module Data.Group where
import Data.Monoid
-- |A 'Group' is a 'Monoid' plus a function, 'invert', such that:
--
-- @a <> invert a == mempty@
class Monoid m => Group m where
invert :: m -> m
instance Group () where
invert () = ()
instance Num a => Group (Sum a) where
invert = Sum . negate . getSum
{-# INLINE invert #-}
instance Fractional a => Group (Product a) where
invert = Product . recip . getProduct
{-# INLINE invert #-}
instance Group a => Group (Dual a) where
invert = Dual . invert . getDual
{-# INLINE invert #-}
instance (Group a, Group b) => Group (a, b) where
invert (a, b) = (invert a, invert b)
instance (Group a, Group b, Group c) => Group (a, b, c) where
invert (a, b, c) = (invert a, invert b, invert c)
instance (Group a, Group b, Group c, Group d) => Group (a, b, c, d) where
invert (a, b, c, d) = (invert a, invert b, invert c, invert d)
instance (Group a, Group b, Group c, Group d, Group e) => Group (a, b, c, d, e) where
invert (a, b, c, d, e) = (invert a, invert b, invert c, invert d, invert e)