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{-
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{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE ExplicitForAll #-}

-- MyMonoid
class MyMonoid s a where
  mmempty :: a
  mmappend :: a -> a -> a

instance (IntegerMonoid s) => MyMonoid s Integer where
  mmempty = im_mmempty @s
  mmappend = im_mmappend @s

instance (BoolMonoid s) => MyMonoid s Bool where
  mmempty = bm_mmempty @s
  mmappend = bm_mmappend @s
--

-- IntegerMonoid
data SumMonoid
data ProdMonoid
class IntegerMonoidSelector s
instance IntegerMonoidSelector SumMonoid
instance IntegerMonoidSelector ProdMonoid

class (IntegerMonoidSelector s) => IntegerMonoid s where
  im_mmempty :: Integer
  im_mmappend :: Integer -> Integer -> Integer

instance IntegerMonoid SumMonoid where
  im_mmempty = 0
  im_mmappend = (+)

instance IntegerMonoid ProdMonoid where
  im_mmempty = 1
  im_mmappend = (*)
--

-- BoolMonoid
data AnyMonoid
data AllMonoid
class BoolMonoidSelector s
instance BoolMonoidSelector AnyMonoid
instance BoolMonoidSelector AllMonoid

class (BoolMonoidSelector s) => BoolMonoid s where
  bm_mmempty :: Bool
  bm_mmappend :: Bool -> Bool -> Bool

instance BoolMonoid AnyMonoid where
  bm_mmempty = False
  bm_mmappend = (||)

instance BoolMonoid AllMonoid where
  bm_mmempty = True
  bm_mmappend = (&&)
--

-- MySemiring
class (MyMonoid m1 a, MyMonoid m2 a) => MySemiring s m1 m2 a where
  add :: a -> a -> a
  zero :: a
  mul :: a -> a -> a
  one :: a

instance (MyMonoid m1 Integer, MyMonoid m2 Integer, IntegerSemiring s m1 m2) => MySemiring s m1 m2 Integer where
  add = is_add @s @m1 @m2
  zero = is_zero @s @m1 @m2
  mul = is_mul @s @m1 @m2
  one = is_one @s @m1 @m2
--

-- IntegerSemiring
class (MyMonoid m1 Integer, MyMonoid m2 Integer) => IntegerSemiring s m1 m2 | s -> m1, s -> m2 where
  is_add :: Integer -> Integer -> Integer
  is_zero :: Integer
  is_mul :: Integer -> Integer -> Integer
  is_one :: Integer

instance IntegerSemiring s SumMonoid ProdMonoid where
  is_add = mmappend @SumMonoid
  is_zero = mmempty @SumMonoid
  is_mul = mmappend @ProdMonoid
  is_one = mmempty @ProdMonoid
--

-- MyOrd
class MyOrd s a where
  my_compare :: a -> a -> Ordering
  lessThan :: a -> a -> Bool
  lessThanEq :: a -> a -> Bool
  greaterThan :: a -> a -> Bool
  greaterThanEq :: a -> a -> Bool
  lessThan x y =
    case (my_compare @s @a) x y of
      LT -> True
      _ -> False
  lessThanEq x y =
    case (my_compare @s @a) x y of
      LT -> True
      EQ -> True
      _ -> False
  greaterThan x y =
    case (my_compare @s @a) x y of
      GT -> True
      _ -> False
  greaterThanEq x y =
    case (my_compare @s @a) x y of
      GT -> True
      EQ -> True
      _ -> False
--

-- IntegerOrd
instance MyOrd s Integer where
  my_compare x y = (compare :: Integer -> Integer -> Ordering) x y
--

-- PairOrd
-- The s selector within LeftFirst 
data LeftFirst sl sr
data RightFirst sl sr
class PairOrdSelector s
instance PairOrdSelector (LeftFirst sl sr)
instance PairOrdSelector (RightFirst sl sr)

instance (PairCompare s (a, b), MyOrd s a, MyOrd s b, PairOrdSelector s) => MyOrd s (a, b) where
  my_compare x y = (pair_compare @s @(a, b)) x y

class PairCompare s p where
  pair_compare :: p -> p -> Ordering

instance (MyOrd sl a, MyOrd sr b) => PairCompare (LeftFirst sl sr) (a, b) where
  pair_compare (x, y) (x', y') =
    case (my_compare @sl @a) x x' of
      EQ -> (my_compare @sr @b) y y'
      result -> result

instance (MyOrd sl a, MyOrd sr b) => PairCompare (RightFirst sl sr) (a, b) where
  pair_compare (x, y) (x', y') =
    case (my_compare @sr @b) y y' of
      EQ -> (my_compare @sl @a) x x'
      result -> result
--

-- Sample usage of ord: quicksort
quicksort :: forall s a . (MyOrd s a) => [a] -> [a]
quicksort [] = []
quicksort (x:xs) = (quicksort @s @a) [y | y <- xs, (lessThanEq @s @a) y x] ++ [x] ++ (quicksort @s @a) [y | y <- xs, (greaterThan @s @a) y x]
--

main = do
  print $ quicksort $ toInteger <$> [5, 4, 3, 6, 9]
  let arrayPair = zip (toInteger <$> [5, 4, 3, 6, 9]) (toInteger <$> [1, 10, 3, -10, 1000])
  print $ (quicksort @(LeftFirst _ _) @_) $ arrayPair
  print $ (quicksort @(RightFirst _ _) @_) $ arrayPair
  print $ (mmappend @SumMonoid) (toInteger 1) (toInteger 2)
  print $ (((mmappend @ProdMonoid) 3 4) :: Integer)
  print $ (mmappend @AnyMonoid) False False
  print $ (mmempty @AllMonoid :: Bool)
  print $ mul (toInteger 7) (toInteger 8)