----------------------------------------------------------------------------
-- 
-- Copyright   :  (C) 2017 Yuchen Pei
-- License     :  GPLv3+
--
-- Maintainer  :  Yuchen Pei
-- Stability   :  experimental
-- Portability :  non-portable
--
----------------------------------------------------------------------------
--import Data.Monoid ((<>))

data SSYT a = S [[a]]
data Wrd a = W [a] deriving Show

-- |Knuth equivalence
instance Ord a => Eq (Wrd a)
  where (W xs) == (W ys) = (reduceWord xs) == (reduceWord ys)

-- |Show a tableau
instance Show a => Show (SSYT a)
  where show (S xs) = "S " ++ (show $ truncList xs)

-- |Plactic monoid
instance Ord a => Monoid (SSYT a)
  where
    mempty = S $ repeat []
    mappend s1 s2 = foldl rowInsert s1 (toRowWord s2)

-- |Convert a nested list to an SSYT
toSSYT :: [[a]] -> SSYT a
toSSYT t = S $ (truncList t) ++ (repeat [])

-- |Row insertion algorithm
rowInsert :: Ord a => SSYT a -> a -> SSYT a
rowInsert (S t) = S . rowInsert' t

-- |Row insertion algorithm on an SSYT as a nested list
rowInsert' :: Ord a => [[a]] -> a -> [[a]]
rowInsert' t x = 
  case break (>x) (head t) of
    (r, []) -> (r ++ [x]):(tail t)
    (r1, r2) -> (r1 ++ x:(tail r2)):(rowInsert' (tail t) (head r2))

-- |The Robinson-Schensted algorithm
robinsonSchensted :: Ord a => [a] -> SSYT a
robinsonSchensted = foldl rowInsert mempty

-- |Truncate a nested list (tableau) by disgarding empty rows
truncList :: [[a]] -> [[a]]
truncList = fst . break null

-- |Convert an SSYT to a row word
toRowWord :: Ord a => SSYT a -> [a]
toRowWord (S t) = concat $ reverse $ truncList t

-- |Whether a word is a row word
isRowWord :: Ord a => [a] -> Bool
isRowWord = isRowWord' [] []

isRowWord' :: Ord a => [a] -> [a] -> [a] -> Bool
isRowWord' _  ys [] = ys == []
isRowWord' [] [] zs = isRowWord' [head zs] [] (tail zs)
isRowWord' xs [] zs = if last xs <= head zs then isRowWord' (xs ++ [head zs]) [] (tail zs) else isRowWord' [] xs zs
isRowWord' xs ys zs = 
  if xs == [] || last xs <= head zs
    then head ys > head zs && (isRowWord' (xs ++ [head zs]) (tail ys) (tail zs))
    else ys == [] && isRowWord' [] xs zs

-- |Reduce a word to a row word
reduceWord :: Ord a => [a] -> [a]
reduceWord xs
  | length xs <= 2 = xs
  | otherwise      = let ys = reduceWord $ init xs in reduceWord' (init $ init ys) (last $ init ys, last ys, last xs) []
  {-- | otherwise      = let ys = reduceWord $ init xs in 
                     let (zs, ws) = splitAt (length ys - 2) ys in
                       reduceWord'' zs (ws ++ [last xs]) --}

reduceWord' :: Ord a => [a] -> (a, a, a) -> [a] -> [a]
reduceWord' [] (u, v, w) ys =
  if isRowWord (u:v:w:ys)
    then u:v:w:ys
    else if w < v && u <= v
      then if u > w then u:w:v:ys
                    else v:u:w:ys
      else u:v:w:ys
reduceWord' xs (u, v, w) ys =
  if isRowWord $ xs ++ (u:v:w:ys)
    then xs ++ (u:v:w:ys)
    else if w < v && u <= v 
      then if u > w then reduceWord' (init xs) (last xs, u, w) (v:ys)
                    else reduceWord' (init xs) (last xs, v, u) (w:ys)
      else reduceWord' (init xs) (last xs, u, v) (w:ys)

-- |QuickCheck properties
prop_ReduceWord :: [Int] -> Bool
prop_ReduceWord = isRowWord . reduceWord

prop_ReduceWord' :: [Int] -> Bool
prop_ReduceWord' xs = (length xs) == (length $ reduceWord xs)

prop_ReduceWord_RobinsonSchensted :: [Int] -> Bool
prop_ReduceWord_RobinsonSchensted xs = (toRowWord $ robinsonSchensted xs) == (reduceWord xs)

-- |Another implementation of reduceWord' in case of performance difference.
reduceWord'' :: Ord a => [a] -> [a] -> [a]
reduceWord'' [] (u:v:w:ys) =
  if isRowWord (u:v:w:ys)
    then u:v:w:ys
    else if w < v && u <= v
      then if u > w then u:w:v:ys
                    else v:u:w:ys
      else u:v:w:ys
reduceWord'' xs (u:v:w:ys) =
  if isRowWord $ xs ++ (u:v:w:ys)
    then xs ++ (u:v:w:ys)
    else if w < v && u <= v 
      then if u > w then reduceWord'' (init xs) (last xs:u:w:v:ys)
                    else reduceWord'' (init xs) (last xs:v:u:w:ys)
      else reduceWord'' (init xs) (last xs:u:v:w:ys)
reduceWord'' xs ys = xs ++ ys