    {-# LANGUAGE MultiParamTypeClasses, FlexibleInstances #-}
    module Numeric.Algebra.Division
      ( Division(..)
      , DivisionAlgebra(..)
      ) where

    import Prelude hiding ((*), recip, (/),(^))
    import Numeric.Algebra.Class
    import Numeric.Algebra.Unital

    infixr 8 ^
    infixl 7 /, \\

    -- A multiplicative group
    class Unital r => Division r where
      recip  :: r -> r
      (/)    :: r -> r -> r
      (\\)   :: r -> r -> r
      (^)    :: Integral n => r -> n -> r
      recip a = one / a
      a / b = a * recip b
      a \\ b = recip a * b
      x0 ^ y0 = case compare y0 0 of
        LT -> f (recip x0) (negate y0)
        EQ -> one
        GT -> f x0 y0
        where
           f x y
             | even y = f (x * x) (y `quot` 2)
             | y == 1 = x
             | otherwise = g (x * x) ((y - 1) `quot` 2) x
           g x y z
             | even y = g (x * x) (y `quot` 2) z
             | y == 1 = x * z
             | otherwise = g (x * x) ((y - 1) `quot` 2) (x * z)

    instance Division () where
      _ / _   = ()
      recip _ = ()
      _ \\ _  = ()
      _ ^ _   = ()

    instance (Division a, Division b) => Division (a,b) where
      recip (a,b) = (recip a, recip b)
      (a,b) / (i,j) = (a/i,b/j)
      (a,b) \\ (i,j) = (a\\i,b\\j)
      (a,b) ^ n = (a^n,b^n)

    instance (Division a, Division b, Division c) => Division (a,b,c) where
      recip (a,b,c) = (recip a, recip b, recip c)
      (a,b,c) / (i,j,k) = (a/i,b/j,c/k)
      (a,b,c) \\ (i,j,k) = (a\\i,b\\j,c\\k)
      (a,b,c) ^ n = (a^n,b^n,c^n)

    instance (Division a, Division b, Division c, Division d) => Division (a,b,c,d) where
      recip (a,b,c,d) = (recip a, recip b, recip c, recip d)
      (a,b,c,d) / (i,j,k,l) = (a/i,b/j,c/k,d/l)
      (a,b,c,d) \\ (i,j,k,l) = (a\\i,b\\j,c\\k,d\\l)
      (a,b,c,d) ^ n = (a^n,b^n,c^n,d^n)

    instance (Division a, Division b, Division c, Division d, Division e) => Division (a,b,c,d,e) where
      recip (a,b,c,d,e) = (recip a, recip b, recip c, recip d, recip e)
      (a,b,c,d,e) / (i,j,k,l,m) = (a/i,b/j,c/k,d/l,e/m)
      (a,b,c,d,e) \\ (i,j,k,l,m) = (a\\i,b\\j,c\\k,d\\l,e\\m)
      (a,b,c,d,e) ^ n = (a^n,b^n,c^n,d^n,e^n)

    class UnitalAlgebra r a => DivisionAlgebra r a where
      recipriocal :: (a -> r) -> a -> r
      -- recipriocal f = one `over` f

    instance (Unital r, DivisionAlgebra r a) => Division (a -> r) where
      recip = recipriocal