- Ввод полинома

inputPoly :: String -> Bool -> Int -> Int -> Int -> [(Int,Int)]
inputPoly [] _ s c p = [(s*c,p)]
inputPoly (x:xs) f s c p | f && (x == 'x') = inputPoly xs True s (if c == 0 then 1 else c) 1
                         |       x == '-'  = (c * s, p) : (inputPoly xs True (-1) 0 0)
                         |       x == '+'  = (c * s, p) : (inputPoly xs True 1 0 0)
                         | f && (x == '^') = inputPoly xs False s c 0
                         | f               = inputPoly xs f s (c * 10 + (read [x] :: Int)) p
                         | otherwise       = inputPoly xs f s c (p * 10 + (read [x]  :: Int))

-- Упрощение

redex :: [(Int, Int)] -> [(Int, Int)]
redex x = filter ((/= 0) . fst) rlist
              where plist = sort $ nub $ map snd x
                    rlist = map (\ n -> (sum (map fst (filter ((== n) . snd) x) ) , n) ) plist

-- Вывод полинома

showPoly :: [(Int, Int)] -> String
showPoly []         = ""
showPoly ((c, p):xs) | p == 0                 = show(c) ++ showPoly xs
                     | (c == 1) && (p == 1)   = "+x" ++ showPoly xs
                     | (c == -1) && (p == 1)  = "-x" ++ showPoly xs
                     | (c == 1) && (p /= 1)   = "+x^" ++ show(p) ++ showPoly xs
                     | (c == -1) && (p /= 1)  = "-x^" ++ show(p) ++ showPoly xs
                     | p /= 1                 = (if c >0 then "+" else "") ++ show(c)++"x^" ++ show(p) ++ showPoly xs
                     | otherwise              = (if c >0 then "+" else "") ++ show(c)++"x"  ++ showPoly xs

-- Производная

derivPoly :: [(Int, Int)] -> [(Int, Int)]
derivPoly = redex . map (\ (c, p) -> (p * c, p-1))

-- Сложение полиномов

addPoly :: [(Int, Int)] -> [(Int, Int)] -> [(Int, Int)]
addPoly x y = redex $ x ++ y

-- Умножение полиномов

multPoly :: [(Int, Int)] -> [(Int, Int)] -> [(Int, Int)]
multPoly x y = redex $ foldl (\ acc c -> map (\ yy -> ( (fst yy) * (fst c), (snd yy)+ (snd c) ) ) y ++ acc) [] x

-- Умножение с вводом в естественном виде

multInput :: String -> String -> String
multInput x y = showPoly $ multPoly (inputPoly x True 1 0 0) (inputPoly y True 1 0 0)

-- Сложение с вводом в естественном виде

addInput :: String -> String -> String
addInput x y = showPoly $ addPoly (inputPoly x True 1 0 0) (inputPoly y True 1 0 0)

-- Дифференцирование с вводом в ест. виде

diffp :: String -> String
diffp x = showPoly $ derivPoly $ inputPoly x True 1 0 0