pull₁ g
+Pull-resp-≈ f≗g {v} = ≡.cong v ∘ f≗g
+
+-- the Pull functor
+Pull : Functor Natop (Setoids 0ℓ 0ℓ)
+F₀ Pull = Values
+F₁ Pull = Pull₁
+identity Pull = Pull-identity
+homomorphism Pull {f = f} {g} {v} = Pull-homomorphism g f {v}
+F-resp-≈ Pull = Pull-resp-≈
diff --git a/Functor/Instance/Nat/Push.agda b/Functor/Instance/Nat/Push.agda
new file mode 100644
index 0000000..53d0bef
--- /dev/null
+++ b/Functor/Instance/Nat/Push.agda
@@ -0,0 +1,64 @@
+{-# 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-≈
diff --git a/Functor/Instance/Nat/System.agda b/Functor/Instance/Nat/System.agda
new file mode 100644
index 0000000..730f90d
--- /dev/null
+++ b/Functor/Instance/Nat/System.agda
@@ -0,0 +1,96 @@
+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Instance.Nat.System {ℓ} where
+
+open import Categories.Category.Instance.Nat using (Nat)
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Functor.Core using (Functor)
+open import Data.Circuit.Value using (Value)
+open import Data.Fin.Base using (Fin)
+open import Data.Nat.Base using (ℕ)
+open import Data.Product.Base using (_,_; _×_)
+open import Data.System using (System; ≤-System; Systemₛ)
+open import Data.System.Values Value using (module ≋)
+open import Function.Bundles using (Func; _⟶ₛ_)
+open import Function.Base using (id; _∘_)
+open import Function.Construct.Setoid using (_∙_)
+open import Functor.Instance.Nat.Pull using (Pull₁; Pull-resp-≈)
+open import Functor.Instance.Nat.Push using (Push₁; Push-identity; Push-homomorphism; Push-resp-≈)
+open import Level using (suc)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
+
+-- import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
+import Function.Construct.Identity as Id
+
+open Func
+open ≤-System
+open Functor
+
+private
+  variable A B C : ℕ
+
+map : (Fin A → Fin B) → System {ℓ} A → System B
+map f X = record
+    { S = S
+    ; fₛ = fₛ ∙ Pull₁ f
+    ; fₒ = Push₁ f ∙ fₒ
+    }
+  where
+    open System X
+
+≤-cong : (f : Fin A → Fin B) {X Y : System A} → ≤-System Y X → ≤-System (map f Y) (map f X)
+⇒S (≤-cong f x≤y) = ⇒S x≤y
+≗-fₛ (≤-cong f x≤y) = ≗-fₛ x≤y ∘ to (Pull₁ f)
+≗-fₒ (≤-cong f x≤y) = cong (Push₁ f) ∘ ≗-fₒ x≤y
+
+System₁ : (Fin A → Fin B) → Systemₛ A ⟶ₛ Systemₛ B
+to (System₁ f) = map f
+cong (System₁ f) (x≤y , y≤x) = ≤-cong f x≤y , ≤-cong f y≤x
+
+id-x≤x : {X : System A} → ≤-System (map id X) X
+⇒S (id-x≤x) = Id.function _
+≗-fₛ (id-x≤x {_} {x}) i s = System.refl x
+≗-fₒ (id-x≤x {A} {x}) s = Push-identity
+
+x≤id-x : {x : System A} → ≤-System x (map id x)
+⇒S x≤id-x = Id.function _
+≗-fₛ (x≤id-x {A} {x}) i s = System.refl x
+≗-fₒ (x≤id-x {A} {x}) s = ≋.sym Push-identity
+
+
+System-homomorphism
+    : {f : Fin A → Fin B}
+      {g : Fin B → Fin C} 
+      {X : System A}
+    → ≤-System (map (g ∘ f) X) (map g (map f X)) × ≤-System (map g (map f X)) (map (g ∘ f) X)
+System-homomorphism {f = f} {g} {X} = left , right
+  where
+    open System X
+    left : ≤-System (map (g ∘ f) X) (map g (map f X))
+    left .⇒S = Id.function S
+    left .≗-fₛ i s = refl
+    left .≗-fₒ s = Push-homomorphism
+    right : ≤-System (map g (map f X)) (map (g ∘ f) X)
+    right .⇒S = Id.function S
+    right .≗-fₛ i s = refl
+    right .≗-fₒ s = ≋.sym Push-homomorphism
+
+System-resp-≈
+    : {f g : Fin A → Fin B}
+    → f ≗ g
+    → {X : System A}
+    → (≤-System (map f X) (map g X)) × (≤-System (map g X) (map f X))
+System-resp-≈ {A} {B} {f = f} {g} f≗g {X} = both f≗g , both (≡.sym ∘ f≗g)
+  where
+    open System X
+    both : {f g : Fin A → Fin B} → f ≗ g → ≤-System (map f X) (map g X)
+    both f≗g .⇒S = Id.function S
+    both f≗g .≗-fₛ i s = cong fₛ (Pull-resp-≈ f≗g {i})
+    both {f} {g} f≗g .≗-fₒ s = Push-resp-≈ f≗g
+
+Sys : Functor Nat (Setoids (suc ℓ) ℓ)
+Sys .F₀ = Systemₛ
+Sys .F₁ = System₁
+Sys .identity = id-x≤x , x≤id-x
+Sys .homomorphism {x = X} = System-homomorphism {X = X}
+Sys .F-resp-≈ = System-resp-≈