module Chapter5.Ornament.Examples.Fin where

open import Function

open import Data.Empty
open import Data.Unit
open import Data.Nat
open import Data.Fin
open import Data.Product

open import Relation.Binary.PropositionalEquality

open import Chapter1.Logic

open import Chapter2.IDesc
open import Chapter2.IDesc.Fixpoint

open import Chapter2.IDesc.Examples.Nat

open import Chapter5.Ornament

u :   
u _ = tt

module Constraint where

  FinO : orn NatD u u
  FinO = orn.mk λ n  
           { zero  insert  λ m 
                        insert (suc m  n) λ _  
                        `1
               ; (suc zero)  insert  λ m  
                              insert (suc m  n) λ _  
                              `var (inv m)  `1
               ; (suc (suc ())) })

  Fin' :   Set
  Fin' = μ  FinO ⟧orn 
  
  fz : ∀{n}  Fin' (suc n)
  fz {n} =  zero , n , refl , tt 
  
  fs : ∀{n}  Fin' n  Fin' (suc n)
  fs {n} k =  suc zero , n , refl , k , tt 

module Compute where

  FinO : orn NatD u u
  FinO = orn.mk λ { zero  insert  ⊥-elim
                  ; (suc n)  
                     λ { zero  `1
                  ; (suc zero)  `var (inv n)  `1
                  ; (suc (suc ())) } }

  Fin' :   Set
  Fin' = μ  FinO ⟧orn 
  
  fz : ∀{n}  Fin' (suc n)
  fz {n} =  zero , tt 
  
  fs : ∀{n}  Fin' n  Fin' (suc n)
  fs {n} k =  suc zero , k , tt