content: -- Standard functions on rational numbers
  
  module  Ratio (
      Ratio, Rational, (%), numerator, denominator, approxRational ) where
  import Prelude
  
  infixl 7  %
  
  prec = 7 :: Int
  
  data  (Integral a)      => Ratio a = !a :% !a  deriving (Eq)
  --instance Eq a => Eq (Ratio a)
  --type  Rational          =  Ratio Integer
  
  intRat x = x % 1
  
  instance  (Show a{-Integral a-})  => Show (Ratio a)  where
      showsPrec p (x:%y)  =  showParen (p > prec)
                                 (shows x . showString " % " . shows y)
  
  --instance Eq a => Eq (Ratio a) where  n1:%d1 == n2:%d2 = n1==n2 && d1==d2
  
  --(%)                     :: (Integral a) => a -> a -> Ratio a
  numerator, denominator  :: (Integral a) => Ratio a -> a
  --approxRational          :: (RealFrac a) => a -> a -> Rational
  
  
  -- "reduce" is a subsidiary function used only in this module.
  -- It normalises a ratio by dividing both numerator
  -- and denominator by their greatest common divisor.
  --
  -- E.g., 12 `reduce` 8    ==  3 :%   2
  --       12 `reduce` (-8) ==  3 :% (-2)
  
  reduce _ 0              =  error "Ratio.% : zero denominator"
  reduce x y              =  (x `quot` d) :% (y `quot` d)
                             where d = gcd x y
  
  x % y                   =  reduce (x * signum y) (abs y)
  
  numerator (x :% _)      =  x
  
  denominator (_ :% y)    =  y
  
  
  instance  (Ord a,Num a{-Integral a-})  => Ord (Ratio a)  where
      (x:%y) <= (x':%y')  =  x * y' <= x' * y
      (x:%y) <  (x':%y')  =  x * y' <  x' * y
  
  instance  (Num a,Integral a)  => Num (Ratio a)  {-where
      (x:%y) + (x':%y')   =  reduce (x*y' + x'*y) (y*y')
      (x:%y) * (x':%y')   =  reduce (x * x') (y * y')
      negate (x:%y)       =  (-x) :% y
      abs (x:%y)          =  abs x :% y
      signum (x:%y)       =  signum x :% 1
      fromInteger x       =  fromInteger x :% 1-}
  
  instance  (Integral a)  => Real (Ratio a)  where
      toRational (x:%y)   =  toInteger x :% toInteger y
  
  instance  (Ord a,Num a,Integral a)  => Fractional (Ratio a){-  where
      (x:%y) / (x':%y')   =  (x*y') % (y*x')
      recip (x:%y)        =  if x < 0 then (-y) :% (-x) else y :% x
      fromRational (x:%y) =  fromInteger x :% fromInteger y-}
  
  {-
  properFractionRatio (x:%y) = case quotRem x y of
  	                      (q,r) -> (fromIntegral q, r:%y)
  -}
  
  
  instance  (Integral a)  => RealFrac (Ratio a) where
      properFraction (x:%y) = case quotRem x y of
  	                      (q,r) -> (fromIntegral q, r:%y)
  
  instance  (Integral a)  => Enum (Ratio a) {- where
      toEnum           =  fromIntegral
      fromEnum         =  fromInteger . truncate -- May overflow
      enumFrom         =  numericEnumFrom -- These numericEnumXXX functions
      enumFromThen     =  numericEnumFromThen -- are as defined in Prelude.hs
      enumFromTo       =  numericEnumFromTo -- but not exported from it!
      enumFromThenTo   =  numericEnumFromThenTo
  -}
  
  instance  (Read a, Integral a)  => Read (Ratio a) {- where
      readsPrec p  =  readParen (p > prec)
                                (\r -> [(x%y,u) | (x,s)   <- reads r,
                                                  ("%",t) <- lex s,
                                                  (y,u)   <- reads t ])
  -}
  
  approxRational x eps    =  simplest (x-eps) (x+eps)
          where simplest x y | y < x      =  simplest y x
                             | x == y     =  xr
                             | x > 0      =  simplest' n d n' d'
                             | y < 0      =  - simplest' (-n') d' (-n) d
                             | otherwise  =  0 :% 1
                                          where xr@(n:%d) = toRational x
                                                (n':%d')  = toRational y
  
                simplest' n d n' d'       -- assumes 0 < n%d < n'%d'
                          | r == 0     =  q :% 1
                          | q /= q'    =  (q+1) :% 1
                          | otherwise  =  (q*n''+d'') :% n''
                                       where (q,r)      =  quotRem n d
                                             (q',r')    =  quotRem n' d'
                                             (n'':%d'') =  simplest' d' r' d r
  