{-@ LIQUID "--expect-any-error" @-}
module RecQSort () where

{-@ type OList a =  [a]<{\fld v -> (v >= fld)}>  @-}

{-@ sort3 :: (Ord a) => [a] -> OList a @-}
sort3 :: (Ord a) => [a] -> [a]
sort3 ls = qsort ls 
  where d = (length ls) 


qsort:: (Ord a) => [a] -> [a]
{-@ qsort:: (Ord a) => xs:[a] -> OList a / [(len xs), 0] @-}
qsort []     = []
qsort (x:xs) = qpart x xs [] [] 

qpart  :: (Ord a) => a -> [a] -> [a] -> [a] -> [a]
{-@ qpart  :: (Ord a) => x:a -> q:[a] -> r:[{v:a | v < x}] -> p:[{v:a | v >= x}] -> OList a / [((len r)+(len q)+(len p)), ((len q)+1)]@-}
qpart x [] rlt rge =
    app x (qsort rlt) (x:qsort rge)
qpart x (y:ys) rlt rge =
    case compare x y of
        GT -> qpart x (y:ys) (y:rlt) rge
        _  -> qpart x ys rlt (y:rge)


{-@ app :: Ord a => x:a -> (OList ({v:a | v < x})) -> (OList ({v:a| v >= x})) -> OList a @-} 
app :: Ord a => a -> [a] -> [a] -> [a]
app k []     ys = ys
app k (x:xs) ys = x : (app k xs ys)