{-# LANGUAGE NoImplicitPrelude #-}
module Algebra.Field (
{- * Class -}
C,
(/),
recip,
fromRational',
fromRational,
(^-),
{- * Properties -}
propDivision,
propReciprocal,
) where
import Number.Ratio (T((:%)), Rational, (%), numerator, denominator, )
import qualified Number.Ratio as Ratio
import qualified Data.Ratio as Ratio98
import qualified Algebra.PrincipalIdealDomain as PID
import qualified Algebra.Ring as Ring
import qualified Algebra.Additive as Additive
import qualified Algebra.ZeroTestable as ZeroTestable
import Algebra.Ring ((*), (^), one, fromInteger)
import Algebra.Additive (zero, negate)
import Algebra.ZeroTestable (isZero)
import PreludeBase
import Prelude (Integer, Float, Double)
import qualified Prelude as P
import Test.QuickCheck ((==>), Property)
infixr 8 ^-
infixl 7 /
{- |
Field again corresponds to a commutative ring.
Division is partially defined and satisfies
> not (isZero b) ==> (a * b) / b === a
> not (isZero a) ==> a * recip a === one
when it is defined.
To safely call division,
the program must take type-specific action;
e.g., the following is appropriate in many cases:
> safeRecip :: (Integral a, Eq a, Field.C a) => a -> Maybe a
> safeRecip x =
> let (q,r) = one `divMod` x
> in toMaybe (isZero r) q
Typical examples include rationals, the real numbers,
and rational functions (ratios of polynomial functions).
An instance should be typically declared
only if most elements are invertible.
Actually, we have also used this type class for non-fields
containing lots of units,
e.g. residue classes with respect to non-primes and power series.
So the restriction @not (isZero a)@ must be better @isUnit a@.
Minimal definition: 'recip' or ('/')
-}
class (Ring.C a) => C a where
(/) :: a -> a -> a
recip :: a -> a
fromRational' :: Rational -> a
(^-) :: a -> Integer -> a
{-# INLINE recip #-}
recip a = one / a
{-# INLINE (/) #-}
a / b = a * recip b
{-# INLINE fromRational' #-}
fromRational' r = fromInteger (numerator r) / fromInteger (denominator r)
{-# INLINE (^-) #-}
a ^- n = if n < zero
then recip (a^(-n))
else a^n
-- a ^ n | n < 0 = reduceRepeated (^) one (recip a) (negate (toInteger n))
-- | True = reduceRepeated (^) one a (toInteger n)
-- | Needed to work around shortcomings in GHC.
{-# INLINE fromRational #-}
fromRational :: (C a) => P.Rational -> a
fromRational x = fromRational' (Ratio98.numerator x :% Ratio98.denominator x)
{- * Instances for atomic types -}
{-
ToDo:
fromRational must be implemented explicitly for Float and Double!
It may be that numerator or denominator cannot be represented as Float
due to size constraints, but the fraction can.
-}
instance C Float where
{-# INLINE (/) #-}
{-# INLINE recip #-}
(/) = (P./)
recip = (P.recip)
instance C Double where
{-# INLINE (/) #-}
{-# INLINE recip #-}
(/) = (P./)
recip = (P.recip)
instance (PID.C a) => C (Ratio.T a) where
{-# INLINE (/) #-}
{-# INLINE recip #-}
{-# INLINE fromRational' #-}
-- (/) = Ratio.liftOrd (%)
(x:%y) / (x':%y') = (x*y') % (y*x')
recip (x:%y) = (y:%x)
fromRational' (x:%y) = fromInteger x % fromInteger y
-- | the restriction on the divisor should be @isUnit a@ instead of @not (isZero a)@
propDivision :: (Eq a, ZeroTestable.C a, C a) => a -> a -> Property
propReciprocal :: (Eq a, ZeroTestable.C a, C a) => a -> Property
propDivision a b = not (isZero b) ==> (a * b) / b == a
propReciprocal a = not (isZero a) ==> a * recip a == one
-- legacy
instance (P.Integral a) => C (Ratio98.Ratio a) where
{-# INLINE (/) #-}
{-# INLINE recip #-}
(/) = (P./)
recip = (P.recip)