----------------------------------------------------------------------------
-- 
-- Copyright   :  (C) 2017 Yuchen Pei
-- License     :  GPLv3+
--
-- Maintainer  :  Yuchen Pei
-- Stability   :  experimental
-- Portability :  non-portable
--
----------------------------------------------------------------------------
--import Math.Combinatorics.RootSystem hiding (s)
import RootSystem hiding (s)
import YoungTableaux
import RobinsonSchensted
--import Math.Algebra.Field.Base (Q)-- for Q
import Math.Algebra.LinearAlgebra
import Prelude hiding ( (*>), Word )
import Test.QuickCheck
import qualified Data.List as L
import Data.Ratio

--pitman :: (Fractional a, Ord a) => Type -> Int -> [a] -> [a]
pitman :: Type -> Int -> [[Q]] -> [[Q]]
pitman t n xs = tail $ foldr s (prependWithZero xs) (longestElement $ simpleSystem t n)

prependWithZero :: [[Q]] -> [[Q]]
prependWithZero [] = []
prependWithZero xs = (replicate (length $ head xs) 0) : xs

s :: [Q] -> [[Q]] -> [[Q]]
--s :: (Fractional a, Ord a) => [a] -> [a] -> [a]
s alpha f = f <<->> fmap (*> alpha) (cumMin $ dynkinIndex alpha <$> f) 

--cumMin :: Ord a => [a] -> [a]
cumMin :: [Q] -> [Q]
cumMin = scanl1 min

cumSum :: [Q] -> [Q]
cumSum = scanl1 (+)

word2Path :: Word Int -> [[Q]]
word2Path (W []) = []
--word2Path (W xs) = L.transpose $ cumSum <$> [0:(indicator (==k) xs) | k <- [1..maximum xs]]
word2Path (W xs) = L.transpose $ cumSum <$> [indicator (==k) xs | k <- [1..maximum xs]]

indicator :: (a -> Bool) -> [a] -> [Q]
indicator f xs = (\x -> if f x then 1 else 0) <$> xs

pitmanA :: [[Q]] -> [[Q]]
pitmanA [] = []
pitmanA xs = 
    let n = length $ head xs in
        if n == 1 then xs else pitman A (n - 1) xs

pitmanAShape :: [[Q]] -> [Q]
pitmanAShape = last . pitmanA

pitmanAGTP :: [[Q]] -> GTP Q
pitmanAGTP = GTP . (pitmanAGTP' []) where 
    pitmanAGTP' :: [[Q]] -> [[Q]] -> [[Q]]
    pitmanAGTP' xs [] = xs
    pitmanAGTP' xs ys = pitmanAGTP' ((pitmanAShape ys):xs) (L.transpose $ init $ L.transpose ys)


prop_Pitman_RobinsonSchensted :: [Int] -> Bool
prop_Pitman_RobinsonSchensted xs = 
    let w = prop_Pitman_RobinsonSchensted_sanitise xs in
        (pitmanAGTP $ word2Path w) == (gTPFromInt $ sSYT2GTP $ robinsonSchensted' w)

gTPFromInt :: GTP Int -> GTP Q
gTPFromInt (GTP xs) = GTP $ fmap (fmap fromIntegral) xs

prop_Pitman_RobinsonSchensted_sanitise :: [Int] -> Word Int
prop_Pitman_RobinsonSchensted_sanitise = W . (fmap (\t -> abs t + 1))


smallRational :: Gen Q
smallRational = do
    x <- smallInt
    y <- smallInt
    return $ (toInteger x) % (toInteger (abs y + 1)) -- this line does not work if 
    --return $ (toInteger x) / (toInteger (abs y + 1)) -- this line does not work for Q = Math.Algebra.Field.Base.Q - Couldn't match type ‘Integer’ with ‘Q’


--smallRational :: Gen Q
--smallRational = getSmall <$> (arbitrary :: Gen (Small Q))

--mediumInt :: Gen Int
--mediumInt = do
    --x <- smallInt
    --y <- smallInt
    --return $ x * y


smallInt :: Gen Int
smallInt = getSmall <$> (arbitrary :: Gen (Small Int))

arbRational :: Gen Q
arbRational = arbitrary

randomQMatrix :: Int -> Gen [[Q]]
randomQMatrix n = vectorOf 20 $ vectorOf n smallRational

prop_Pitman_WeylChamber :: Int -> Property
prop_Pitman_WeylChamber m = let (t, n) = int2TypeInt m in
    forAll (randomQMatrix $ dimensionOfHostSpace t n) (\xs -> isInWeylChamber (simpleSystem t n) (last $ pitman t n xs))

prop_PitmanA_WeylChamber :: Int -> Property
prop_PitmanA_WeylChamber n = let m = (n `mod` 2) + 1 in
    forAll (randomQMatrix $ m + 1) (\xs -> isInWeylChamber (simpleSystem A m) (last $ pitman A m xs))

--prop_PitmanA_WeylChamber :: Property
--prop_PitmanA_WeylChamber = let m = (1 `mod` 2) + 1 in
    --forAll (randomQMatrix $ m + 1) (\xs -> isInWeylChamber (simpleSystem A m) (last $ pitman A m xs))