Safe Haskell  Trustworthy 

The Haskell 2010 Prelude: a standard module imported by default into all Haskell modules. For more documentation, see the Haskell 2010 Report http://www.haskell.org/onlinereport/.
 data Bool
 (&&) :: Bool > Bool > Bool
 () :: Bool > Bool > Bool
 not :: Bool > Bool
 otherwise :: Bool
 data Maybe a
 maybe :: b > (a > b) > Maybe a > b
 data Either a b
 either :: (a > c) > (b > c) > Either a b > c
 data Ordering
 data Char
 type String = [Char]
 fst :: (a, b) > a
 snd :: (a, b) > b
 curry :: ((a, b) > c) > a > b > c
 uncurry :: (a > b > c) > (a, b) > c
 class Eq a where
 class Eq a => Ord a where
 class Enum a where
 succ :: a > a
 pred :: a > a
 toEnum :: Int > a
 fromEnum :: a > Int
 enumFrom :: a > [a]
 enumFromThen :: a > a > [a]
 enumFromTo :: a > a > [a]
 enumFromThenTo :: a > a > a > [a]
 class Bounded a where
 data Int
 data Integer
 data Float
 data Double
 type Rational = Ratio Integer
 class Num a where
 class (Num a, Ord a) => Real a where
 toRational :: a > Rational
 class (Real a, Enum a) => Integral a where
 class Num a => Fractional a where
 (/) :: a > a > a
 recip :: a > a
 fromRational :: Rational > a
 class Fractional a => Floating a where
 class (Real a, Fractional a) => RealFrac a where
 class (RealFrac a, Floating a) => RealFloat a where
 floatRadix :: a > Integer
 floatDigits :: a > Int
 floatRange :: a > (Int, Int)
 decodeFloat :: a > (Integer, Int)
 encodeFloat :: Integer > Int > a
 exponent :: a > Int
 significand :: a > a
 scaleFloat :: Int > a > a
 isNaN :: a > Bool
 isInfinite :: a > Bool
 isDenormalized :: a > Bool
 isNegativeZero :: a > Bool
 isIEEE :: a > Bool
 atan2 :: a > a > a
 subtract :: Num a => a > a > a
 even :: Integral a => a > Bool
 odd :: Integral a => a > Bool
 gcd :: Integral a => a > a > a
 lcm :: Integral a => a > a > a
 (^) :: (Num a, Integral b) => a > b > a
 (^^) :: (Fractional a, Integral b) => a > b > a
 fromIntegral :: (Integral a, Num b) => a > b
 realToFrac :: (Real a, Fractional b) => a > b
 class Monad m where
 class Functor f where
 fmap :: (a > b) > f a > f b
 mapM :: Monad m => (a > m b) > [a] > m [b]
 mapM_ :: Monad m => (a > m b) > [a] > m ()
 sequence :: Monad m => [m a] > m [a]
 sequence_ :: Monad m => [m a] > m ()
 (=<<) :: Monad m => (a > m b) > m a > m b
 id :: a > a
 const :: a > b > a
 (.) :: (b > c) > (a > b) > a > c
 flip :: (a > b > c) > b > a > c
 ($) :: (a > b) > a > b
 until :: (a > Bool) > (a > a) > a > a
 asTypeOf :: a > a > a
 error :: [Char] > a
 undefined :: a
 seq :: a > b > b
 ($!) :: (a > b) > a > b
 map :: (a > b) > [a] > [b]
 (++) :: [a] > [a] > [a]
 filter :: (a > Bool) > [a] > [a]
 head :: [a] > a
 last :: [a] > a
 tail :: [a] > [a]
 init :: [a] > [a]
 null :: [a] > Bool
 length :: [a] > Int
 (!!) :: [a] > Int > a
 reverse :: [a] > [a]
 foldl :: (a > b > a) > a > [b] > a
 foldl1 :: (a > a > a) > [a] > a
 foldr :: (a > b > b) > b > [a] > b
 foldr1 :: (a > a > a) > [a] > a
 and :: [Bool] > Bool
 or :: [Bool] > Bool
 any :: (a > Bool) > [a] > Bool
 all :: (a > Bool) > [a] > Bool
 sum :: Num a => [a] > a
 product :: Num a => [a] > a
 concat :: [[a]] > [a]
 concatMap :: (a > [b]) > [a] > [b]
 maximum :: Ord a => [a] > a
 minimum :: Ord a => [a] > a
 scanl :: (a > b > a) > a > [b] > [a]
 scanl1 :: (a > a > a) > [a] > [a]
 scanr :: (a > b > b) > b > [a] > [b]
 scanr1 :: (a > a > a) > [a] > [a]
 iterate :: (a > a) > a > [a]
 repeat :: a > [a]
 replicate :: Int > a > [a]
 cycle :: [a] > [a]
 take :: Int > [a] > [a]
 drop :: Int > [a] > [a]
 splitAt :: Int > [a] > ([a], [a])
 takeWhile :: (a > Bool) > [a] > [a]
 dropWhile :: (a > Bool) > [a] > [a]
 span :: (a > Bool) > [a] > ([a], [a])
 break :: (a > Bool) > [a] > ([a], [a])
 elem :: Eq a => a > [a] > Bool
 notElem :: Eq a => a > [a] > Bool
 lookup :: Eq a => a > [(a, b)] > Maybe b
 zip :: [a] > [b] > [(a, b)]
 zip3 :: [a] > [b] > [c] > [(a, b, c)]
 zipWith :: (a > b > c) > [a] > [b] > [c]
 zipWith3 :: (a > b > c > d) > [a] > [b] > [c] > [d]
 unzip :: [(a, b)] > ([a], [b])
 unzip3 :: [(a, b, c)] > ([a], [b], [c])
 lines :: String > [String]
 words :: String > [String]
 unlines :: [String] > String
 unwords :: [String] > String
 type ShowS = String > String
 class Show a where
 shows :: Show a => a > ShowS
 showChar :: Char > ShowS
 showString :: String > ShowS
 showParen :: Bool > ShowS > ShowS
 type ReadS a = String > [(a, String)]
 class Read a where
 reads :: Read a => ReadS a
 readParen :: Bool > ReadS a > ReadS a
 read :: Read a => String > a
 lex :: ReadS String
 data IO a
 putChar :: Char > IO ()
 putStr :: String > IO ()
 putStrLn :: String > IO ()
 print :: Show a => a > IO ()
 getChar :: IO Char
 getLine :: IO String
 getContents :: IO String
 interact :: (String > String) > IO ()
 type FilePath = String
 readFile :: FilePath > IO String
 writeFile :: FilePath > String > IO ()
 appendFile :: FilePath > String > IO ()
 readIO :: Read a => String > IO a
 readLn :: Read a => IO a
 type IOError = IOException
 ioError :: IOError > IO a
 userError :: String > IOError
 catch :: IO a > (IOError > IO a) > IO a
Standard types, classes and related functions
Basic data types
data Bool
data Maybe a
The Maybe
type encapsulates an optional value. A value of type
either contains a value of type Maybe
aa
(represented as
),
or it is empty (represented as Just
aNothing
). Using Maybe
is a good way to
deal with errors or exceptional cases without resorting to drastic
measures such as error
.
The Maybe
type is also a monad. It is a simple kind of error
monad, where all errors are represented by Nothing
. A richer
error monad can be built using the Either
type.
data Either a b
The Either
type represents values with two possibilities: a value of
type
is either Either
a b
or Left
a
.
Right
b
The Either
type is sometimes used to represent a value which is
either correct or an error; by convention, the Left
constructor is
used to hold an error value and the Right
constructor is used to
hold a correct value (mnemonic: "right" also means "correct").
data Ordering
data Char
The character type Char
is an enumeration whose values represent
Unicode (or equivalently ISO/IEC 10646) characters (see
http://www.unicode.org/ for details). This set extends the ISO 88591
(Latin1) character set (the first 256 characters), which is itself an extension
of the ASCII character set (the first 128 characters). A character literal in
Haskell has type Char
.
To convert a Char
to or from the corresponding Int
value defined
by Unicode, use toEnum
and fromEnum
from the
Enum
class respectively (or equivalently ord
and chr
).
Tuples
fst :: (a, b) > a
Extract the first component of a pair.
snd :: (a, b) > b
Extract the second component of a pair.
Basic type classes
class Eq a where
The Eq
class defines equality (==
) and inequality (/=
).
All the basic datatypes exported by the Prelude are instances of Eq
,
and Eq
may be derived for any datatype whose constituents are also
instances of Eq
.
Eq Bool  
Eq Char  
Eq Double  
Eq Float  
Eq Int  
Eq Int8  
Eq Int16  
Eq Int32  
Eq Int64  
Eq Integer  
Eq Ordering  
Eq Word  
Eq Word8  
Eq Word16  
Eq Word32  
Eq Word64  
Eq ()  
Eq Handle  
Eq Finalizers  
Eq HandlePosn  
Eq Errno  
Eq AsyncException  
Eq ArrayException  
Eq ExitCode  
Eq IOErrorType  
Eq BufferMode  
Eq Newline  
Eq NewlineMode  
Eq WordPtr  
Eq IntPtr  
Eq GeneralCategory  
Eq CChar  
Eq CSChar  
Eq CUChar  
Eq CShort  
Eq CUShort  
Eq CInt  
Eq CUInt  
Eq CLong  
Eq CULong  
Eq CLLong  
Eq CULLong  
Eq CFloat  
Eq CDouble  
Eq CPtrdiff  
Eq CSize  
Eq CWchar  
Eq CSigAtomic  
Eq CClock  
Eq CTime  
Eq CUSeconds  
Eq CSUSeconds  
Eq CIntPtr  
Eq CUIntPtr  
Eq CIntMax  
Eq CUIntMax  
Eq IODeviceType  
Eq SeekMode  
Eq IOMode  
Eq MaskingState  
Eq IOException  
Eq ArithException  
Eq a => Eq [a]  
Eq a => Eq (Ratio a)  
Eq (StablePtr a)  
Eq (Ptr a)  
Eq (FunPtr a)  
Eq (ForeignPtr a)  
Eq a => Eq (Complex a)  
Eq a => Eq (Maybe a)  
(Eq a, Eq b) => Eq (Either a b)  
(Eq a, Eq b) => Eq (a, b)  
(Ix i, Eq e) => Eq (Array i e)  
(Eq a, Eq b, Eq c) => Eq (a, b, c)  
Eq (STArray s i e)  
(Eq a, Eq b, Eq c, Eq d) => Eq (a, b, c, d)  
(Eq a, Eq b, Eq c, Eq d, Eq e) => Eq (a, b, c, d, e)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f) => Eq (a, b, c, d, e, f)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g) => Eq (a, b, c, d, e, f, g)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h) => Eq (a, b, c, d, e, f, g, h)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i) => Eq (a, b, c, d, e, f, g, h, i)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j) => Eq (a, b, c, d, e, f, g, h, i, j)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k) => Eq (a, b, c, d, e, f, g, h, i, j, k)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k, Eq l) => Eq (a, b, c, d, e, f, g, h, i, j, k, l)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k, Eq l, Eq m) => Eq (a, b, c, d, e, f, g, h, i, j, k, l, m)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k, Eq l, Eq m, Eq n) => Eq (a, b, c, d, e, f, g, h, i, j, k, l, m, n)  
(Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k, Eq l, Eq m, Eq n, Eq o) => Eq (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) 
The Ord
class is used for totally ordered datatypes.
Instances of Ord
can be derived for any userdefined
datatype whose constituent types are in Ord
. The declared order
of the constructors in the data declaration determines the ordering
in derived Ord
instances. The Ordering
datatype allows a single
comparison to determine the precise ordering of two objects.
Minimal complete definition: either compare
or <=
.
Using compare
can be more efficient for complex types.
Ord Bool  
Ord Char  
Ord Double  
Ord Float  
Ord Int  
Ord Int8  
Ord Int16  
Ord Int32  
Ord Int64  
Ord Integer  
Ord Ordering  
Ord Word  
Ord Word8  
Ord Word16  
Ord Word32  
Ord Word64  
Ord ()  
Ord AsyncException  
Ord ArrayException  
Ord ExitCode  
Ord BufferMode  
Ord Newline  
Ord NewlineMode  
Ord WordPtr  
Ord IntPtr  
Ord GeneralCategory  
Ord CChar  
Ord CSChar  
Ord CUChar  
Ord CShort  
Ord CUShort  
Ord CInt  
Ord CUInt  
Ord CLong  
Ord CULong  
Ord CLLong  
Ord CULLong  
Ord CFloat  
Ord CDouble  
Ord CPtrdiff  
Ord CSize  
Ord CWchar  
Ord CSigAtomic  
Ord CClock  
Ord CTime  
Ord CUSeconds  
Ord CSUSeconds  
Ord CIntPtr  
Ord CUIntPtr  
Ord CIntMax  
Ord CUIntMax  
Ord SeekMode  
Ord IOMode  
Ord ArithException  
(Eq [a], Ord a) => Ord [a]  
(Eq (Ratio a), Integral a) => Ord (Ratio a)  
Eq (Ptr a) => Ord (Ptr a)  
Eq (FunPtr a) => Ord (FunPtr a)  
Eq (ForeignPtr a) => Ord (ForeignPtr a)  
(Eq (Maybe a), Ord a) => Ord (Maybe a)  
(Eq (Either a b), Ord a, Ord b) => Ord (Either a b)  
(Eq (a, b), Ord a, Ord b) => Ord (a, b)  
(Eq (Array i e), Ix i, Ord e) => Ord (Array i e)  
(Eq (a, b, c), Ord a, Ord b, Ord c) => Ord (a, b, c)  
(Eq (a, b, c, d), Ord a, Ord b, Ord c, Ord d) => Ord (a, b, c, d)  
(Eq (a, b, c, d, e), Ord a, Ord b, Ord c, Ord d, Ord e) => Ord (a, b, c, d, e)  
(Eq (a, b, c, d, e, f), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f) => Ord (a, b, c, d, e, f)  
(Eq (a, b, c, d, e, f, g), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g) => Ord (a, b, c, d, e, f, g)  
(Eq (a, b, c, d, e, f, g, h), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h) => Ord (a, b, c, d, e, f, g, h)  
(Eq (a, b, c, d, e, f, g, h, i), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i) => Ord (a, b, c, d, e, f, g, h, i)  
(Eq (a, b, c, d, e, f, g, h, i, j), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j) => Ord (a, b, c, d, e, f, g, h, i, j)  
(Eq (a, b, c, d, e, f, g, h, i, j, k), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k) => Ord (a, b, c, d, e, f, g, h, i, j, k)  
(Eq (a, b, c, d, e, f, g, h, i, j, k, l), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k, Ord l) => Ord (a, b, c, d, e, f, g, h, i, j, k, l)  
(Eq (a, b, c, d, e, f, g, h, i, j, k, l, m), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k, Ord l, Ord m) => Ord (a, b, c, d, e, f, g, h, i, j, k, l, m)  
(Eq (a, b, c, d, e, f, g, h, i, j, k, l, m, n), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k, Ord l, Ord m, Ord n) => Ord (a, b, c, d, e, f, g, h, i, j, k, l, m, n)  
(Eq (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o), Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k, Ord l, Ord m, Ord n, Ord o) => Ord (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) 
class Enum a where
Class Enum
defines operations on sequentially ordered types.
The enumFrom
... methods are used in Haskell's translation of
arithmetic sequences.
Instances of Enum
may be derived for any enumeration type (types
whose constructors have no fields). The nullary constructors are
assumed to be numbered lefttoright by fromEnum
from 0
through n1
.
See Chapter 10 of the Haskell Report for more details.
For any type that is an instance of class Bounded
as well as Enum
,
the following should hold:
 The calls
andsucc
maxBound
should result in a runtime error.pred
minBound

fromEnum
andtoEnum
should give a runtime error if the result value is not representable in the result type. For example,
is an error.toEnum
7 ::Bool

enumFrom
andenumFromThen
should be defined with an implicit bound, thus:
enumFrom x = enumFromTo x maxBound enumFromThen x y = enumFromThenTo x y bound where bound  fromEnum y >= fromEnum x = maxBound  otherwise = minBound
succ :: a > a
the successor of a value. For numeric types, succ
adds 1.
pred :: a > a
the predecessor of a value. For numeric types, pred
subtracts 1.
Convert from an Int
.
Convert to an Int
.
It is implementationdependent what fromEnum
returns when
applied to a value that is too large to fit in an Int
.
enumFrom :: a > [a]
Used in Haskell's translation of [n..]
.
enumFromThen :: a > a > [a]
Used in Haskell's translation of [n,n'..]
.
enumFromTo :: a > a > [a]
Used in Haskell's translation of [n..m]
.
enumFromThenTo :: a > a > a > [a]
Used in Haskell's translation of [n,n'..m]
.
class Bounded a where
The Bounded
class is used to name the upper and lower limits of a
type. Ord
is not a superclass of Bounded
since types that are not
totally ordered may also have upper and lower bounds.
The Bounded
class may be derived for any enumeration type;
minBound
is the first constructor listed in the data
declaration
and maxBound
is the last.
Bounded
may also be derived for singleconstructor datatypes whose
constituent types are in Bounded
.
Numbers
Numeric types
data Int
data Integer
Arbitraryprecision integers.
data Float
Singleprecision floating point numbers. It is desirable that this type be at least equal in range and precision to the IEEE singleprecision type.
data Double
Doubleprecision floating point numbers. It is desirable that this type be at least equal in range and precision to the IEEE doubleprecision type.
Numeric type classes
class Num a where
Basic numeric class.
Minimal complete definition: all except negate
or ()
(+) :: a > a > a
(*) :: a > a > a
() :: a > a > a
negate :: a > a
Unary negation.
abs :: a > a
Absolute value.
signum :: a > a
Sign of a number.
The functions abs
and signum
should satisfy the law:
abs x * signum x == x
For real numbers, the signum
is either 1
(negative), 0
(zero)
or 1
(positive).
fromInteger :: Integer > a
Conversion from an Integer
.
An integer literal represents the application of the function
fromInteger
to the appropriate value of type Integer
,
so such literals have type (
.
Num
a) => a
Num Double  
Num Float  
Num Int  
Num Int8  
Num Int16  
Num Int32  
Num Int64  
Num Integer  
Num Word  
Num Word8  
Num Word16  
Num Word32  
Num Word64  
Num WordPtr  
Num IntPtr  
Num CChar  
Num CSChar  
Num CUChar  
Num CShort  
Num CUShort  
Num CInt  
Num CUInt  
Num CLong  
Num CULong  
Num CLLong  
Num CULLong  
Num CFloat  
Num CDouble  
Num CPtrdiff  
Num CSize  
Num CWchar  
Num CSigAtomic  
Num CClock  
Num CTime  
Num CUSeconds  
Num CSUSeconds  
Num CIntPtr  
Num CUIntPtr  
Num CIntMax  
Num CUIntMax  
Integral a => Num (Ratio a)  
RealFloat a => Num (Complex a) 
class (Num a, Ord a) => Real a where
toRational :: a > Rational
the rational equivalent of its real argument with full precision
class (Real a, Enum a) => Integral a where
quot :: a > a > a
integer division truncated toward zero
rem :: a > a > a
integer remainder, satisfying
(x `quot` y)*y + (x `rem` y) == x
div :: a > a > a
integer division truncated toward negative infinity
mod :: a > a > a
integer modulus, satisfying
(x `div` y)*y + (x `mod` y) == x
quotRem :: a > a > (a, a)
divMod :: a > a > (a, a)
conversion to Integer
class Num a => Fractional a where
Fractional numbers, supporting real division.
Minimal complete definition: fromRational
and (recip
or (
)
/
)
(/) :: a > a > a
fractional division
recip :: a > a
reciprocal fraction
fromRational :: Rational > a
Conversion from a Rational
(that is
).
A floating literal stands for an application of Ratio
Integer
fromRational
to a value of type Rational
, so such literals have type
(
.
Fractional
a) => a
Fractional Double  
Fractional Float  
Fractional CFloat  
Fractional CDouble  
(Num (Ratio a), Integral a) => Fractional (Ratio a)  
(Num (Complex a), RealFloat a) => Fractional (Complex a) 
class Fractional a => Floating a where
Trigonometric and hyperbolic functions and related functions.
Minimal complete definition:
pi
, exp
, log
, sin
, cos
, sinh
, cosh
,
asin
, acos
, atan
, asinh
, acosh
and atanh
class (Real a, Fractional a) => RealFrac a where
Extracting components of fractions.
Minimal complete definition: properFraction
properFraction :: Integral b => a > (b, a)
The function properFraction
takes a real fractional number x
and returns a pair (n,f)
such that x = n+f
, and:

n
is an integral number with the same sign asx
; and 
f
is a fraction with the same type and sign asx
, and with absolute value less than1
.
The default definitions of the ceiling
, floor
, truncate
and round
functions are in terms of properFraction
.
truncate :: Integral b => a > b
returns the integer nearest truncate
xx
between zero and x
returns the nearest integer to round
xx
;
the even integer if x
is equidistant between two integers
ceiling :: Integral b => a > b
returns the least integer not less than ceiling
xx
returns the greatest integer not greater than floor
xx
class (RealFrac a, Floating a) => RealFloat a where
Efficient, machineindependent access to the components of a floatingpoint number.
Minimal complete definition:
all except exponent
, significand
, scaleFloat
and atan2
floatRadix :: a > Integer
a constant function, returning the radix of the representation
(often 2
)
floatDigits :: a > Int
a constant function, returning the number of digits of
floatRadix
in the significand
floatRange :: a > (Int, Int)
a constant function, returning the lowest and highest values the exponent may assume
decodeFloat :: a > (Integer, Int)
The function decodeFloat
applied to a real floatingpoint
number returns the significand expressed as an Integer
and an
appropriately scaled exponent (an Int
). If
yields decodeFloat
x(m,n)
, then x
is equal in value to m*b^^n
, where b
is the floatingpoint radix, and furthermore, either m
and n
are both zero or else b^(d1) <=
, where abs
m < b^dd
is
the value of
.
In particular, floatDigits
x
. If the type
contains a negative zero, also decodeFloat
0 = (0,0)
.
The result of decodeFloat
(0.0) = (0,0)
is unspecified if either of
decodeFloat
x
or isNaN
x
is isInfinite
xTrue
.
encodeFloat :: Integer > Int > a
encodeFloat
performs the inverse of decodeFloat
in the
sense that for finite x
with the exception of 0.0
,
.
uncurry
encodeFloat
(decodeFloat
x) = x
is one of the two closest representable
floatingpoint numbers to encodeFloat
m nm*b^^n
(or ±Infinity
if overflow
occurs); usually the closer, but if m
contains too many bits,
the result may be rounded in the wrong direction.
exponent
corresponds to the second component of decodeFloat
.
and for finite nonzero exponent
0 = 0x
,
.
If exponent
x = snd (decodeFloat
x) + floatDigits
xx
is a finite floatingpoint number, it is equal in value to
, where significand
x * b ^^ exponent
xb
is the
floatingpoint radix.
The behaviour is unspecified on infinite or NaN
values.
significand :: a > a
The first component of decodeFloat
, scaled to lie in the open
interval (1
,1
), either 0.0
or of absolute value >= 1/b
,
where b
is the floatingpoint radix.
The behaviour is unspecified on infinite or NaN
values.
scaleFloat :: Int > a > a
multiplies a floatingpoint number by an integer power of the radix
True
if the argument is an IEEE "notanumber" (NaN) value
isInfinite :: a > Bool
True
if the argument is an IEEE infinity or negative infinity
isDenormalized :: a > Bool
True
if the argument is too small to be represented in
normalized format
isNegativeZero :: a > Bool
True
if the argument is an IEEE negative zero
True
if the argument is an IEEE floating point number
atan2 :: a > a > a
a version of arctangent taking two real floatingpoint arguments.
For real floating x
and y
,
computes the angle
(from the positive xaxis) of the vector from the origin to the
point atan2
y x(x,y)
.
returns a value in the range [atan2
y xpi
,
pi
]. It follows the Common Lisp semantics for the origin when
signed zeroes are supported.
, with atan2
y 1y
in a type
that is RealFloat
, should return the same value as
.
A default definition of atan
yatan2
is provided, but implementors
can provide a more accurate implementation.
Numeric functions
(^^) :: (Fractional a, Integral b) => a > b > a
raise a number to an integral power
fromIntegral :: (Integral a, Num b) => a > b
general coercion from integral types
realToFrac :: (Real a, Fractional b) => a > b
general coercion to fractional types
Monads and functors
class Monad m where
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, Maybe
and IO
defined in the Prelude satisfy these laws.
(>>=) :: m a > (a > m b) > m b
Sequentially compose two actions, passing any value produced by the first as an argument to the second.
(>>) :: m 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.
return :: a > m a
Inject a value into the monadic type.
Fail with a message. This operation is not part of the
mathematical definition of a monad, but is invoked on patternmatch
failure in a do
expression.
class Functor f where
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, Maybe
and IO
satisfy these laws.
fmap :: (a > b) > f a > f b
sequence :: Monad m => [m a] > m [a]
Evaluate each action in the sequence from left to right, and collect the results.
sequence_ :: Monad m => [m a] > m ()
Evaluate each action in the sequence from left to right, and ignore the results.
Miscellaneous functions
id :: a > a
Identity function.
const :: a > b > a
Constant function.
(.) :: (b > c) > (a > b) > a > c
Function composition.
flip :: (a > b > c) > b > a > c
takes its (first) two arguments in the reverse order of flip
ff
.
($) :: (a > b) > a > b
Application operator. This operator is redundant, since ordinary
application (f x)
means the same as (f
. However, $
x)$
has
low, rightassociative 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 higherorder situations, such as
,
or map
($
0) xs
.
zipWith
($
) fs xs
asTypeOf :: a > a > a
undefined :: a
seq :: a > b > b
Evaluates its first argument to head normal form, and then returns its second argument as the result.
List operations
map :: (a > b) > [a] > [b]
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, ...]
(++) :: [a] > [a] > [a]
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.
filter :: (a > Bool) > [a] > [a]
filter
, applied to a predicate and a list, returns the list of
those elements that satisfy the predicate; i.e.,
filter p xs = [ x  x < xs, p x]
head :: [a] > a
Extract the first element of a list, which must be nonempty.
last :: [a] > a
Extract the last element of a list, which must be finite and nonempty.
tail :: [a] > [a]
Extract the elements after the head of a list, which must be nonempty.
init :: [a] > [a]
Return all the elements of a list except the last one. The list must be nonempty.
O(n). length
returns the length of a finite list as an Int
.
It is an instance of the more general genericLength
,
the result type of which may be any kind of number.
List index (subscript) operator, starting from 0.
It is an instance of the more general genericIndex
,
which takes an index of any integral type.
Reducing lists (folds)
foldl :: (a > b > a) > a > [b] > a
foldl
, applied to a binary operator, a starting value (typically
the leftidentity of the operator), and a list, reduces the list
using the binary operator, from left to right:
foldl f z [x1, x2, ..., xn] == (...((z `f` x1) `f` x2) `f`...) `f` xn
The list must be finite.
foldl1 :: (a > a > a) > [a] > a
foldr :: (a > b > b) > b > [a] > b
foldr
, applied to a binary operator, a starting value (typically
the rightidentity 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)...)
foldr1 :: (a > a > a) > [a] > a
Special folds
concat :: [[a]] > [a]
Concatenate a list of lists.
concatMap :: (a > [b]) > [a] > [b]
Map a function over a list and concatenate the results.
Building lists
Scans
scanl :: (a > b > a) > a > [b] > [a]
scanl1 :: (a > a > a) > [a] > [a]
scanr :: (a > b > b) > b > [a] > [b]
scanr1 :: (a > a > a) > [a] > [a]
Infinite lists
iterate :: (a > a) > a > [a]
iterate
f x
returns an infinite list of repeated applications
of f
to x
:
iterate f x == [x, f x, f (f x), ...]
replicate
n x
is a list of length n
with x
the value of
every element.
It is an instance of the more general genericReplicate
,
in which n
may be of any integral type.
cycle :: [a] > [a]
cycle
ties a finite list into a circular one, or equivalently,
the infinite repetition of the original list. It is the identity
on infinite lists.
Sublists
take
n
, applied to a list xs
, returns the prefix of xs
of length n
, or xs
itself if n >
:
length
xs
take 5 "Hello World!" == "Hello" take 3 [1,2,3,4,5] == [1,2,3] take 3 [1,2] == [1,2] take 3 [] == [] take (1) [1,2] == [] take 0 [1,2] == []
It is an instance of the more general genericTake
,
in which n
may be of any integral type.
drop
n xs
returns the suffix of xs
after the first n
elements, or []
if n >
:
length
xs
drop 6 "Hello World!" == "World!" drop 3 [1,2,3,4,5] == [4,5] drop 3 [1,2] == [] drop 3 [] == [] drop (1) [1,2] == [1,2] drop 0 [1,2] == [1,2]
It is an instance of the more general genericDrop
,
in which n
may be of any integral type.
splitAt :: Int > [a] > ([a], [a])Source
splitAt
n xs
returns a tuple where first element is xs
prefix of
length n
and second element is the remainder of the list:
splitAt 6 "Hello World!" == ("Hello ","World!") splitAt 3 [1,2,3,4,5] == ([1,2,3],[4,5]) splitAt 1 [1,2,3] == ([1],[2,3]) splitAt 3 [1,2,3] == ([1,2,3],[]) splitAt 4 [1,2,3] == ([1,2,3],[]) splitAt 0 [1,2,3] == ([],[1,2,3]) splitAt (1) [1,2,3] == ([],[1,2,3])
It is equivalent to (
.
take
n xs, drop
n xs)splitAt
is an instance of the more general genericSplitAt
,
in which n
may be of any integral type.
takeWhile :: (a > Bool) > [a] > [a]
takeWhile
, applied to a predicate p
and a list xs
, returns the
longest prefix (possibly empty) of xs
of elements that satisfy p
:
takeWhile (< 3) [1,2,3,4,1,2,3,4] == [1,2] takeWhile (< 9) [1,2,3] == [1,2,3] takeWhile (< 0) [1,2,3] == []
span :: (a > Bool) > [a] > ([a], [a])
span
, applied to a predicate p
and a list xs
, returns a tuple where
first element is longest prefix (possibly empty) of xs
of elements that
satisfy p
and second element is the remainder of the list:
span (< 3) [1,2,3,4,1,2,3,4] == ([1,2],[3,4,1,2,3,4]) span (< 9) [1,2,3] == ([1,2,3],[]) span (< 0) [1,2,3] == ([],[1,2,3])
break :: (a > Bool) > [a] > ([a], [a])
break
, applied to a predicate p
and a list xs
, returns a tuple where
first element is longest prefix (possibly empty) of xs
of elements that
do not satisfy p
and second element is the remainder of the list:
break (> 3) [1,2,3,4,1,2,3,4] == ([1,2,3],[4,1,2,3,4]) break (< 9) [1,2,3] == ([],[1,2,3]) break (> 9) [1,2,3] == ([1,2,3],[])
Searching lists
Zipping and unzipping lists
zip :: [a] > [b] > [(a, b)]
zip
takes two lists and returns a list of corresponding pairs.
If one input list is short, excess elements of the longer list are
discarded.
zip3 :: [a] > [b] > [c] > [(a, b, c)]
zipWith :: (a > b > c) > [a] > [b] > [c]
zipWith3 :: (a > b > c > d) > [a] > [b] > [c] > [d]
unzip :: [(a, b)] > ([a], [b])
unzip
transforms a list of pairs into a list of first components
and a list of second components.
unzip3 :: [(a, b, c)] > ([a], [b], [c])
Functions on strings
lines
breaks a string up into a list of strings at newline
characters. The resulting strings do not contain newlines.
words
breaks a string up into a list of words, which were delimited
by white space.
Converting to and from String
Converting to String
class Show a where
Conversion of values to readable String
s.
Minimal complete definition: showsPrec
or show
.
Derived instances of Show
have the following properties, which
are compatible with derived instances of Read
:
 The result of
show
is a syntactically correct Haskell expression containing only constants, given the fixity declarations in force at the point where the type is declared. It contains only the constructor names defined in the data type, parentheses, and spaces. When labelled constructor fields are used, braces, commas, field names, and equal signs are also used.  If the constructor is defined to be an infix operator, then
showsPrec
will produce infix applications of the constructor.  the representation will be enclosed in parentheses if the
precedence of the toplevel constructor in
x
is less thand
(associativity is ignored). Thus, ifd
is0
then the result is never surrounded in parentheses; ifd
is11
it is always surrounded in parentheses, unless it is an atomic expression.  If the constructor is defined using record syntax, then
show
will produce the recordsyntax form, with the fields given in the same order as the original declaration.
For example, given the declarations
infixr 5 :^: data Tree a = Leaf a  Tree a :^: Tree a
the derived instance of Show
is equivalent to
instance (Show a) => Show (Tree a) where showsPrec d (Leaf m) = showParen (d > app_prec) $ showString "Leaf " . showsPrec (app_prec+1) m where app_prec = 10 showsPrec d (u :^: v) = showParen (d > up_prec) $ showsPrec (up_prec+1) u . showString " :^: " . showsPrec (up_prec+1) v where up_prec = 5
Note that rightassociativity of :^:
is ignored. For example,

produces the stringshow
(Leaf 1 :^: Leaf 2 :^: Leaf 3)"Leaf 1 :^: (Leaf 2 :^: Leaf 3)"
.
:: Int  the operator precedence of the enclosing
context (a number from 
> a  the value to be converted to a 
> ShowS 
Convert a value to a readable String
.
showsPrec
should satisfy the law
showsPrec d x r ++ s == showsPrec d x (r ++ s)
Derived instances of Read
and Show
satisfy the following:
That is, readsPrec
parses the string produced by
showsPrec
, and delivers the value that showsPrec
started with.
utility function converting a Char
to a show function that
simply prepends the character unchanged.
showString :: String > ShowS
utility function converting a String
to a show function that
simply prepends the string unchanged.
Converting from String
class Read a where
Parsing of String
s, producing values.
Minimal complete definition: readsPrec
(or, for GHC only, readPrec
)
Derived instances of Read
make the following assumptions, which
derived instances of Show
obey:
 If the constructor is defined to be an infix operator, then the
derived
Read
instance will parse only infix applications of the constructor (not the prefix form).  Associativity is not used to reduce the occurrence of parentheses, although precedence may be.
 If the constructor is defined using record syntax, the derived
Read
will parse only the recordsyntax form, and furthermore, the fields must be given in the same order as the original declaration.  The derived
Read
instance allows arbitrary Haskell whitespace between tokens of the input string. Extra parentheses are also allowed.
For example, given the declarations
infixr 5 :^: data Tree a = Leaf a  Tree a :^: Tree a
the derived instance of Read
in Haskell 98 is equivalent to
instance (Read a) => Read (Tree a) where readsPrec d r = readParen (d > app_prec) (\r > [(Leaf m,t)  ("Leaf",s) < lex r, (m,t) < readsPrec (app_prec+1) s]) r ++ readParen (d > up_prec) (\r > [(u:^:v,w)  (u,s) < readsPrec (up_prec+1) r, (":^:",t) < lex s, (v,w) < readsPrec (up_prec+1) t]) r where app_prec = 10 up_prec = 5
Note that rightassociativity of :^:
is unused.
The derived instance in GHC is equivalent to
instance (Read a) => Read (Tree a) where readPrec = parens $ (prec app_prec $ do Ident "Leaf" < lexP m < step readPrec return (Leaf m)) +++ (prec up_prec $ do u < step readPrec Symbol ":^:" < lexP v < step readPrec return (u :^: v)) where app_prec = 10 up_prec = 5 readListPrec = readListPrecDefault
:: Int  the operator precedence of the enclosing
context (a number from 
> ReadS a 
attempts to parse a value from the front of the string, returning a list of (parsed value, remaining string) pairs. If there is no successful parse, the returned list is empty.
Derived instances of Read
and Show
satisfy the following:
That is, readsPrec
parses the string produced by
showsPrec
, and delivers the value that
showsPrec
started with.
The read
function reads input from a string, which must be
completely consumed by the input process.
The lex
function reads a single lexeme from the input, discarding
initial white space, and returning the characters that constitute the
lexeme. If the input string contains only white space, lex
returns a
single successful `lexeme' consisting of the empty string. (Thus
.) If there is no legal lexeme at the
beginning of the input string, lex
"" = [("","")]lex
fails (i.e. returns []
).
This lexer is not completely faithful to the Haskell lexical syntax in the following respects:
 Qualified names are not handled properly
 Octal and hexadecimal numerics are not recognized as a single token
 Comments are not treated properly
Basic Input and output
data IO a
A value of type
is a computation which, when performed,
does some I/O before returning a value of type IO
aa
.
There is really only one way to "perform" an I/O action: bind it to
Main.main
in your program. When your program is run, the I/O will
be performed. It isn't possible to perform I/O from an arbitrary
function, unless that function is itself in the IO
monad and called
at some point, directly or indirectly, from Main.main
.
IO
is a monad, so IO
actions can be combined using either the donotation
or the >>
and >>=
operations from the Monad
class.
Simple I/O operations
Output functions
The print
function outputs a value of any printable type to the
standard output device.
Printable types are those that are instances of class Show
; print
converts values to strings for output using the show
operation and
adds a newline.
For example, a program to print the first 20 integers and their powers of 2 could be written as:
main = print ([(n, 2^n)  n < [0..19]])
Input functions
getContents :: IO String
The getContents
operation returns all user input as a single string,
which is read lazily as it is needed
(same as hGetContents
stdin
).
interact :: (String > String) > IO ()
The interact
function takes a function of type String>String
as its argument. The entire input from the standard input device is
passed to this function as its argument, and the resulting string is
output on the standard output device.
Files
File and directory names are values of type String
, whose precise
meaning is operating system dependent. Files can be opened, yielding a
handle which can then be used to operate on the contents of that file.
readFile :: FilePath > IO String
The readFile
function reads a file and
returns the contents of the file as a string.
The file is read lazily, on demand, as with getContents
.
writeFile :: FilePath > String > IO ()
The computation writeFile
file str
function writes the string str
,
to the file file
.
appendFile :: FilePath > String > IO ()
The computation appendFile
file str
function appends the string str
,
to the file file
.
Note that writeFile
and appendFile
write a literal string
to a file. To write a value of any printable type, as with print
,
use the show
function to convert the value to a string first.
main = appendFile "squares" (show [(x,x*x)  x < [0,0.1..2]])
Exception handling in the I/O monad
type IOError = IOException
The Haskell 98 type for exceptions in the IO
monad.
Any I/O operation may raise an IOError
instead of returning a result.
For a more general type of exception, including also those that arise
in pure code, see Control.Exception.Exception.
In Haskell 98, this is an opaque type.
catch :: IO a > (IOError > IO a) > IO aSource
The catch
function establishes a handler that receives any
IOError
raised in the action protected by catch
.
An IOError
is caught by
the most recent handler established by one of the exception handling
functions. These handlers are
not selective: all IOError
s are caught. Exception propagation
must be explicitly provided in a handler by reraising any unwanted
exceptions. For example, in
f = catch g (\e > if IO.isEOFError e then return [] else ioError e)
the function f
returns []
when an endoffile exception
(cf. isEOFError
) occurs in g
; otherwise, the
exception is propagated to the next outer handler.
When an exception propagates outside the main program, the Haskell
system prints the associated IOError
value and exits the program.
NonI/O exceptions are not caught by this variant; to catch all
exceptions, use catch
from Control.Exception.