content: \section[GHC.Base]{Module @GHC.Base@}
  
  The overall structure of the GHC Prelude is a bit tricky.
  
    a) We want to avoid "orphan modules", i.e. ones with instance
          decls that don't belong either to a tycon or a class
          defined in the same module
  
    b) We want to avoid giant modules
  
  So the rough structure is as follows, in (linearised) dependency order
  
  
  GHC.Prim                Has no implementation.  It defines built-in things, and
                  by importing it you bring them into scope.
                  The source file is GHC.Prim.hi-boot, which is just
                  copied to make GHC.Prim.hi
  
  GHC.Base        Classes: Eq, Ord, Functor, Monad
                  Types:   list, (), Int, Bool, Ordering, Char, String
  
  Data.Tuple      Types: tuples, plus instances for GHC.Base classes
  
  GHC.Show        Class: Show, plus instances for GHC.Base/GHC.Tup types
  
  GHC.Enum        Class: Enum,  plus instances for GHC.Base/GHC.Tup types
  
  Data.Maybe      Type: Maybe, plus instances for GHC.Base classes
  
  GHC.List        List functions
  
  GHC.Num         Class: Num, plus instances for Int
                  Type:  Integer, plus instances for all classes so far (Eq, Ord, Num, Show)
  
                  Integer is needed here because it is mentioned in the signature
                  of 'fromInteger' in class Num
  
  GHC.Real        Classes: Real, Integral, Fractional, RealFrac
                           plus instances for Int, Integer
                  Types:  Ratio, Rational
                          plus intances for classes so far
  
                  Rational is needed here because it is mentioned in the signature
                  of 'toRational' in class Real
  
  GHC.ST  The ST monad, instances and a few helper functions
  
  Ix              Classes: Ix, plus instances for Int, Bool, Char, Integer, Ordering, tuples
  
  GHC.Arr         Types: Array, MutableArray, MutableVar
  
                  Arrays are used by a function in GHC.Float
  
  GHC.Float       Classes: Floating, RealFloat
                  Types:   Float, Double, plus instances of all classes so far
  
                  This module contains everything to do with floating point.
                  It is a big module (900 lines)
                  With a bit of luck, many modules can be compiled without ever reading GHC.Float.hi
  
  
  Other Prelude modules are much easier with fewer complex dependencies.
  
  \begin{code}
  {-# LANGUAGE Unsafe #-}
  {-# LANGUAGE CPP
             , NoImplicitPrelude
             , BangPatterns
             , ExplicitForAll
             , MagicHash
             , UnboxedTuples
             , ExistentialQuantification
             , Rank2Types
    #-}
  -- -fno-warn-orphans is needed for things like:
  -- Orphan rule: "x# -# x#" ALWAYS forall x# :: Int# -# x# x# = 0
  {-# OPTIONS_GHC -fno-warn-orphans #-}
  {-# OPTIONS_HADDOCK hide #-}
  
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  GHC.Base
  -- Copyright   :  (c) The University of Glasgow, 1992-2002
  -- License     :  see libraries/base/LICENSE
  -- 
  -- Maintainer  :  cvs-ghc@haskell.org
  -- Stability   :  internal
  -- Portability :  non-portable (GHC extensions)
  --
  -- Basic data types and classes.
  -- 
  -----------------------------------------------------------------------------
  
  #include "MachDeps.h"
  
  -- #hide
  module GHC.Base
          (
          module GHC.Base,
          module GHC.Classes,
          module GHC.CString,
          module GHC.Types,
          module GHC.Prim,        -- Re-export GHC.Prim and GHC.Err, to avoid lots
          module GHC.Err          -- of people having to import it explicitly
    ) 
          where
  
  import GHC.Types
  import GHC.Classes
  import GHC.CString
  import GHC.Prim
  import {-# SOURCE #-} GHC.Show
  import {-# SOURCE #-} GHC.Err
  
  -- This is not strictly speaking required by this module, but is an
  -- implicit dependency whenever () or tuples are mentioned, so adding it
  -- as an import here helps to get the dependencies right in the new
  -- build system.
  import GHC.Tuple ()
  -- Likewise we need Integer when deriving things like Eq instances, and
  -- this is a convenient place to force it to be built
  import GHC.Integer ()
  
  infixr 9  .
  infixr 5  ++
  infixl 4  <$
  infixl 1  >>, >>=
  infixr 0  $
  
  default ()              -- Double isn't available yet
  \end{code}
  
  
  %*********************************************************
  %*                                                      *
  \subsection{DEBUGGING STUFF}
  %*  (for use when compiling GHC.Base itself doesn't work)
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  {-
  data  Bool  =  False | True
  data Ordering = LT | EQ | GT 
  data Char = C# Char#
  type  String = [Char]
  data Int = I# Int#
  data  ()  =  ()
  data [] a = MkNil
  
  not True = False
  (&&) True True = True
  otherwise = True
  
  build = error "urk"
  foldr = error "urk"
  -}
  \end{code}
  
  
  %*********************************************************
  %*                                                      *
  \subsection{Monadic classes @Functor@, @Monad@ }
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  {- | The 'Functor' class is used for types that can be mapped over.
  Instances of 'Functor' should satisfy the following laws:
  
  > fmap id  ==  id
  > fmap (f . g)  ==  fmap f . fmap g
  
  The instances of 'Functor' for lists, 'Data.Maybe.Maybe' and 'System.IO.IO'
  satisfy these laws.
  -}
  
  class  Functor f  where
      fmap        :: (a -> b) -> f a -> f b
  
      -- | Replace all locations in the input with the same value.
      -- The default definition is @'fmap' . 'const'@, but this may be
      -- overridden with a more efficient version.
      (<$)        :: a -> f b -> f a
      (<$)        =  fmap . const
  
  {- | The 'Monad' class defines the basic operations over a /monad/,
  a concept from a branch of mathematics known as /category theory/.
  From the perspective of a Haskell programmer, however, it is best to
  think of a monad as an /abstract datatype/ of actions.
  Haskell's @do@ expressions provide a convenient syntax for writing
  monadic expressions.
  
  Minimal complete definition: '>>=' and 'return'.
  
  Instances of 'Monad' should satisfy the following laws:
  
  > return a >>= k  ==  k a
  > m >>= return  ==  m
  > m >>= (\x -> k x >>= h)  ==  (m >>= k) >>= h
  
  Instances of both 'Monad' and 'Functor' should additionally satisfy the law:
  
  > fmap f xs  ==  xs >>= return . f
  
  The instances of 'Monad' for lists, 'Data.Maybe.Maybe' and 'System.IO.IO'
  defined in the "Prelude" satisfy these laws.
  -}
  
  class  Monad m  where
      -- | Sequentially compose two actions, passing any value produced
      -- by the first as an argument to the second.
      (>>=)       :: forall a b. m a -> (a -> m b) -> m b
      -- | Sequentially compose two actions, discarding any value produced
      -- by the first, like sequencing operators (such as the semicolon)
      -- in imperative languages.
      (>>)        :: forall a b. m a -> m b -> m b
          -- Explicit for-alls so that we know what order to
          -- give type arguments when desugaring
  
      -- | Inject a value into the monadic type.
      return      :: a -> m a
      -- | Fail with a message.  This operation is not part of the
      -- mathematical definition of a monad, but is invoked on pattern-match
      -- failure in a @do@ expression.
      fail        :: String -> m a
  
      {-# INLINE (>>) #-}
      m >> k      = m >>= \_ -> k
      fail s      = error s
  \end{code}
  
  
  %*********************************************************
  %*                                                      *
  \subsection{The list type}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  instance Functor [] where
      fmap = map
  
  instance  Monad []  where
      m >>= k             = foldr ((++) . k) [] m
      m >> k              = foldr ((++) . (\ _ -> k)) [] m
      return x            = [x]
  \end{code}
  
  A few list functions that appear here because they are used here.
  The rest of the prelude list functions are in GHC.List.
  
  ----------------------------------------------
  --      foldr/build/augment
  ----------------------------------------------
    
  \begin{code}
  -- | 'foldr', applied to a binary operator, a starting value (typically
  -- the right-identity of the operator), and a list, reduces the list
  -- using the binary operator, from right to left:
  --
  -- > foldr f z [x1, x2, ..., xn] == x1 `f` (x2 `f` ... (xn `f` z)...)
  
  foldr            :: (a -> b -> b) -> b -> [a] -> b
  -- foldr _ z []     =  z
  -- foldr f z (x:xs) =  f x (foldr f z xs)
  {-# INLINE [0] foldr #-}
  -- Inline only in the final stage, after the foldr/cons rule has had a chance
  -- Also note that we inline it when it has *two* parameters, which are the 
  -- ones we are keen about specialising!
  foldr k z = go
            where
              go []     = z
              go (y:ys) = y `k` go ys
  
  -- | A list producer that can be fused with 'foldr'.
  -- This function is merely
  --
  -- >    build g = g (:) []
  --
  -- but GHC's simplifier will transform an expression of the form
  -- @'foldr' k z ('build' g)@, which may arise after inlining, to @g k z@,
  -- which avoids producing an intermediate list.
  
  build   :: forall a. (forall b. (a -> b -> b) -> b -> b) -> [a]
  {-# INLINE [1] build #-}
          -- The INLINE is important, even though build is tiny,
          -- because it prevents [] getting inlined in the version that
          -- appears in the interface file.  If [] *is* inlined, it
          -- won't match with [] appearing in rules in an importing module.
          --
          -- The "1" says to inline in phase 1
  
  build g = g (:) []
  
  -- | A list producer that can be fused with 'foldr'.
  -- This function is merely
  --
  -- >    augment g xs = g (:) xs
  --
  -- but GHC's simplifier will transform an expression of the form
  -- @'foldr' k z ('augment' g xs)@, which may arise after inlining, to
  -- @g k ('foldr' k z xs)@, which avoids producing an intermediate list.
  
  augment :: forall a. (forall b. (a->b->b) -> b -> b) -> [a] -> [a]
  {-# INLINE [1] augment #-}
  augment g xs = g (:) xs
  
  {-# RULES
  "fold/build"    forall k z (g::forall b. (a->b->b) -> b -> b) . 
                  foldr k z (build g) = g k z
  
  "foldr/augment" forall k z xs (g::forall b. (a->b->b) -> b -> b) . 
                  foldr k z (augment g xs) = g k (foldr k z xs)
  
  "foldr/id"                        foldr (:) [] = \x  -> x
  "foldr/app"     [1] forall ys. foldr (:) ys = \xs -> xs ++ ys
          -- Only activate this from phase 1, because that's
          -- when we disable the rule that expands (++) into foldr
  
  -- The foldr/cons rule looks nice, but it can give disastrously
  -- bloated code when commpiling
  --      array (a,b) [(1,2), (2,2), (3,2), ...very long list... ]
  -- i.e. when there are very very long literal lists
  -- So I've disabled it for now. We could have special cases
  -- for short lists, I suppose.
  -- "foldr/cons" forall k z x xs. foldr k z (x:xs) = k x (foldr k z xs)
  
  "foldr/single"  forall k z x. foldr k z [x] = k x z
  "foldr/nil"     forall k z.   foldr k z []  = z 
  
  "augment/build" forall (g::forall b. (a->b->b) -> b -> b)
                         (h::forall b. (a->b->b) -> b -> b) .
                         augment g (build h) = build (\c n -> g c (h c n))
  "augment/nil"   forall (g::forall b. (a->b->b) -> b -> b) .
                          augment g [] = build g
   #-}
  
  -- This rule is true, but not (I think) useful:
  --      augment g (augment h t) = augment (\cn -> g c (h c n)) t
  \end{code}
  
  
  ----------------------------------------------
  --              map     
  ----------------------------------------------
  
  \begin{code}
  -- | 'map' @f xs@ is the list obtained by applying @f@ to each element
  -- of @xs@, i.e.,
  --
  -- > map f [x1, x2, ..., xn] == [f x1, f x2, ..., f xn]
  -- > map f [x1, x2, ...] == [f x1, f x2, ...]
  
  map :: (a -> b) -> [a] -> [b]
  map _ []     = []
  map f (x:xs) = f x : map f xs
  
  -- Note eta expanded
  mapFB ::  (elt -> lst -> lst) -> (a -> elt) -> a -> lst -> lst
  {-# INLINE [0] mapFB #-}
  mapFB c f = \x ys -> c (f x) ys
  
  -- The rules for map work like this.
  -- 
  -- Up to (but not including) phase 1, we use the "map" rule to
  -- rewrite all saturated applications of map with its build/fold 
  -- form, hoping for fusion to happen.
  -- In phase 1 and 0, we switch off that rule, inline build, and
  -- switch on the "mapList" rule, which rewrites the foldr/mapFB
  -- thing back into plain map.  
  --
  -- It's important that these two rules aren't both active at once 
  -- (along with build's unfolding) else we'd get an infinite loop 
  -- in the rules.  Hence the activation control below.
  --
  -- The "mapFB" rule optimises compositions of map.
  --
  -- This same pattern is followed by many other functions: 
  -- e.g. append, filter, iterate, repeat, etc.
  
  {-# RULES
  "map"       [~1] forall f xs.   map f xs                = build (\c n -> foldr (mapFB c f) n xs)
  "mapList"   [1]  forall f.      foldr (mapFB (:) f) []  = map f
  "mapFB"     forall c f g.       mapFB (mapFB c f) g     = mapFB c (f.g) 
    #-}
  \end{code}
  
  
  ----------------------------------------------
  --              append  
  ----------------------------------------------
  \begin{code}
  -- | Append two lists, i.e.,
  --
  -- > [x1, ..., xm] ++ [y1, ..., yn] == [x1, ..., xm, y1, ..., yn]
  -- > [x1, ..., xm] ++ [y1, ...] == [x1, ..., xm, y1, ...]
  --
  -- If the first list is not finite, the result is the first list.
  
  (++) :: [a] -> [a] -> [a]
  (++) []     ys = ys
  (++) (x:xs) ys = x : xs ++ ys
  
  {-# RULES
  "++"    [~1] forall xs ys. xs ++ ys = augment (\c n -> foldr c n xs) ys
    #-}
  
  \end{code}
  
  
  %*********************************************************
  %*                                                      *
  \subsection{Type @Bool@}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  -- |'otherwise' is defined as the value 'True'.  It helps to make
  -- guards more readable.  eg.
  --
  -- >  f x | x < 0     = ...
  -- >      | otherwise = ...
  otherwise               :: Bool
  otherwise               =  True
  \end{code}
  
  %*********************************************************
  %*                                                      *
  \subsection{Type @Char@ and @String@}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  -- | A 'String' is a list of characters.  String constants in Haskell are values
  -- of type 'String'.
  --
  type String = [Char]
  
  {-# RULES
  "x# `eqChar#` x#" forall x#. x# `eqChar#` x# = True
  "x# `neChar#` x#" forall x#. x# `neChar#` x# = False
  "x# `gtChar#` x#" forall x#. x# `gtChar#` x# = False
  "x# `geChar#` x#" forall x#. x# `geChar#` x# = True
  "x# `leChar#` x#" forall x#. x# `leChar#` x# = True
  "x# `ltChar#` x#" forall x#. x# `ltChar#` x# = False
    #-}
  
  -- | The 'Prelude.toEnum' method restricted to the type 'Data.Char.Char'.
  chr :: Int -> Char
  chr i@(I# i#)
   | int2Word# i# `leWord#` int2Word# 0x10FFFF# = C# (chr# i#)
   | otherwise
      = error ("Prelude.chr: bad argument: " ++ showSignedInt (I# 9#) i "")
  
  unsafeChr :: Int -> Char
  unsafeChr (I# i#) = C# (chr# i#)
  
  -- | The 'Prelude.fromEnum' method restricted to the type 'Data.Char.Char'.
  ord :: Char -> Int
  ord (C# c#) = I# (ord# c#)
  \end{code}
  
  String equality is used when desugaring pattern-matches against strings.
  
  \begin{code}
  eqString :: String -> String -> Bool
  eqString []       []       = True
  eqString (c1:cs1) (c2:cs2) = c1 == c2 && cs1 `eqString` cs2
  eqString _        _        = False
  
  {-# RULES "eqString" (==) = eqString #-}
  -- eqString also has a BuiltInRule in PrelRules.lhs:
  --      eqString (unpackCString# (Lit s1)) (unpackCString# (Lit s2) = s1==s2
  \end{code}
  
  
  %*********************************************************
  %*                                                      *
  \subsection{Type @Int@}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  zeroInt, oneInt, twoInt, maxInt, minInt :: Int
  zeroInt = I# 0#
  oneInt  = I# 1#
  twoInt  = I# 2#
  
  {- Seems clumsy. Should perhaps put minInt and MaxInt directly into MachDeps.h -}
  #if WORD_SIZE_IN_BITS == 31
  minInt  = I# (-0x40000000#)
  maxInt  = I# 0x3FFFFFFF#
  #elif WORD_SIZE_IN_BITS == 32
  minInt  = I# (-0x80000000#)
  maxInt  = I# 0x7FFFFFFF#
  #else 
  minInt  = I# (-0x8000000000000000#)
  maxInt  = I# 0x7FFFFFFFFFFFFFFF#
  #endif
  \end{code}
  
  
  %*********************************************************
  %*                                                      *
  \subsection{The function type}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  -- | Identity function.
  id                      :: a -> a
  id x                    =  x
  
  -- | The call '(lazy e)' means the same as 'e', but 'lazy' has a 
  -- magical strictness property: it is lazy in its first argument, 
  -- even though its semantics is strict.
  lazy :: a -> a
  lazy x = x
  -- Implementation note: its strictness and unfolding are over-ridden
  -- by the definition in MkId.lhs; in both cases to nothing at all.
  -- That way, 'lazy' does not get inlined, and the strictness analyser
  -- sees it as lazy.  Then the worker/wrapper phase inlines it.
  -- Result: happiness
  
  -- Assertion function.  This simply ignores its boolean argument.
  -- The compiler may rewrite it to @('assertError' line)@.
  
  -- | If the first argument evaluates to 'True', then the result is the
  -- second argument.  Otherwise an 'AssertionFailed' exception is raised,
  -- containing a 'String' with the source file and line number of the
  -- call to 'assert'.
  --
  -- Assertions can normally be turned on or off with a compiler flag
  -- (for GHC, assertions are normally on unless optimisation is turned on 
  -- with @-O@ or the @-fignore-asserts@
  -- option is given).  When assertions are turned off, the first
  -- argument to 'assert' is ignored, and the second argument is
  -- returned as the result.
  
  --      SLPJ: in 5.04 etc 'assert' is in GHC.Prim,
  --      but from Template Haskell onwards it's simply
  --      defined here in Base.lhs
  assert :: Bool -> a -> a
  assert _pred r = r
  
  breakpoint :: a -> a
  breakpoint r = r
  
  breakpointCond :: Bool -> a -> a
  breakpointCond _ r = r
  
  data Opaque = forall a. O a
  
  -- | Constant function.
  const                   :: a -> b -> a
  const x _               =  x
  
  -- | Function composition.
  {-# INLINE (.) #-}
  -- Make sure it has TWO args only on the left, so that it inlines
  -- when applied to two functions, even if there is no final argument
  (.)    :: (b -> c) -> (a -> b) -> a -> c
  (.) f g = \x -> f (g x)
  
  -- | @'flip' f@ takes its (first) two arguments in the reverse order of @f@.
  flip                    :: (a -> b -> c) -> b -> a -> c
  flip f x y              =  f y x
  
  -- | Application operator.  This operator is redundant, since ordinary
  -- application @(f x)@ means the same as @(f '$' x)@. However, '$' has
  -- low, right-associative binding precedence, so it sometimes allows
  -- parentheses to be omitted; for example:
  --
  -- >     f $ g $ h x  =  f (g (h x))
  --
  -- It is also useful in higher-order situations, such as @'map' ('$' 0) xs@,
  -- or @'Data.List.zipWith' ('$') fs xs@.
  {-# INLINE ($) #-}
  ($)                     :: (a -> b) -> a -> b
  f $ x                   =  f x
  
  -- | @'until' p f@ yields the result of applying @f@ until @p@ holds.
  until                   :: (a -> Bool) -> (a -> a) -> a -> a
  until p f x | p x       =  x
              | otherwise =  until p f (f x)
  
  -- | 'asTypeOf' is a type-restricted version of 'const'.  It is usually
  -- used as an infix operator, and its typing forces its first argument
  -- (which is usually overloaded) to have the same type as the second.
  asTypeOf                :: a -> a -> a
  asTypeOf                =  const
  \end{code}
  
  %*********************************************************
  %*                                                      *
  \subsection{@Functor@ and @Monad@ instances for @IO@}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  instance  Functor IO where
     fmap f x = x >>= (return . f)
  
  instance  Monad IO  where
      {-# INLINE return #-}
      {-# INLINE (>>)   #-}
      {-# INLINE (>>=)  #-}
      m >> k    = m >>= \ _ -> k
      return    = returnIO
      (>>=)     = bindIO
  
  returnIO :: a -> IO a
  returnIO x = IO $ \ s -> (# s, x #)
  
  bindIO :: IO a -> (a -> IO b) -> IO b
  bindIO (IO m) k = IO $ \ s -> case m s of (# new_s, a #) -> unIO (k a) new_s
  
  thenIO :: IO a -> IO b -> IO b
  thenIO (IO m) k = IO $ \ s -> case m s of (# new_s, _ #) -> unIO k new_s
  
  unIO :: IO a -> (State# RealWorld -> (# State# RealWorld, a #))
  unIO (IO a) = a
  \end{code}
  
  %*********************************************************
  %*                                                      *
  \subsection{@getTag@}
  %*                                                      *
  %*********************************************************
  
  Returns the 'tag' of a constructor application; this function is used
  by the deriving code for Eq, Ord and Enum.
  
  The primitive dataToTag# requires an evaluated constructor application
  as its argument, so we provide getTag as a wrapper that performs the
  evaluation before calling dataToTag#.  We could have dataToTag#
  evaluate its argument, but we prefer to do it this way because (a)
  dataToTag# can be an inline primop if it doesn't need to do any
  evaluation, and (b) we want to expose the evaluation to the
  simplifier, because it might be possible to eliminate the evaluation
  in the case when the argument is already known to be evaluated.
  
  \begin{code}
  {-# INLINE getTag #-}
  getTag :: a -> Int#
  getTag x = x `seq` dataToTag# x
  \end{code}
  
  %*********************************************************
  %*                                                      *
  \subsection{Numeric primops}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  divInt# :: Int# -> Int# -> Int#
  x# `divInt#` y#
          -- Be careful NOT to overflow if we do any additional arithmetic
          -- on the arguments...  the following  previous version of this
          -- code has problems with overflow:
  --    | (x# ># 0#) && (y# <# 0#) = ((x# -# y#) -# 1#) `quotInt#` y#
  --    | (x# <# 0#) && (y# ># 0#) = ((x# -# y#) +# 1#) `quotInt#` y#
      | (x# ># 0#) && (y# <# 0#) = ((x# -# 1#) `quotInt#` y#) -# 1#
      | (x# <# 0#) && (y# ># 0#) = ((x# +# 1#) `quotInt#` y#) -# 1#
      | otherwise                = x# `quotInt#` y#
  
  modInt# :: Int# -> Int# -> Int#
  x# `modInt#` y#
      | (x# ># 0#) && (y# <# 0#) ||
        (x# <# 0#) && (y# ># 0#)    = if r# /=# 0# then r# +# y# else 0#
      | otherwise                   = r#
      where
      !r# = x# `remInt#` y#
  \end{code}
  
  Definitions of the boxed PrimOps; these will be
  used in the case of partial applications, etc.
  
  \begin{code}
  {-# INLINE plusInt #-}
  {-# INLINE minusInt #-}
  {-# INLINE timesInt #-}
  {-# INLINE quotInt #-}
  {-# INLINE remInt #-}
  {-# INLINE negateInt #-}
  
  plusInt, minusInt, timesInt, quotInt, remInt, divInt, modInt :: Int -> Int -> Int
  (I# x) `plusInt`  (I# y) = I# (x +# y)
  (I# x) `minusInt` (I# y) = I# (x -# y)
  (I# x) `timesInt` (I# y) = I# (x *# y)
  (I# x) `quotInt`  (I# y) = I# (x `quotInt#` y)
  (I# x) `remInt`   (I# y) = I# (x `remInt#`  y)
  (I# x) `divInt`   (I# y) = I# (x `divInt#`  y)
  (I# x) `modInt`   (I# y) = I# (x `modInt#`  y)
  
  {-# RULES
  "x# +# 0#" forall x#. x# +# 0# = x#
  "0# +# x#" forall x#. 0# +# x# = x#
  "x# -# 0#" forall x#. x# -# 0# = x#
  "x# -# x#" forall x#. x# -# x# = 0#
  "x# *# 0#" forall x#. x# *# 0# = 0#
  "0# *# x#" forall x#. 0# *# x# = 0#
  "x# *# 1#" forall x#. x# *# 1# = x#
  "1# *# x#" forall x#. 1# *# x# = x#
    #-}
  
  negateInt :: Int -> Int
  negateInt (I# x) = I# (negateInt# x)
  
  {-# RULES
  "x# ># x#"  forall x#. x# >#  x# = False
  "x# >=# x#" forall x#. x# >=# x# = True
  "x# ==# x#" forall x#. x# ==# x# = True
  "x# /=# x#" forall x#. x# /=# x# = False
  "x# <# x#"  forall x#. x# <#  x# = False
  "x# <=# x#" forall x#. x# <=# x# = True
    #-}
  
  {-# RULES
  "plusFloat x 0.0"   forall x#. plusFloat#  x#   0.0# = x#
  "plusFloat 0.0 x"   forall x#. plusFloat#  0.0# x#   = x#
  "minusFloat x 0.0"  forall x#. minusFloat# x#   0.0# = x#
  "timesFloat x 1.0"  forall x#. timesFloat# x#   1.0# = x#
  "timesFloat 1.0 x"  forall x#. timesFloat# 1.0# x#   = x#
  "divideFloat x 1.0" forall x#. divideFloat# x#  1.0# = x#
    #-}
  
  {-# RULES
  "plusDouble x 0.0"   forall x#. (+##) x#    0.0## = x#
  "plusDouble 0.0 x"   forall x#. (+##) 0.0## x#    = x#
  "minusDouble x 0.0"  forall x#. (-##) x#    0.0## = x#
  "timesDouble x 1.0"  forall x#. (*##) x#    1.0## = x#
  "timesDouble 1.0 x"  forall x#. (*##) 1.0## x#    = x#
  "divideDouble x 1.0" forall x#. (/##) x#    1.0## = x#
    #-}
  
  {-
  We'd like to have more rules, but for example:
  
  This gives wrong answer (0) for NaN - NaN (should be NaN):
      "minusDouble x x"    forall x#. (-##) x#    x#    = 0.0##
  
  This gives wrong answer (0) for 0 * NaN (should be NaN):
      "timesDouble 0.0 x"  forall x#. (*##) 0.0## x#    = 0.0##
  
  This gives wrong answer (0) for NaN * 0 (should be NaN):
      "timesDouble x 0.0"  forall x#. (*##) x#    0.0## = 0.0##
  
  These are tested by num014.
  
  Similarly for Float (#5178):
  
  "minusFloat x x"    forall x#. minusFloat# x#   x#   = 0.0#
  "timesFloat0.0 x"   forall x#. timesFloat# 0.0# x#   = 0.0#
  "timesFloat x 0.0"  forall x#. timesFloat# x#   0.0# = 0.0#
  -}
  
  -- Wrappers for the shift operations.  The uncheckedShift# family are
  -- undefined when the amount being shifted by is greater than the size
  -- in bits of Int#, so these wrappers perform a check and return
  -- either zero or -1 appropriately.
  --
  -- Note that these wrappers still produce undefined results when the
  -- second argument (the shift amount) is negative.
  
  -- | Shift the argument left by the specified number of bits
  -- (which must be non-negative).
  shiftL# :: Word# -> Int# -> Word#
  a `shiftL#` b   | b >=# WORD_SIZE_IN_BITS# = int2Word# 0#
                  | otherwise                = a `uncheckedShiftL#` b
  
  -- | Shift the argument right by the specified number of bits
  -- (which must be non-negative).
  shiftRL# :: Word# -> Int# -> Word#
  a `shiftRL#` b  | b >=# WORD_SIZE_IN_BITS# = int2Word# 0#
                  | otherwise                = a `uncheckedShiftRL#` b
  
  -- | Shift the argument left by the specified number of bits
  -- (which must be non-negative).
  iShiftL# :: Int# -> Int# -> Int#
  a `iShiftL#` b  | b >=# WORD_SIZE_IN_BITS# = 0#
                  | otherwise                = a `uncheckedIShiftL#` b
  
  -- | Shift the argument right (signed) by the specified number of bits
  -- (which must be non-negative).
  iShiftRA# :: Int# -> Int# -> Int#
  a `iShiftRA#` b | b >=# WORD_SIZE_IN_BITS# = if a <# 0# then (-1#) else 0#
                  | otherwise                = a `uncheckedIShiftRA#` b
  
  -- | Shift the argument right (unsigned) by the specified number of bits
  -- (which must be non-negative).
  iShiftRL# :: Int# -> Int# -> Int#
  a `iShiftRL#` b | b >=# WORD_SIZE_IN_BITS# = 0#
                  | otherwise                = a `uncheckedIShiftRL#` b
  
  #if WORD_SIZE_IN_BITS == 32
  {-# RULES
  "narrow32Int#"  forall x#. narrow32Int#   x# = x#
  "narrow32Word#" forall x#. narrow32Word#   x# = x#
     #-}
  #endif
  
  {-# RULES
  "int2Word2Int"  forall x#. int2Word# (word2Int# x#) = x#
  "word2Int2Word" forall x#. word2Int# (int2Word# x#) = x#
    #-}
  
  
  -- Rules for C strings (the functions themselves are now in GHC.CString)
  {-# RULES
  "unpack"       [~1] forall a   . unpackCString# a             = build (unpackFoldrCString# a)
  "unpack-list"  [1]  forall a   . unpackFoldrCString# a (:) [] = unpackCString# a
  "unpack-append"     forall a n . unpackFoldrCString# a (:) n  = unpackAppendCString# a n
  
  -- There's a built-in rule (in PrelRules.lhs) for
  --      unpackFoldr "foo" c (unpackFoldr "baz" c n)  =  unpackFoldr "foobaz" c n
  
    #-}
  \end{code}
  
  
  #ifdef __HADDOCK__
  \begin{code}
  -- | A special argument for the 'Control.Monad.ST.ST' type constructor,
  -- indexing a state embedded in the 'Prelude.IO' monad by
  -- 'Control.Monad.ST.stToIO'.
  data RealWorld
  \end{code}
  #endif
  
  