    -- |
    -- Module      :  Data.Functor.Product
    -- Copyright   :  (c) Ross Paterson 2010
    -- License     :  BSD-style (see the file LICENSE)
    --
    -- Maintainer  :  ross@soi.city.ac.uk
    -- Stability   :  experimental
    -- Portability :  portable
    --
    -- Products, lifted to functors.

    module Data.Functor.Product (
        Product(..),
       ) where

    import Control.Applicative
    import Control.Monad (MonadPlus(..))
    import Control.Monad.Fix (MonadFix(..))
    import Data.Foldable (Foldable(foldMap))
    import Data.Monoid (mappend)
    import Data.Traversable (Traversable(traverse))

    -- | Lifted product of functors.
    data Product f g a = Pair (f a) (g a)

    instance (Functor f, Functor g) => Functor (Product f g) where
        fmap f (Pair x y) = Pair (fmap f x) (fmap f y)

    instance (Foldable f, Foldable g) => Foldable (Product f g) where
        foldMap f (Pair x y) = foldMap f x `mappend` foldMap f y

    instance (Traversable f, Traversable g) => Traversable (Product f g) where
        traverse f (Pair x y) = Pair <$> traverse f x <*> traverse f y

    instance (Applicative f, Applicative g) => Applicative (Product f g) where
        pure x = Pair (pure x) (pure x)
        Pair f g <*> Pair x y = Pair (f <*> x) (g <*> y)

    instance (Alternative f, Alternative g) => Alternative (Product f g) where
        empty = Pair empty empty
        Pair x1 y1 <|> Pair x2 y2 = Pair (x1 <|> x2) (y1 <|> y2)

    instance (Monad f, Monad g) => Monad (Product f g) where
        return x = Pair (return x) (return x)
        Pair m n >>= f = Pair (m >>= fstP . f) (n >>= sndP . f)
          where
            fstP (Pair a _) = a
            sndP (Pair _ b) = b

    instance (MonadPlus f, MonadPlus g) => MonadPlus (Product f g) where
        mzero = Pair mzero mzero
        Pair x1 y1 `mplus` Pair x2 y2 = Pair (x1 `mplus` x2) (y1 `mplus` y2)

    instance (MonadFix f, MonadFix g) => MonadFix (Product f g) where
        mfix f = Pair (mfix (fstP . f)) (mfix (sndP . f))
          where
            fstP (Pair a _) = a
            sndP (Pair _ b) = b