```{-# 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

```