module Chapter8.Ornament.Examples.List where

open import Level
  renaming ( zero to zeroL
           ; suc to sucL )
open import Function

open import Data.Unit
open import Data.Nat
open import Data.Fin hiding (lift)
open import Data.Product

open import Chapter2.Logic

open import Chapter5.IDesc
open import Chapter5.IDesc.Fixpoint

open import Chapter5.IDesc.Examples.Nat

open import Chapter8.Ornament


ListO : Set → orn NatD id id 
ListO A = orn.mk λ _ → 
           `σ (λ { zero → `1 
                 ; (suc zero) → insert A (λ _ → `var (inv tt) `× `1) 
                 ; (suc (suc ())) })

List : Set → Set
List A = μ ⟦ ListO A ⟧orn tt

nil : ∀{A} → List A
nil = ⟨ zero , lift tt ⟩

cons : ∀{A} → A → List A → List A
cons a xs = ⟨ suc zero , a , xs , lift tt ⟩