module Chapter8.Ornament.Examples.Vec where

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

open import Data.Empty
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 Chapter2.Logic

import Chapter4.Desc.Examples.List as Desc

open import Chapter5.IDesc
open import Chapter5.IDesc.Fixpoint
open import Chapter5.IDesc.Examples.ToIDesc


open import Chapter8.Ornament

u : ℕ → ⊤
u _ = tt

ListD : Set → func zeroL ⊤ ⊤
ListD A = func.mk λ _ → toIDesc (Desc.ListD A)

module Constraint {A : Set} where

  VecO : orn (ListD A) u u
  VecO = orn.mk λ n → 
         `Σ {S = Fin }
             λ { zero → insert ( ≡ n) λ _ → 
                        `1
               ; (suc zero) → insert ℕ λ m → 
                              insert (suc m ≡ n) λ _ → 
                              `Σ λ _ →
                              `var (inv m) `× `1
               ; (suc (suc ())) }

  Vec : ℕ → Set
  Vec = μ ⟦ VecO ⟧orn 
  
  nil : Vec 
  nil = ⟨ zero , refl , lift tt ⟩
  
  cons : ∀{n} → A → Vec n → Vec (suc n)
  cons {n} a xs = ⟨ suc zero , n , refl , a , xs , lift tt ⟩

module Compute {A : Set} where

  VecO : orn (ListD A) u u
  VecO = orn.mk λ { zero → deleteΣ zero `1
                  ; (suc n) → deleteΣ (suc zero) (`Σ λ _ → `var (inv n) `× `1) }

  Vec : ℕ → Set
  Vec = μ ⟦ VecO ⟧orn 
  
  nil : Vec 
  nil = ⟨ lift tt ⟩

  cons : ∀{n} → A → Vec n → Vec (suc n)
  cons a xs = ⟨ a , xs , lift tt ⟩