{-# OPTIONS -fno-implicit-prelude #-} 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 NumericPrelude.List(reduceRepeated) 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 recip a = one / a a / b = a * recip b fromRational' r = fromInteger (numerator r) / fromInteger (denominator r) 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. 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 (/) = (P./) recip = (P.recip) instance C Double where (/) = (P./) recip = (P.recip) instance (PID.C a) => C (Ratio.T a) where -- (/) = 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 (/) = (P./)