Safe Haskell | Trustworthy |
---|---|

Language | Haskell2010 |

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 :: (b -> a -> b) -> b -> [a] -> b
- 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 :: (b -> a -> b) -> b -> [a] -> [b]
- 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`

a`a`

(represented as

),
or it is empty (represented as `Just`

a`Nothing`

). 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 8859-1
(Latin-1) 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 HandlePosn | |

Eq IOMode | |

Eq Errno | |

Eq AsyncException | |

Eq ArrayException | |

Eq ExitCode | |

Eq IOErrorType | |

Eq BufferMode | |

Eq Newline | |

Eq NewlineMode | |

Eq IODeviceType | |

Eq SeekMode | |

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 MaskingState | |

Eq IOException | |

Eq ErrorCall | |

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 (ZipList 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 user-defined
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 IOMode | |

Ord AsyncException | |

Ord ArrayException | |

Ord ExitCode | |

Ord BufferMode | |

Ord Newline | |

Ord NewlineMode | |

Ord SeekMode | |

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 ErrorCall | |

Ord ArithException | |

Ord a => Ord [a] | |

Integral a => Ord (Ratio a) | |

Ord (Ptr a) | |

Ord (FunPtr a) | |

Ord (ForeignPtr a) | |

Ord a => Ord (ZipList a) | |

Ord a => Ord (Maybe a) | |

(Ord a, Ord b) => Ord (Either a b) | |

(Ord a, Ord b) => Ord (a, b) | |

(Ix i, Ord e) => Ord (Array i e) | |

(Ord a, Ord b, Ord c) => Ord (a, b, c) | |

(Ord a, Ord b, Ord c, Ord d) => Ord (a, b, c, d) | |

(Ord a, Ord b, Ord c, Ord d, Ord e) => Ord (a, b, c, d, e) | |

(Ord a, Ord b, Ord c, Ord d, Ord e, Ord f) => Ord (a, b, c, d, e, f) | |

(Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g) => Ord (a, b, c, d, e, f, g) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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 left-to-right by `fromEnum`

from `0`

through `n-1`

.
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

and`succ`

`maxBound`

should result in a runtime error.`pred`

`minBound`

`fromEnum`

and`toEnum`

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`

and`enumFromThen`

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 implementation-dependent 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 single-constructor datatypes whose
constituent types are in `Bounded`

.

## Numbers

### Numeric types

data Int :: *

data Integer :: *

Arbitrary-precision integers.

data Float :: *

Single-precision floating point numbers. It is desirable that this type be at least equal in range and precision to the IEEE single-precision type.

data Double :: *

Double-precision floating point numbers. It is desirable that this type be at least equal in range and precision to the IEEE double-precision type.

### Numeric type classes

class Num a where

Basic numeric class.

Minimal complete definition: all except `negate`

or `(-)`

(+) :: a -> a -> a infixl 6

(*) :: a -> a -> a infixl 7

(-) :: a -> a -> a infixl 6

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 infixl 7

integer division truncated toward zero

rem :: a -> a -> a infixl 7

integer remainder, satisfying

(x `quot` y)*y + (x `rem` y) == x

div :: a -> a -> a infixl 7

integer division truncated toward negative infinity

mod :: a -> a -> a infixl 7

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 `(`

)`/`

)

fromRational, (recip | (/))

(/) :: a -> a -> a infixl 7

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 | |

Integral a => Fractional (Ratio 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 as`x`

; and`f`

is a fraction with the same type and sign as`x`

, and with absolute value less than`1`

.

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`

x`x`

between zero and `x`

returns the nearest integer to `round`

x`x`

;
the even integer if `x`

is equidistant between two integers

ceiling :: Integral b => a -> b

returns the least integer not less than `ceiling`

x`x`

returns the greatest integer not greater than `floor`

x`x`

class (RealFrac a, Floating a) => RealFloat a where

Efficient, machine-independent access to the components of a floating-point number.

Minimal complete definition:
all except `exponent`

, `significand`

, `scaleFloat`

and `atan2`

floatRadix, floatDigits, floatRange, decodeFloat, encodeFloat, isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE

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 floating-point
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 floating-point radix, and furthermore, either `m`

and `n`

are both zero or else `b^(d-1) <= `

, where `abs`

m < b^d`d`

is
the value of

.
In particular, `floatDigits`

x

. If the type
contains a negative zero, also `decodeFloat`

0 = (0,0)

.
`decodeFloat`

(-0.0) = (0,0)*The result of* `decodeFloat`

x*is unspecified if either of*
`isNaN`

x*or* `isInfinite`

x*is* `True`

.

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
floating-point numbers to `encodeFloat`

m n`m*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 = 0`x`

,

.
If `exponent`

x = snd (`decodeFloat`

x) + `floatDigits`

x`x`

is a finite floating-point number, it is equal in value to

, where `significand`

x * b ^^ `exponent`

x`b`

is the
floating-point 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 floating-point radix.
The behaviour is unspecified on infinite or `NaN`

values.

scaleFloat :: Int -> a -> a

multiplies a floating-point number by an integer power of the radix

`True`

if the argument is an IEEE "not-a-number" (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 floating-point arguments.
For real floating `x`

and `y`

,

computes the angle
(from the positive x-axis) of the vector from the origin to the
point `atan2`

y x`(x,y)`

.

returns a value in the range [`atan2`

y x`-pi`

,
`pi`

]. It follows the Common Lisp semantics for the origin when
signed zeroes are supported.

, with `atan2`

y 1`y`

in a type
that is `RealFloat`

, should return the same value as

.
A default definition of `atan`

y`atan2`

is provided, but implementors
can provide a more accurate implementation.

### Numeric functions

(^^) :: (Fractional a, Integral b) => a -> b -> a infixr 8

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 infixl 1

Sequentially compose two actions, passing any value produced by the first as an argument to the second.

(>>) :: m a -> m b -> m b infixl 1

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 pattern-match
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.

(=<<) :: Monad m => (a -> m b) -> m a -> m b infixr 1

Same as `>>=`

, but with the arguments interchanged.

## Miscellaneous functions

id :: a -> a

Identity function.

const :: a -> b -> a

Constant function.

(.) :: (b -> c) -> (a -> b) -> a -> c infixr 9

Function composition.

flip :: (a -> b -> c) -> b -> a -> c

takes its (first) two arguments in the reverse order of `flip`

f`f`

.

($) :: (a -> b) -> a -> b infixr 0

Application operator. This operator is redundant, since ordinary
application `(f x)`

means the same as `(f `

. However, `$`

x)`$`

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

,
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.

($!) :: (a -> b) -> a -> b infixr 0 Source

Strict (call-by-value) application, defined in terms of `seq`

.

# 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] infixr 5

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 non-empty.

last :: [a] -> a

Extract the last element of a list, which must be finite and non-empty.

tail :: [a] -> [a]

Extract the elements after the head of a list, which must be non-empty.

init :: [a] -> [a]

Return all the elements of a list except the last one. The list must be non-empty.

*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.

(!!) :: [a] -> Int -> a infixl 9

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 :: (b -> a -> b) -> b -> [a] -> b

`foldl`

, applied to a binary operator, a starting value (typically
the left-identity 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 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)...)

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 :: (b -> a -> b) -> b -> [a] -> [b]

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 top-level constructor in
`x`

is less than`d`

(associativity is ignored). Thus, if`d`

is`0`

then the result is never surrounded in parentheses; if`d`

is`11`

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 record-syntax 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 right-associativity of `:^:`

is ignored. For example,

produces the string`show`

(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 record-syntax 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 2010 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 right-associativity 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`

a`a`

.

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 do-notation
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 2010 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 2010, this is an opaque type.

catch :: IO a -> (IOError -> IO a) -> IO a Source

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 re-raising 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 end-of-file 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.

Non-I/O exceptions are not caught by this variant; to catch all
exceptions, use `catch`

from Control.Exception.