module Chapter5.IDesc.Examples.Vec where

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

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

open import Relation.Binary.PropositionalEquality

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

module Compute where
  
  VecD : (A : Set) → func zeroL ℕ ℕ
  VecD A = func.mk (λ { zero → `1 
                      ; (suc n) → `Σ A λ _ → `var n `× `1 })

  Vec : (A : Set) → ℕ → Set
  Vec A = μ (VecD A)

  nil : ∀{A} → Vec A 
  nil = ⟨ lift tt ⟩

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

module Constraint where

  VecD : (A : Set) → func zeroL ℕ ℕ
  VecD A = func.mk (λ n → `Σ (Fin ) 
                          λ { zero → `Σ (n ≡ ) λ _ → `1  
                            ; (suc zero) → `Σ A λ _ → 
                                           `Σ ℕ λ m → 
                                           `Σ (n ≡ suc m) λ _ → 
                                           `var m `× `1
                            ; (suc (suc ())) } )

  Vec : (A : Set) → ℕ → Set
  Vec A = μ (VecD A)

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

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

module TaggedCompute where

  VecD : (A : Set) → tagDesc zeroL ℕ
  VecD A = ( , λ n → []) ,
           (λ { zero →  
              ; (suc n) →  }) ,
           (λ { zero → `1 ∷ [] 
              ; (suc n) → (`Σ A λ _ → `var n `× `1) ∷ [] })

  Vec : (A : Set) → ℕ → Set
  Vec A = μ (toDesc (VecD A))

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

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

module TaggedConstraint where

  VecD : (A : Set) → tagDesc zeroL ℕ
  VecD A = ( , λ n → 
              (`Σ (n ≡ ) λ _ → `1) ∷ 
              (`Σ A λ _ → 
               `Σ ℕ λ m → 
               `Σ (n ≡ suc m) λ _ → 
               `var m `× `1) ∷ []) ,
           ((λ n → ) , λ n → [])

  Vec : (A : Set) → ℕ → Set
  Vec A = μ (toDesc (VecD A))

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

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