```----------------------------------------------------------------------------
-- |
-- Module     : Data.Group
-- Copyright  : 2007-2009 Edward Kmett
-- License    : BSD
--
-- Maintainer  : Edward Kmett <ekmett@gmail.com>
-- Stability   : experimental
-- Portability : portable
--
-- Extends 'Monoid' to support 'Group' operations
-----------------------------------------------------------------------------

module Data.Group
( Group
, gnegate
, gsubtract
, minus
, MultiplicativeGroup
, over
, under
, grecip
) where

import Data.Monoid (Monoid, Sum(..), Product(..), Dual(..))
import Data.Monoid.Additive (plus, zero)
import Data.Monoid.Multiplicative (Multiplicative, one, times, Log(..), Exp(..))
import Data.Monoid.Self (Self(Self,getSelf))

infixl 6 `minus`

-- | Minimal complete definition: 'gnegate' or 'minus'
class Monoid a => Group a where
-- additive inverse
gnegate :: a -> a
minus :: a -> a -> a
gsubtract :: a -> a -> a

gnegate = minus zero
a `minus` b = a `plus` gnegate b
a `gsubtract` b = gnegate a `plus` b

instance Num a => Group (Sum a) where
gnegate = Sum . negate . getSum
Sum a `minus` Sum b = Sum (a - b)

instance Fractional a => Group (Product a) where
gnegate = Product . negate . getProduct
Product a `minus` Product b = Product (a / b)

instance Group a => Group (Dual a) where
gnegate = Dual . gnegate . getDual

instance Group a => Group (Self a) where
gnegate = Self . gnegate . getSelf
Self a `minus` Self b = Self (a `minus` b)

-- | Minimal definition over or grecip
class Multiplicative g => MultiplicativeGroup g where
-- | @x / y@
over :: g -> g -> g
-- | @x \ y@
under :: g -> g -> g
grecip :: g -> g

x `under` y = grecip x `times` y
x `over` y = x `times` grecip y
grecip x = one `over` x

instance MultiplicativeGroup g => Group (Log g) where
Log x `minus` Log y = Log (x `over` y)
Log x `gsubtract` Log y = Log (x `under` y)
gnegate (Log x) = Log (grecip x)

instance Group g => MultiplicativeGroup (Exp g) where
Exp x `over` Exp y = Exp (x `minus` y)
Exp x `under` Exp y = Exp (x `gsubtract` y)
grecip (Exp x) = Exp (gnegate x)

instance MultiplicativeGroup g => MultiplicativeGroup (Self g) where
Self x `over` Self y = Self (x `over` y)
Self x `under` Self y = Self (x `under` y)
grecip (Self x) = Self (grecip x)

instance MultiplicativeGroup a => MultiplicativeGroup (Dual a) where
grecip = Dual . grecip . getDual

```