content: ------------------------------------------------------------------------------
  -- | An implementation of Merge Sort, where LH verifies:
  --   1. Termination (Totality) 
  --   2. The output is indeed in non-decreasing order 
  ------------------------------------------------------------------------------
  
  module MergeSort (sort) where
  
  {-@ type OList a    = [a]<{\fld v -> (v >= fld)}> @-}
  
  {-@ type OListN a N = {v:OList a | len v == N} @-}
  
  -- | The top level `sort` function. Proved:
  --   (a) terminating, 
  --   (b) ordered, and 
  --   (c) of same size as input.
  
  {-@ sort :: (Ord a) => xs:[a] -> OListN a {len xs} @-}
  sort :: Ord a => [a] -> [a]
  sort []   = []
  sort [x]  = [x]
  sort xs   = merge (sort xs1) (sort xs2) 
    where 
      (xs1, xs2) = split xs
  
  -- Fun fact: if you delete the singleton case above,
  -- the resulting function is, in fact, non-terminating!
  
  -- | A type describing two `Halves` of a list `Xs` 
  
  {-@ type Halves a Xs = {v: (Half a Xs, Half a Xs) | len (fst v) + len (snd v) == len Xs} @-}
  
  -- | Each `Half` is empty or smaller than the input:
  
  {-@ type Half a Xs  = {v:[a] | (len v > 1) => (len v < len Xs)} @-}
  
  -- | The `split` function breaks its list into two `Halves`:
  
  {-@ split :: xs:[a] -> Halves a xs @-}
  split :: [a] -> ([a], [a])
  split (x:(y:zs)) = (x:xs, y:ys) where (xs, ys) = split zs
  split xs         = (xs, [])
  
  -- | Finally, the `merge` function combines two ordered lists into a single ordered result.
  
  {-@ merge :: Ord a => xs:OList a -> ys:OList a -> OListN a {len xs + len ys} / [(len xs + len ys)] @-}
  merge :: Ord a => [a] -> [a] ->  [a]
  merge xs []         = xs
  merge [] ys         = ys
  merge (x:xs) (y:ys)
    | x <= y          = x : merge xs (y:ys)
    | otherwise       = y : merge (x:xs) ys
  