------------------------------------------------------------------------
-- A universe for stream programs
------------------------------------------------------------------------

module Stream.Programs where

open import Coinduction renaming ( to ∞_)
import Stream as S
open S using (Stream; _≈_; _≺_; head; tail)
open import Relation.Binary.PropositionalEquality
open import Data.Vec using (Vec; []; _∷_)

------------------------------------------------------------------------
-- Stream programs

infix  8 _∞
infixr 7 _·_
infix  6 _⟨_⟩_
infixr 5 _≺_ _⋎_ _≺≺_

data Prog (A : Set) : Set1 where
  _≺_     : (x : A) (xs :  (Prog A))  Prog A
  _∞      : (x : A)  Prog A
  _·_     :  {B} (f : B  A) (xs : Prog B)  Prog A
  _⟨_⟩_   :  {B C} (xs : Prog B) (_∙_ : B  C  A) (ys : Prog C) 
            Prog A
  _⋎_     : (xs ys : Prog A)  Prog A
  iterate : (f : A  A) (x : A)  Prog A
  _≺≺_    :  {n} (xs : Vec A n) (ys : Prog A)  Prog A

data WHNF A : Set1 where
  _≺_ : (x : A) (xs : Prog A)  WHNF A

------------------------------------------------------------------------
-- Conversion

whnf :  {A}  Prog A  WHNF A
whnf (x  xs)         = x   xs
whnf (x )            = x  x 
whnf (f · xs)         with whnf xs
whnf (f · xs)         | x  xs′ = f x  f · xs′
whnf (xs  _∙_  ys)  with whnf xs | whnf ys
whnf (xs  _∙_  ys)  | x  xs′ | y  ys′ = (x  y)  xs′  _∙_  ys′
whnf (xs  ys)        with whnf xs
whnf (xs  ys)        | x  xs′ = x  ys  xs′
whnf (iterate f x)    = x  iterate f (f x)
whnf ([]       ≺≺ ys) = whnf ys
whnf ((x  xs) ≺≺ ys) = x  xs ≺≺ ys

mutual

  value :  {A}  WHNF A  Stream A
  value (x  xs) = x    xs 

  ⟦_⟧ :  {A}  Prog A  Stream A
   xs  = value (whnf xs)

fromStream :  {A}  Stream A  Prog A
fromStream (x  xs) = x   fromStream ( xs)

lift :  {A}  (Prog A  Prog A)  Stream A  Stream A
lift f xs =  f (fromStream xs) 

------------------------------------------------------------------------
-- Some abbreviations

infixr 5 _≺♯_

_≺♯_ :  {A}  A  Prog A  Prog A
x ≺♯ xs = x   xs

headP :  {A}  Prog A  A
headP xs = head  xs 

tailP :  {A}  Prog A  Prog A
tailP xs with whnf xs
tailP xs | x  xs′ = xs′