module Chapter5.IDesc.Examples.Walk where

open import Level 
  hiding (suc)
  renaming (zero to zeroL)

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

open import Relation.Binary.PropositionalEquality

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


WalkD : tagDesc zeroL ℕ
WalkD = ( , (λ n → `var (suc n) `× `1 ∷ [])) ,
        ((λ { zero →  
            ; (suc n) →  }) , 
          λ { zero → `1 ∷ [] 
            ; (suc n) → `var n `× `1 ∷ [] })

Walk : ℕ → Set
Walk = μ (toDesc WalkD)

up : ∀{n} → Walk (suc n) → Walk n
up w = ⟨ lift zero , w , lift tt ⟩

stop : Walk 
stop = ⟨ lift (suc zero) , lift tt ⟩

down : ∀{n} → Walk n → Walk (suc n)
down w = ⟨ lift (suc zero) , w , lift tt ⟩