module Chapter8.AlgebraicOrnament.Examples.Interval where

open import Level hiding (zero ; suc)

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

open import Relation.Binary.PropositionalEquality

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

open import Chapter5.IDesc.Examples.Nat

import  Chapter8.AlgebraicOrnament

α+m : Nat → Alg NatD (λ _ → Nat)
α+m m {tt} (zero , lift tt) = m
α+m m {tt} (suc zero , n , lift tt) = su n
α+m m {tt} (suc (suc ()) , _)


[_∶_] : Nat → Nat → Set
[ m ∶ n ] = μ algOrnD (tt , n)
  where open Chapter8.AlgebraicOrnament.Func NatD (α+m m)

ize : ∀{m} → [ m ∶ m ] 
ize = ⟨ (zero , lift tt) , refl , lift tt ⟩

isu : ∀{m n} → [ m ∶ n ] → [ m ∶ su n ]
isu {m}{n} ik = ⟨ (suc zero , n , lift tt) , refl , ik , lift tt ⟩