prop_sqr a = sqr a == a*a

prop_sqr a = sqr a == a * a
prop_pow :: NonZero Nimber -> Integer -> Bool
prop_pow NonZero {getNonZero = a} n = pow a n == a ^^ n
prop_pow_zero :: NonNegative Integer -> Bool
prop_pow_zero NonNegative {getNonNegative = n} = 0 `pow` n == if n == 0 then 1 else 0

--   This module implements /finite/ nimbers.  The set of finite nimbers is the quadratic closure of the field with two elements.

--   This module implements /finite/ nimbers, which form the smallest quadratically closed field of characteristic 2.

-- | Raise a @'Nimber'@ to an integral power.  Faster than using '^' or '^^'.
pow :: (Integral a) => Nimber -> a -> Nimber
_ `pow` 0 = 1
a `pow` n
  | n < 0 = recip a `pow` negate n
  | otherwise = if even n then sqr a `pow` quot n 2 else powAcc (sqr a) (quot n 2) a
  where
    powAcc _ 0 x = x
    powAcc b m x = if even m then powAcc (sqr b) (quot m 2) x else powAcc (sqr b) (quot m 2) (b * x)