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-% ------------------------------------------------------------------------------
-% $Id: PrelReal.lhs,v 1.16 2001/09/26 16:27:04 simonpj Exp $
-%
-% (c) The University of Glasgow, 1994-2000
-%
-
-\section[PrelReal]{Module @PrelReal@}
-
-The types
-
-	Ratio, Rational
-
-and the classes
-
-	Real
-	Integral
-	Fractional
-	RealFrac
-
-
-\begin{code}
-{-# OPTIONS -fno-implicit-prelude #-}
-
-module PrelReal where
-
-import {-# SOURCE #-} PrelErr
-import PrelBase
-import PrelNum
-import PrelList
-import PrelEnum
-import PrelShow
-
-infixr 8  ^, ^^
-infixl 7  /, `quot`, `rem`, `div`, `mod`
-
-default ()		-- Double isn't available yet, 
-			-- and we shouldn't be using defaults anyway
-\end{code}
-
-
-%*********************************************************
-%*							*
-\subsection{The @Ratio@ and @Rational@ types}
-%*							*
-%*********************************************************
-
-\begin{code}
-data  (Integral a)	=> Ratio a = !a :% !a  deriving (Eq)
-type  Rational		=  Ratio Integer
-\end{code}
-
-
-\begin{code}
-{-# SPECIALISE (%) :: Integer -> Integer -> Rational #-}
-(%)			:: (Integral a) => a -> a -> Ratio a
-numerator, denominator	:: (Integral a) => Ratio a -> a
-\end{code}
-
-\tr{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.
-
-\begin{code}
-reduce ::  (Integral a) => a -> a -> Ratio a
-{-# SPECIALISE reduce :: Integer -> Integer -> Rational #-}
-reduce _ 0		=  error "Ratio.%: zero denominator"
-reduce x y		=  (x `quot` d) :% (y `quot` d)
-			   where d = gcd x y
-\end{code}
-
-\begin{code}
-x % y			=  reduce (x * signum y) (abs y)
-
-numerator   (x :% _)	=  x
-denominator (_ :% y)	=  y
-\end{code}
-
-
-%*********************************************************
-%*							*
-\subsection{Standard numeric classes}
-%*							*
-%*********************************************************
-
-\begin{code}
-class  (Num a, Ord a) => Real a  where
-    toRational		::  a -> Rational
-
-class  (Real a, Enum a) => Integral a  where
-    quot, rem, div, mod	:: a -> a -> a
-    quotRem, divMod	:: a -> a -> (a,a)
-    toInteger		:: a -> Integer
-
-    n `quot` d		=  q  where (q,_) = quotRem n d
-    n `rem` d		=  r  where (_,r) = quotRem n d
-    n `div` d		=  q  where (q,_) = divMod n d
-    n `mod` d		=  r  where (_,r) = divMod n d
-    divMod n d 		=  if signum r == negate (signum d) then (q-1, r+d) else qr
-			   where qr@(q,r) = quotRem n d
-
-class  (Num a) => Fractional a  where
-    (/)			:: a -> a -> a
-    recip		:: a -> a
-    fromRational	:: Rational -> a
-
-    recip x		=  1 / x
-    x / y		= x * recip y
-
-class  (Real a, Fractional a) => RealFrac a  where
-    properFraction	:: (Integral b) => a -> (b,a)
-    truncate, round	:: (Integral b) => a -> b
-    ceiling, floor	:: (Integral b) => a -> b
-
-    truncate x		=  m  where (m,_) = properFraction x
-    
-    round x		=  let (n,r) = properFraction x
-    			       m     = if r < 0 then n - 1 else n + 1
-    			   in case signum (abs r - 0.5) of
-    				-1 -> n
-    			 	0  -> if even n then n else m
-    				1  -> m
-    
-    ceiling x		=  if r > 0 then n + 1 else n
-    			   where (n,r) = properFraction x
-    
-    floor x		=  if r < 0 then n - 1 else n
-    			   where (n,r) = properFraction x
-\end{code}
-
-
-These 'numeric' enumerations come straight from the Report
-
-\begin{code}
-numericEnumFrom		:: (Fractional a) => a -> [a]
-numericEnumFrom		=  iterate (+1)
-
-numericEnumFromThen	:: (Fractional a) => a -> a -> [a]
-numericEnumFromThen n m	=  iterate (+(m-n)) n
-
-numericEnumFromTo       :: (Ord a, Fractional a) => a -> a -> [a]
-numericEnumFromTo n m   = takeWhile (<= m + 1/2) (numericEnumFrom n)
-
-numericEnumFromThenTo   :: (Ord a, Fractional a) => a -> a -> a -> [a]
-numericEnumFromThenTo e1 e2 e3 = takeWhile pred (numericEnumFromThen e1 e2)
-				where
-				 mid = (e2 - e1) / 2
-				 pred | e2 > e1   = (<= e3 + mid)
-				      | otherwise = (>= e3 + mid)
-\end{code}
-
-
-%*********************************************************
-%*							*
-\subsection{Instances for @Int@}
-%*							*
-%*********************************************************
-
-\begin{code}
-instance  Real Int  where
-    toRational x	=  toInteger x % 1
-
-instance  Integral Int	where
-    toInteger i = int2Integer i  -- give back a full-blown Integer
-
-    -- Following chks for zero divisor are non-standard (WDP)
-    a `quot` b	=  if b /= 0
-		   then a `quotInt` b
-		   else error "Prelude.Integral.quot{Int}: divide by 0"
-    a `rem` b	=  if b /= 0
-		   then a `remInt` b
-		   else error "Prelude.Integral.rem{Int}: divide by 0"
-
-    x `div` y = x `divInt` y
-    x `mod` y = x `modInt` y
-
-    a `quotRem` b = a `quotRemInt` b
-    a `divMod`  b = a `divModInt`  b
-\end{code}
-
-
-%*********************************************************
-%*							*
-\subsection{Instances for @Integer@}
-%*							*
-%*********************************************************
-
-\begin{code}
-instance  Real Integer  where
-    toRational x	=  x % 1
-
-instance  Integral Integer where
-    toInteger n	     = n
-
-    n `quot` d = n `quotInteger` d
-    n `rem`  d = n `remInteger`  d
-
-    n `div` d	=  q  where (q,_) = divMod n d
-    n `mod` d	=  r  where (_,r) = divMod n d
-
-    a `divMod` b = a `divModInteger` b
-    a `quotRem` b = a `quotRemInteger` b
-\end{code}
-
-
-%*********************************************************
-%*							*
-\subsection{Instances for @Ratio@}
-%*							*
-%*********************************************************
-
-\begin{code}
-instance  (Integral a)	=> Ord (Ratio a)  where
-    {-# SPECIALIZE instance Ord Rational #-}
-    (x:%y) <= (x':%y')	=  x * y' <= x' * y
-    (x:%y) <  (x':%y')	=  x * y' <  x' * y
-
-instance  (Integral a)	=> Num (Ratio a)  where
-    {-# SPECIALIZE instance Num Rational #-}
-    (x:%y) + (x':%y')	=  reduce (x*y' + x'*y) (y*y')
-    (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:%_)	=  signum x :% 1
-    fromInteger x	=  fromInteger x :% 1
-
-instance  (Integral a)	=> Fractional (Ratio a)  where
-    {-# SPECIALIZE instance Fractional Rational #-}
-    (x:%y) / (x':%y')	=  (x*y') % (y*x')
-    recip (x:%y)	=  y % x
-    fromRational (x:%y) =  fromInteger x :% fromInteger y
-
-instance  (Integral a)	=> Real (Ratio a)  where
-    {-# SPECIALIZE instance Real Rational #-}
-    toRational (x:%y)	=  toInteger x :% toInteger y
-
-instance  (Integral a)	=> RealFrac (Ratio a)  where
-    {-# SPECIALIZE instance RealFrac Rational #-}
-    properFraction (x:%y) = (fromInteger (toInteger q), r:%y)
-			  where (q,r) = quotRem x y
-
-instance  (Integral a)  => Show (Ratio a)  where
-    {-# SPECIALIZE instance Show Rational #-}
-    showsPrec p (x:%y)	=  showParen (p > ratio_prec)
-    	    	    	       (shows x . showString " % " . shows y)
-
-ratio_prec :: Int
-ratio_prec = 7
-
-instance  (Integral a)	=> Enum (Ratio a)  where
-    {-# SPECIALIZE instance Enum Rational #-}
-    succ x	        =  x + 1
-    pred x	        =  x - 1
-
-    toEnum n            =  fromInteger (int2Integer n) :% 1
-    fromEnum            =  fromInteger . truncate
-
-    enumFrom		=  numericEnumFrom
-    enumFromThen 	=  numericEnumFromThen
-    enumFromTo		=  numericEnumFromTo
-    enumFromThenTo	=  numericEnumFromThenTo
-\end{code}
-
-
-%*********************************************************
-%*							*
-\subsection{Coercions}
-%*							*
-%*********************************************************
-
-\begin{code}
-fromIntegral :: (Integral a, Num b) => a -> b
-fromIntegral = fromInteger . toInteger
-
-{-# RULES
-"fromIntegral/Int->Int" fromIntegral = id :: Int -> Int
-    #-}
-
-realToFrac :: (Real a, Fractional b) => a -> b
-realToFrac = fromRational . toRational
-
-{-# RULES
-"realToFrac/Int->Int" realToFrac = id :: Int -> Int
-    #-}
-
--- For backward compatibility
-{-# DEPRECATED fromInt "use fromIntegral instead" #-}
-fromInt :: Num a => Int -> a
-fromInt = fromIntegral
-
--- For backward compatibility
-{-# DEPRECATED toInt "use fromIntegral instead" #-}
-toInt :: Integral a => a -> Int
-toInt = fromIntegral
-\end{code}
-
-%*********************************************************
-%*							*
-\subsection{Overloaded numeric functions}
-%*							*
-%*********************************************************
-
-\begin{code}
-showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS
-showSigned showPos p x 
-   | x < 0     = showParen (p > 6) (showChar '-' . showPos (-x))
-   | otherwise = showPos x
-
-even, odd	:: (Integral a) => a -> Bool
-even n		=  n `rem` 2 == 0
-odd		=  not . even
-
--------------------------------------------------------
-{-# SPECIALISE (^) ::
-	Integer -> Integer -> Integer,
-	Integer -> Int -> Integer,
-	Int -> Int -> Int #-}
-(^)		:: (Num a, Integral b) => a -> b -> a
-_ ^ 0		=  1
-x ^ n | n > 0	=  f x (n-1) x
-		   where f _ 0 y = y
-		         f a d y = g a d  where
-			           g b i | even i  = g (b*b) (i `quot` 2)
-				         | otherwise = f b (i-1) (b*y)
-_ ^ _		= error "Prelude.^: negative exponent"
-
-{-# SPECIALISE (^^) ::
-	Rational -> Int -> Rational #-}
-(^^)		:: (Fractional a, Integral b) => a -> b -> a
-x ^^ n		=  if n >= 0 then x^n else recip (x^(negate n))
-
-
--------------------------------------------------------
-gcd		:: (Integral a) => a -> a -> a
-gcd 0 0		=  error "Prelude.gcd: gcd 0 0 is undefined"
-gcd x y		=  gcd' (abs x) (abs y)
-		   where gcd' a 0  =  a
-			 gcd' a b  =  gcd' b (a `rem` b)
-
-lcm		:: (Integral a) => a -> a -> a
-{-# SPECIALISE lcm :: Int -> Int -> Int #-}
-lcm _ 0		=  0
-lcm 0 _		=  0
-lcm x y		=  abs ((x `quot` (gcd x y)) * y)
-
-
-{-# RULES
-"gcd/Int->Int->Int"             gcd = gcdInt
-"gcd/Integer->Integer->Integer" gcd = gcdInteger
-"lcm/Integer->Integer->Integer" lcm = lcmInteger
- #-}
-
-integralEnumFrom :: (Integral a, Bounded a) => a -> [a]
-integralEnumFrom n = map fromInteger [toInteger n .. toInteger (maxBound `asTypeOf` n)]
-
-integralEnumFromThen :: (Integral a, Bounded a) => a -> a -> [a]
-integralEnumFromThen n1 n2
-  | i_n2 >= i_n1  = map fromInteger [i_n1, i_n2 .. toInteger (maxBound `asTypeOf` n1)]
-  | otherwise     = map fromInteger [i_n1, i_n2 .. toInteger (minBound `asTypeOf` n1)]
-  where
-    i_n1 = toInteger n1
-    i_n2 = toInteger n2
-
-integralEnumFromTo :: Integral a => a -> a -> [a]
-integralEnumFromTo n m = map fromInteger [toInteger n .. toInteger m]
-
-integralEnumFromThenTo :: Integral a => a -> a -> a -> [a]
-integralEnumFromThenTo n1 n2 m
-  = map fromInteger [toInteger n1, toInteger n2 .. toInteger m]
-\end{code}