| Copyright | (c) The University of Glasgow 2001 |
|---|---|
| License | BSD-style (see the file libraries/base/LICENSE) |
| Maintainer | libraries@haskell.org |
| Stability | stable |
| Portability | portable |
| Safe Haskell | Trustworthy |
| Language | Haskell2010 |
Prelude
Contents
Description
The Prelude: a standard module. The Prelude is imported by default into all Haskell modules unless either there is an explicit import statement for it, or the NoImplicitPrelude extension is enabled.
- data Bool :: TYPE Lifted
- (&&) :: 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 :: TYPE Lifted
- data Char :: TYPE Lifted
- 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
- class Bounded a where
- data Int :: TYPE Lifted
- data Integer :: TYPE Lifted
- data Float :: TYPE Lifted
- data Double :: TYPE Lifted
- type Rational = Ratio Integer
- data Word :: TYPE Lifted
- class Num a where
- class (Num a, Ord a) => Real a where
- class (Real a, Enum a) => Integral a where
- class Num a => Fractional a where
- class Fractional a => Floating a where
- class (Real a, Fractional a) => RealFrac a where
- class (RealFrac a, Floating a) => RealFloat a where
- 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 Monoid a where
- class Functor f where
- (<$>) :: Functor f => (a -> b) -> f a -> f b
- class Functor f => Applicative f where
- class Applicative m => Monad m where
- mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m ()
- sequence_ :: (Foldable t, Monad m) => t (m a) -> m ()
- (=<<) :: Monad m => (a -> m b) -> m a -> m b
- class Foldable t where
- class (Functor t, Foldable t) => Traversable t where
- id :: a -> a
- const :: a -> b -> a
- (.) :: (b -> c) -> (a -> b) -> a -> c
- flip :: (a -> b -> c) -> b -> a -> c
- ($) :: (a -> b) -> a -> b
- until :: (a -> Bool) -> (a -> a) -> a -> a
- asTypeOf :: a -> a -> a
- error :: forall v. forall a. HasCallStack => [Char] -> a
- errorWithoutStackTrace :: forall v. forall a. [Char] -> a
- undefined :: forall v. forall a. HasCallStack => 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 :: Foldable t => t a -> Bool
- length :: Foldable t => t a -> Int
- (!!) :: [a] -> Int -> a
- reverse :: [a] -> [a]
- and :: Foldable t => t Bool -> Bool
- or :: Foldable t => t Bool -> Bool
- any :: Foldable t => (a -> Bool) -> t a -> Bool
- all :: Foldable t => (a -> Bool) -> t a -> Bool
- concat :: Foldable t => t [a] -> [a]
- concatMap :: Foldable t => (a -> [b]) -> t a -> [b]
- 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])
- notElem :: (Foldable t, Eq a) => a -> t 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 :: TYPE Lifted -> TYPE Lifted
- 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
Standard types, classes and related functions
Basic data types
data Bool :: TYPE Lifted Source
Instances
| Bounded Bool | |
| Enum Bool | |
| Eq Bool | |
| Data Bool | |
| Ord Bool | |
| Read Bool | |
| Show Bool | |
| Ix Bool | |
| Generic Bool | |
| FiniteBits Bool | |
| Bits Bool | |
| Storable Bool | |
| type Rep Bool = D1 (MetaData "Bool" "GHC.Types" "ghc-prim" False) ((:+:) (C1 (MetaCons "False" PrefixI False) U1) (C1 (MetaCons "True" PrefixI False) U1)) | |
| type (==) Bool a b | |
The Maybe type encapsulates an optional value. A value of type
Maybe aa (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.
Instances
maybe :: b -> (a -> b) -> Maybe a -> b Source
The maybe function takes a default value, a function, and a Maybe
value. If the Maybe value is Nothing, the function returns the
default value. Otherwise, it applies the function to the value inside
the Just and returns the result.
Examples
Basic usage:
>>>maybe False odd (Just 3)True
>>>maybe False odd NothingFalse
Read an integer from a string using readMaybe. If we succeed,
return twice the integer; that is, apply (*2) to it. If instead
we fail to parse an integer, return 0 by default:
>>>import Text.Read ( readMaybe )>>>maybe 0 (*2) (readMaybe "5")10>>>maybe 0 (*2) (readMaybe "")0
Apply show to a Maybe Int. If we have Just n, we want to show
the underlying Int n. But if we have Nothing, we return the
empty string instead of (for example) "Nothing":
>>>maybe "" show (Just 5)"5">>>maybe "" show Nothing""
The Either type represents values with two possibilities: a value of
type Either a bLeft aRight 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").
Examples
The type Either String IntString or an Int. The Left constructor can be used only on
Strings, and the Right constructor can be used only on Ints:
>>>let s = Left "foo" :: Either String Int>>>sLeft "foo">>>let n = Right 3 :: Either String Int>>>nRight 3>>>:type ss :: Either String Int>>>:type nn :: Either String Int
The fmap from our Functor instance will ignore Left values, but
will apply the supplied function to values contained in a Right:
>>>let s = Left "foo" :: Either String Int>>>let n = Right 3 :: Either String Int>>>fmap (*2) sLeft "foo">>>fmap (*2) nRight 6
The Monad instance for Either allows us to chain together multiple
actions which may fail, and fail overall if any of the individual
steps failed. First we'll write a function that can either parse an
Int from a Char, or fail.
>>>import Data.Char ( digitToInt, isDigit )>>>:{let parseEither :: Char -> Either String Int parseEither c | isDigit c = Right (digitToInt c) | otherwise = Left "parse error">>>:}
The following should work, since both '1' and '2' can be
parsed as Ints.
>>>:{let parseMultiple :: Either String Int parseMultiple = do x <- parseEither '1' y <- parseEither '2' return (x + y)>>>:}
>>>parseMultipleRight 3
But the following should fail overall, since the first operation where
we attempt to parse 'm' as an Int will fail:
>>>:{let parseMultiple :: Either String Int parseMultiple = do x <- parseEither 'm' y <- parseEither '2' return (x + y)>>>:}
>>>parseMultipleLeft "parse error"
Instances
either :: (a -> c) -> (b -> c) -> Either a b -> c Source
Case analysis for the Either type.
If the value is Left aa;
if it is Right bb.
Examples
We create two values of type Either String IntLeft constructor and another using the Right constructor. Then
we apply "either" the length function (if we have a String)
or the "times-two" function (if we have an Int):
>>>let s = Left "foo" :: Either String Int>>>let n = Right 3 :: Either String Int>>>either length (*2) s3>>>either length (*2) n6
data Ordering :: TYPE Lifted Source
Instances
| Bounded Ordering | |
| Enum Ordering | |
| Eq Ordering | |
| Data Ordering | |
| Ord Ordering | |
| Read Ordering | |
| Show Ordering | |
| Ix Ordering | |
| Generic Ordering | |
| Semigroup Ordering | |
| Monoid Ordering | |
| type Rep Ordering = D1 (MetaData "Ordering" "GHC.Types" "ghc-prim" False) ((:+:) (C1 (MetaCons "LT" PrefixI False) U1) ((:+:) (C1 (MetaCons "EQ" PrefixI False) U1) (C1 (MetaCons "GT" PrefixI False) U1))) | |
| type (==) Ordering a b | |
data Char :: TYPE Lifted Source
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).
Instances
| Bounded Char | |
| Enum Char | |
| Eq Char | |
| Data Char | |
| Ord Char | |
| Read Char | |
| Show Char | |
| Ix Char | |
| Storable Char | |
| IsChar Char | |
| PrintfArg Char | |
| Eq (URec Char p) | |
| Ord (URec Char p) | |
| Show (URec Char p) | |
| Generic (URec Char p) | |
| data URec Char = UChar {} | Used for marking occurrences of |
| type Rep (URec Char p) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UChar" PrefixI True) (S1 (MetaSel (Just Symbol "uChar#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) UChar)) | |
Tuples
curry :: ((a, b) -> c) -> a -> b -> c Source
curry converts an uncurried function to a curried function.
uncurry :: (a -> b -> c) -> (a, b) -> c Source
uncurry converts a curried function to a function on pairs.
Basic type classes
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.
Instances
| 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 TypeRep | |
| Eq () | |
| Eq TyCon | |
| Eq BigNat | |
| Eq GeneralCategory | |
| Eq Number | |
| Eq Lexeme | |
| Eq SomeSymbol | |
| Eq SomeNat | |
| Eq DecidedStrictness | |
| Eq SourceStrictness | |
| Eq SourceUnpackedness | |
| Eq Associativity | |
| Eq Fixity | |
| Eq Any | |
| Eq All | |
| Eq IOMode | |
| Eq Fingerprint | |
| Eq ArithException | |
| Eq ErrorCall | |
| Eq IOException | |
| Eq MaskingState | |
| Eq CUIntMax | |
| Eq CIntMax | |
| Eq CUIntPtr | |
| Eq CIntPtr | |
| Eq CSUSeconds | |
| Eq CUSeconds | |
| Eq CTime | |
| Eq CClock | |
| Eq CSigAtomic | |
| Eq CWchar | |
| Eq CSize | |
| Eq CPtrdiff | |
| Eq CDouble | |
| Eq CFloat | |
| Eq CULLong | |
| Eq CLLong | |
| Eq CULong | |
| Eq CLong | |
| Eq CUInt | |
| Eq CInt | |
| Eq CUShort | |
| Eq CShort | |
| Eq CUChar | |
| Eq CSChar | |
| Eq CChar | |
| Eq IntPtr | |
| Eq WordPtr | |
| Eq BufferState | |
| Eq CodingProgress | |
| Eq SeekMode | |
| Eq IODeviceType | |
| Eq NewlineMode | |
| Eq Newline | |
| Eq BufferMode | |
| Eq Handle | |
| Eq IOErrorType | |
| Eq ExitCode | |
| Eq ArrayException | |
| Eq AsyncException | |
| Eq Errno | |
| Eq Fd | |
| Eq CRLim | |
| Eq CTcflag | |
| Eq CSpeed | |
| Eq CCc | |
| Eq CUid | |
| Eq CNlink | |
| Eq CGid | |
| Eq CSsize | |
| Eq CPid | |
| Eq COff | |
| Eq CMode | |
| Eq CIno | |
| Eq CDev | |
| Eq Lifetime | |
| Eq Event | |
| Eq ThreadStatus | |
| Eq BlockReason | |
| Eq ThreadId | |
| Eq FdKey | |
| Eq TimeoutKey | |
| Eq HandlePosn | |
| Eq Version | |
| Eq Fixity | |
| Eq ConstrRep | |
| Eq DataRep | |
| Eq Constr | Equality of constructors |
| Eq Natural | |
| Eq SpecConstrAnnotation | |
| Eq Void | |
| Eq Unique | |
| Eq a => Eq [a] | |
| Eq a => Eq (Maybe a) | |
| Eq a => Eq (Ratio a) | |
| Eq (StablePtr a) | |
| Eq (Ptr a) | |
| Eq (FunPtr a) | |
| Eq (U1 p) | |
| Eq p => Eq (Par1 p) | |
| Eq (MVar a) | |
| Eq a => Eq (Down a) | |
| Eq a => Eq (Last a) | |
| Eq a => Eq (First a) | |
| Eq a => Eq (Product a) | |
| Eq a => Eq (Sum a) | |
| Eq a => Eq (Dual a) | |
| Eq (IORef a) | |
| Eq (ForeignPtr a) | |
| Eq (TVar a) | |
| Eq a => Eq (ZipList a) | |
| Eq (Chan a) | |
| Eq a => Eq (Complex a) | |
| Eq (Fixed a) | |
| Eq a => Eq (NonEmpty a) | |
| Eq a => Eq (Option a) | |
| Eq m => Eq (WrappedMonoid m) | |
| Eq a => Eq (Last a) | |
| Eq a => Eq (First a) | |
| Eq a => Eq (Max a) | |
| Eq a => Eq (Min a) | |
| Eq a => Eq (Identity a) | |
| Eq (StableName a) | |
| (Eq a, Eq b) => Eq (Either a b) | |
| Eq (f p) => Eq (Rec1 f p) | |
| Eq (URec Char p) | |
| Eq (URec Double p) | |
| Eq (URec Float p) | |
| Eq (URec Int p) | |
| Eq (URec Word p) | |
| Eq (URec (Ptr ()) p) | |
| (Eq a, Eq b) => Eq (a, b) | |
| Eq (STRef s a) | |
| Eq (Proxy k s) | |
| Eq a => Eq (Arg a b) | |
| Eq c => Eq (K1 i c p) | |
| (Eq (f p), Eq (g p)) => Eq ((:+:) f g p) | |
| (Eq (f p), Eq (g p)) => Eq ((:*:) f g p) | |
| Eq (f (g p)) => Eq ((:.:) f g p) | |
| (Eq a, Eq b, Eq c) => Eq (a, b, c) | |
| Eq ((:~:) k a b) | |
| Eq (Coercion k a b) | |
| Eq (f a) => Eq (Alt k f a) | |
| Eq a => Eq (Const k a b) | |
| Eq (f p) => Eq (M1 i c f p) | |
| (Eq a, Eq b, Eq c, Eq d) => Eq (a, b, c, d) | |
| (Eq1 f, Eq1 g, Eq a) => Eq (Product (TYPE Lifted) f g a) | |
| (Eq1 f, Eq1 g, Eq a) => Eq (Sum (TYPE Lifted) f g a) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e) => Eq (a, b, c, d, e) | |
| (Eq1 f, Eq1 g, Eq a) => Eq (Compose (TYPE Lifted) (TYPE Lifted) f g a) | |
| (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) | |
class Eq a => Ord a where Source
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.
Instances
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
succmaxBoundpredminBound fromEnumandtoEnumshould give a runtime error if the result value is not representable in the result type. For example,toEnum7 ::BoolenumFromandenumFromThenshould 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 = minBoundMethods
the successor of a value. For numeric types, succ adds 1.
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.
Used in Haskell's translation of [n..].
enumFromThen :: a -> a -> [a] Source
Used in Haskell's translation of [n,n'..].
enumFromTo :: a -> a -> [a] Source
Used in Haskell's translation of [n..m].
enumFromThenTo :: a -> a -> a -> [a] Source
Used in Haskell's translation of [n,n'..m].
Instances
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.
Instances
Numbers
Numeric types
data Int :: TYPE Lifted Source
A fixed-precision integer type with at least the range [-2^29 .. 2^29-1].
The exact range for a given implementation can be determined by using
minBound and maxBound from the Bounded class.
Instances
| Bounded Int | |
| Enum Int | |
| Eq Int | |
| Integral Int | |
| Data Int | |
| Num Int | |
| Ord Int | |
| Read Int | |
| Real Int | |
| Show Int | |
| Ix Int | |
| FiniteBits Int | |
| Bits Int | |
| Storable Int | |
| PrintfArg Int | |
| Eq (URec Int p) | |
| Ord (URec Int p) | |
| Show (URec Int p) | |
| Generic (URec Int p) | |
| data URec Int = UInt {} | Used for marking occurrences of |
| type Rep (URec Int p) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UInt" PrefixI True) (S1 (MetaSel (Just Symbol "uInt#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) UInt)) | |
data Float :: TYPE Lifted Source
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.
Instances
| Eq Float | |
| Floating Float | |
| Data Float | |
| Ord Float | |
| Read Float | |
| RealFloat Float | |
| Storable Float | |
| PrintfArg Float | |
| Eq (URec Float p) | |
| Ord (URec Float p) | |
| Show (URec Float p) | |
| Generic (URec Float p) | |
| data URec Float = UFloat {} | Used for marking occurrences of |
| type Rep (URec Float p) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UFloat" PrefixI True) (S1 (MetaSel (Just Symbol "uFloat#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) UFloat)) | |
data Double :: TYPE Lifted Source
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.
Instances
| Eq Double | |
| Floating Double | |
| Data Double | |
| Ord Double | |
| Read Double | |
| RealFloat Double | |
| Storable Double | |
| PrintfArg Double | |
| Eq (URec Double p) | |
| Ord (URec Double p) | |
| Show (URec Double p) | |
| Generic (URec Double p) | |
| data URec Double = UDouble {} | Used for marking occurrences of |
| type Rep (URec Double p) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UDouble" PrefixI True) (S1 (MetaSel (Just Symbol "uDouble#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) UDouble)) | |
data Word :: TYPE Lifted Source
Instances
| Bounded Word | |
| Enum Word | |
| Eq Word | |
| Integral Word | |
| Data Word | |
| Num Word | |
| Ord Word | |
| Read Word | |
| Real Word | |
| Show Word | |
| Ix Word | |
| FiniteBits Word | |
| Bits Word | |
| Storable Word | |
| PrintfArg Word | |
| Eq (URec Word p) | |
| Ord (URec Word p) | |
| Show (URec Word p) | |
| Generic (URec Word p) | |
| data URec Word = UWord {} | Used for marking occurrences of |
| type Rep (URec Word p) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UWord" PrefixI True) (S1 (MetaSel (Just Symbol "uWord#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) UWord)) | |
Numeric type classes
Basic numeric class.
Methods
(+), (-), (*) :: a -> a -> a infixl 7 *infixl 6 +, - Source
Unary negation.
Absolute value.
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 Source
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
Instances
| Num Int | |
| Num Int8 | |
| Num Int16 | |
| Num Int32 | |
| Num Int64 | |
| Num Integer | |
| Num Word | |
| Num Word8 | |
| Num Word16 | |
| Num Word32 | |
| Num Word64 | |
| Num CUIntMax | |
| Num CIntMax | |
| Num CUIntPtr | |
| Num CIntPtr | |
| Num CSUSeconds | |
| Num CUSeconds | |
| Num CTime | |
| Num CClock | |
| Num CSigAtomic | |
| Num CWchar | |
| Num CSize | |
| Num CPtrdiff | |
| Num CDouble | |
| Num CFloat | |
| Num CULLong | |
| Num CLLong | |
| Num CULong | |
| Num CLong | |
| Num CUInt | |
| Num CInt | |
| Num CUShort | |
| Num CShort | |
| Num CUChar | |
| Num CSChar | |
| Num CChar | |
| Num IntPtr | |
| Num WordPtr | |
| Num Fd | |
| Num CRLim | |
| Num CTcflag | |
| Num CSpeed | |
| Num CCc | |
| Num CUid | |
| Num CNlink | |
| Num CGid | |
| Num CSsize | |
| Num CPid | |
| Num COff | |
| Num CMode | |
| Num CIno | |
| Num CDev | |
| Num Natural | |
| Integral a => Num (Ratio a) | |
| Num a => Num (Product a) | |
| Num a => Num (Sum a) | |
| RealFloat a => Num (Complex a) | |
| HasResolution a => Num (Fixed a) | |
| Num a => Num (Max a) | |
| Num a => Num (Min a) | |
| Num (f a) => Num (Alt k f a) | |
class (Num a, Ord a) => Real a where Source
Minimal complete definition
Methods
toRational :: a -> Rational Source
the rational equivalent of its real argument with full precision
Instances
class (Real a, Enum a) => Integral a where Source
Integral numbers, supporting integer division.
Methods
quot :: a -> a -> a infixl 7 Source
integer division truncated toward zero
rem :: a -> a -> a infixl 7 Source
integer remainder, satisfying
(x `quot` y)*y + (x `rem` y) == x
div :: a -> a -> a infixl 7 Source
integer division truncated toward negative infinity
mod :: a -> a -> a infixl 7 Source
integer modulus, satisfying
(x `div` y)*y + (x `mod` y) == x
quotRem :: a -> a -> (a, a) Source
divMod :: a -> a -> (a, a) Source
toInteger :: a -> Integer Source
conversion to Integer
Instances
class Num a => Fractional a where Source
Fractional numbers, supporting real division.
Minimal complete definition
fromRational, (recip | (/))
Methods
(/) :: a -> a -> a infixl 7 Source
fractional division
reciprocal fraction
fromRational :: Rational -> a Source
Conversion from a Rational (that is Ratio IntegerfromRational
to a value of type Rational, so such literals have type
(.Fractional a) => a
Instances
| Fractional CDouble | |
| Fractional CFloat | |
| Integral a => Fractional (Ratio a) | |
| RealFloat a => Fractional (Complex a) | |
| HasResolution a => Fractional (Fixed a) | |
class Fractional a => Floating a where Source
Trigonometric and hyperbolic functions and related functions.
Minimal complete definition
pi, exp, log, sin, cos, asin, acos, atan, sinh, cosh, asinh, acosh, atanh
class (Real a, Fractional a) => RealFrac a where Source
Extracting components of fractions.
Minimal complete definition
Methods
properFraction :: Integral b => a -> (b, a) Source
The function properFraction takes a real fractional number x
and returns a pair (n,f) such that x = n+f, and:
nis an integral number with the same sign asx; andfis 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 Source
truncate xx between zero and x
round :: Integral b => a -> b Source
round xx;
the even integer if x is equidistant between two integers
ceiling :: Integral b => a -> b Source
ceiling xx
floor :: Integral b => a -> b Source
floor xx
class (RealFrac a, Floating a) => RealFloat a where Source
Efficient, machine-independent access to the components of a floating-point number.
Minimal complete definition
floatRadix, floatDigits, floatRange, decodeFloat, encodeFloat, isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE
Methods
floatRadix :: a -> Integer Source
a constant function, returning the radix of the representation
(often 2)
floatDigits :: a -> Int Source
a constant function, returning the number of digits of
floatRadix in the significand
floatRange :: a -> (Int, Int) Source
a constant function, returning the lowest and highest values the exponent may assume
decodeFloat :: a -> (Integer, Int) Source
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 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^dd is
the value of floatDigits xdecodeFloat 0 = (0,0)decodeFloat (-0.0) = (0,0)decodeFloat xisNaN xisInfinite xTrue.
encodeFloat :: Integer -> Int -> a Source
encodeFloat performs the inverse of decodeFloat in the
sense that for finite x with the exception of -0.0,
uncurry encodeFloat (decodeFloat x) = xencodeFloat 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.
exponent 0 = 0x,
exponent x = snd (decodeFloat x) + floatDigits xx is a finite floating-point number, it is equal in value to
significand x * b ^^ exponent xb is the
floating-point radix.
The behaviour is unspecified on infinite or NaN values.
significand :: a -> a Source
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 Source
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 Source
True if the argument is an IEEE infinity or negative infinity
isDenormalized :: a -> Bool Source
True if the argument is too small to be represented in
normalized format
isNegativeZero :: a -> Bool Source
True if the argument is an IEEE negative zero
True if the argument is an IEEE floating point number
a version of arctangent taking two real floating-point arguments.
For real floating x and y, atan2 y x(x,y). atan2 y x-pi,
pi]. It follows the Common Lisp semantics for the origin when
signed zeroes are supported. atan2 y 1y in a type
that is RealFloat, should return the same value as atan yatan2 is provided, but implementors
can provide a more accurate implementation.
Numeric functions
gcd :: Integral a => a -> a -> a Source
gcd x yx and y of which
every common factor of x and y is also a factor; for example
gcd 4 2 = 2gcd (-4) 6 = 2gcd 0 44. gcd 0 00.
(That is, the common divisor that is "greatest" in the divisibility
preordering.)
Note: Since for signed fixed-width integer types, abs minBound < 0minBound0 or minBound
lcm :: Integral a => a -> a -> a Source
lcm x yx and y divide.
(^) :: (Num a, Integral b) => a -> b -> a infixr 8 Source
raise a number to a non-negative integral power
(^^) :: (Fractional a, Integral b) => a -> b -> a infixr 8 Source
raise a number to an integral power
fromIntegral :: (Integral a, Num b) => a -> b Source
general coercion from integral types
realToFrac :: (Real a, Fractional b) => a -> b Source
general coercion to fractional types
Monoids
The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following laws:
mappend mempty x = x
mappend x mempty = x
mappend x (mappend y z) = mappend (mappend x y) z
mconcat =
foldrmappend mempty
The method names refer to the monoid of lists under concatenation, but there are many other instances.
Some types can be viewed as a monoid in more than one way,
e.g. both addition and multiplication on numbers.
In such cases we often define newtypes and make those instances
of Monoid, e.g. Sum and Product.
Methods
Identity of mappend
An associative operation
Fold a list using the monoid.
For most types, the default definition for mconcat will be
used, but the function is included in the class definition so
that an optimized version can be provided for specific types.
Instances
| Monoid Ordering | |
| Monoid () | |
| Monoid Any | |
| Monoid All | |
| Monoid Lifetime |
|
| Monoid Event | |
| Monoid [a] | |
| Monoid a => Monoid (Maybe a) | Lift a semigroup into |
| Monoid a => Monoid (IO a) | |
| Monoid (Last a) | |
| Monoid (First a) | |
| Num a => Monoid (Product a) | |
| Num a => Monoid (Sum a) | |
| Monoid (Endo a) | |
| Monoid a => Monoid (Dual a) | |
| Semigroup a => Monoid (Option a) | |
| Monoid m => Monoid (WrappedMonoid m) | |
| (Ord a, Bounded a) => Monoid (Max a) | |
| (Ord a, Bounded a) => Monoid (Min a) | |
| Monoid a => Monoid (Identity a) | |
| Monoid b => Monoid (a -> b) | |
| (Monoid a, Monoid b) => Monoid (a, b) | |
| Monoid (Proxy k s) | |
| (Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) | |
| Alternative f => Monoid (Alt (TYPE Lifted) f a) | |
| Monoid a => Monoid (Const k a b) | |
| (Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d) | |
| (Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e) | |
Monads and functors
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.
Minimal complete definition
Instances
| Functor [] | |
| Functor Maybe | |
| Functor IO | |
| Functor ReadP | |
| Functor ReadPrec | |
| Functor Last | |
| Functor First | |
| Functor Product | |
| Functor Sum | |
| Functor Dual | |
| Functor STM | |
| Functor Handler | |
| Functor ZipList | |
| Functor Complex | |
| Functor NonEmpty | |
| Functor Option | |
| Functor Last | |
| Functor First | |
| Functor Max | |
| Functor Min | |
| Functor Identity | |
| Functor ArgDescr | |
| Functor OptDescr | |
| Functor ArgOrder | |
| Functor ((->) r) | |
| Functor (Either a) | |
| Functor ((,) a) | |
| Functor (ST s) | |
| Functor (Proxy (TYPE Lifted)) | |
| Arrow a => Functor (ArrowMonad a) | |
| Monad m => Functor (WrappedMonad m) | |
| Functor (ST s) | |
| Functor (Arg a) | |
| Functor f => Functor (Alt (TYPE Lifted) f) | |
| Functor (Const (TYPE Lifted) m) | |
| Arrow a => Functor (WrappedArrow a b) | |
| (Functor f, Functor g) => Functor (Product (TYPE Lifted) f g) | |
| (Functor f, Functor g) => Functor (Sum (TYPE Lifted) f g) | |
| (Functor f, Functor g) => Functor (Compose (TYPE Lifted) (TYPE Lifted) f g) | |
(<$>) :: Functor f => (a -> b) -> f a -> f b infixl 4 Source
An infix synonym for fmap.
Examples
Convert from a Maybe IntMaybe Stringshow:
>>>show <$> NothingNothing>>>show <$> Just 3Just "3"
Convert from an Either Int IntEither IntString using show:
>>>show <$> Left 17Left 17>>>show <$> Right 17Right "17"
Double each element of a list:
>>>(*2) <$> [1,2,3][2,4,6]
Apply even to the second element of a pair:
>>>even <$> (2,2)(2,True)
class Functor f => Applicative f where Source
A functor with application, providing operations to
A minimal complete definition must include implementations of these functions satisfying the following laws:
- identity
pureid<*>v = v- composition
pure(.)<*>u<*>v<*>w = u<*>(v<*>w)- homomorphism
puref<*>purex =pure(f x)- interchange
u
<*>purey =pure($y)<*>u
The other methods have the following default definitions, which may be overridden with equivalent specialized implementations:
As a consequence of these laws, the Functor instance for f will satisfy
If f is also a Monad, it should satisfy
(which implies that pure and <*> satisfy the applicative functor laws).
Methods
Lift a value.
(<*>) :: f (a -> b) -> f a -> f b infixl 4 Source
Sequential application.
(*>) :: f a -> f b -> f b infixl 4 Source
Sequence actions, discarding the value of the first argument.
(<*) :: f a -> f b -> f a infixl 4 Source
Sequence actions, discarding the value of the second argument.
Instances
| Applicative [] | |
| Applicative Maybe | |
| Applicative IO | |
| Applicative ReadP | |
| Applicative ReadPrec | |
| Applicative Last | |
| Applicative First | |
| Applicative Product | |
| Applicative Sum | |
| Applicative Dual | |
| Applicative STM | |
| Applicative ZipList | |
| Applicative Complex | |
| Applicative NonEmpty | |
| Applicative Option | |
| Applicative Last | |
| Applicative First | |
| Applicative Max | |
| Applicative Min | |
| Applicative Identity | |
| Applicative ((->) a) | |
| Applicative (Either e) | |
| Monoid a => Applicative ((,) a) | |
| Applicative (ST s) | |
| Applicative (Proxy (TYPE Lifted)) | |
| Arrow a => Applicative (ArrowMonad a) | |
| Monad m => Applicative (WrappedMonad m) | |
| Applicative (ST s) | |
| Applicative f => Applicative (Alt (TYPE Lifted) f) | |
| Monoid m => Applicative (Const (TYPE Lifted) m) | |
| Arrow a => Applicative (WrappedArrow a b) | |
| (Applicative f, Applicative g) => Applicative (Product (TYPE Lifted) f g) | |
| (Applicative f, Applicative g) => Applicative (Compose (TYPE Lifted) (TYPE Lifted) f g) | |
class Applicative m => Monad m where Source
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.
Instances of Monad should satisfy the following laws:
Furthermore, the Monad and Applicative operations should relate as follows:
The above laws imply:
and that pure and (<*>) satisfy the applicative functor laws.
The instances of Monad for lists, Maybe and IO
defined in the Prelude satisfy these laws.
Minimal complete definition
Methods
(>>=) :: forall a b. m a -> (a -> m b) -> m b infixl 1 Source
Sequentially compose two actions, passing any value produced by the first as an argument to the second.
(>>) :: forall a b. m a -> m b -> m b infixl 1 Source
Sequentially compose two actions, discarding any value produced by the first, like sequencing operators (such as the semicolon) in imperative languages.
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.
As part of the MonadFail proposal (MFP), this function is moved
to its own class MonadFail (see Control.Monad.Fail for more
details). The definition here will be removed in a future
release.
Instances
| Monad [] | |
| Monad Maybe | |
| Monad IO | |
| Monad ReadP | |
| Monad ReadPrec | |
| Monad Last | |
| Monad First | |
| Monad Product | |
| Monad Sum | |
| Monad Dual | |
| Monad STM | |
| Monad Complex | |
| Monad NonEmpty | |
| Monad Option | |
| Monad Last | |
| Monad First | |
| Monad Max | |
| Monad Min | |
| Monad Identity | |
| Monad ((->) r) | |
| Monad (Either e) | |
| Monoid a => Monad ((,) a) | |
| Monad (ST s) | |
| Monad (Proxy (TYPE Lifted)) | |
| ArrowApply a => Monad (ArrowMonad a) | |
| Monad m => Monad (WrappedMonad m) | |
| Monad (ST s) | |
| Monad f => Monad (Alt (TYPE Lifted) f) | |
| (Monad f, Monad g) => Monad (Product (TYPE Lifted) f g) | |
sequence_ :: (Foldable t, Monad m) => t (m a) -> m () Source
Evaluate each monadic action in the structure from left to right,
and ignore the results. For a version that doesn't ignore the
results see sequence.
As of base 4.8.0.0, sequence_ is just sequenceA_, specialized
to Monad.
(=<<) :: Monad m => (a -> m b) -> m a -> m b infixr 1 Source
Same as >>=, but with the arguments interchanged.
Folds and traversals
Data structures that can be folded.
For example, given a data type
data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
a suitable instance would be
instance Foldable Tree where foldMap f Empty = mempty foldMap f (Leaf x) = f x foldMap f (Node l k r) = foldMap f l `mappend` f k `mappend` foldMap f r
This is suitable even for abstract types, as the monoid is assumed
to satisfy the monoid laws. Alternatively, one could define foldr:
instance Foldable Tree where foldr f z Empty = z foldr f z (Leaf x) = f x z foldr f z (Node l k r) = foldr f (f k (foldr f z r)) l
Foldable instances are expected to satisfy the following laws:
foldr f z t = appEndo (foldMap (Endo . f) t ) z
foldl f z t = appEndo (getDual (foldMap (Dual . Endo . flip f) t)) z
fold = foldMap id
sum, product, maximum, and minimum should all be essentially
equivalent to foldMap forms, such as
sum = getSum . foldMap Sum
but may be less defined.
If the type is also a Functor instance, it should satisfy
foldMap f = fold . fmap f
which implies that
foldMap f . fmap g = foldMap (f . g)
Methods
foldMap :: Monoid m => (a -> m) -> t a -> m Source
Map each element of the structure to a monoid, and combine the results.
foldr :: (a -> b -> b) -> b -> t a -> b Source
Right-associative fold of a structure.
In the case of lists, foldr, when 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)...)
Note that, since the head of the resulting expression is produced by
an application of the operator to the first element of the list,
foldr can produce a terminating expression from an infinite list.
For a general Foldable structure this should be semantically identical
to,
foldr f z =foldrf z .toList
foldl :: (b -> a -> b) -> b -> t a -> b Source
Left-associative fold of a structure.
In the case of lists, foldl, when 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
Note that to produce the outermost application of the operator the
entire input list must be traversed. This means that foldl' will
diverge if given an infinite list.
Also note that if you want an efficient left-fold, you probably want to
use foldl' instead of foldl. The reason for this is that latter does
not force the "inner" results (e.g. z in the above example)
before applying them to the operator (e.g. to f x1(). This results
in a thunk chain f x2)O(n) elements long, which then must be evaluated from
the outside-in.
For a general Foldable structure this should be semantically identical
to,
foldl f z =foldlf z .toList
foldr1 :: (a -> a -> a) -> t a -> a Source
A variant of foldr that has no base case,
and thus may only be applied to non-empty structures.
foldr1f =foldr1f .toList
foldl1 :: (a -> a -> a) -> t a -> a Source
A variant of foldl that has no base case,
and thus may only be applied to non-empty structures.
foldl1f =foldl1f .toList
Test whether the structure is empty. The default implementation is optimized for structures that are similar to cons-lists, because there is no general way to do better.
Returns the size/length of a finite structure as an Int. The
default implementation is optimized for structures that are similar to
cons-lists, because there is no general way to do better.
elem :: Eq a => a -> t a -> Bool infix 4 Source
Does the element occur in the structure?
maximum :: forall a. Ord a => t a -> a Source
The largest element of a non-empty structure.
minimum :: forall a. Ord a => t a -> a Source
The least element of a non-empty structure.
sum :: Num a => t a -> a Source
The sum function computes the sum of the numbers of a structure.
product :: Num a => t a -> a Source
The product function computes the product of the numbers of a
structure.
Instances
| Foldable [] | |
| Foldable Maybe | |
| Foldable Last | |
| Foldable First | |
| Foldable Product | |
| Foldable Sum | |
| Foldable Dual | |
| Foldable ZipList | |
| Foldable Complex | |
| Foldable NonEmpty | |
| Foldable Option | |
| Foldable Last | |
| Foldable First | |
| Foldable Max | |
| Foldable Min | |
| Foldable Identity | |
| Foldable (Either a) | |
| Foldable ((,) a) | |
| Foldable (Proxy (TYPE Lifted)) | |
| Foldable (Arg a) | |
| Foldable (Const (TYPE Lifted) m) | |
| (Foldable f, Foldable g) => Foldable (Product (TYPE Lifted) f g) | |
| (Foldable f, Foldable g) => Foldable (Sum (TYPE Lifted) f g) | |
| (Foldable f, Foldable g) => Foldable (Compose (TYPE Lifted) (TYPE Lifted) f g) | |
class (Functor t, Foldable t) => Traversable t where Source
Functors representing data structures that can be traversed from left to right.
A definition of traverse must satisfy the following laws:
- naturality
t .for every applicative transformationtraversef =traverse(t . f)t- identity
traverseIdentity = Identity- composition
traverse(Compose .fmapg . f) = Compose .fmap(traverseg) .traversef
A definition of sequenceA must satisfy the following laws:
- naturality
t .for every applicative transformationsequenceA=sequenceA.fmaptt- identity
sequenceA.fmapIdentity = Identity- composition
sequenceA.fmapCompose = Compose .fmapsequenceA.sequenceA
where an applicative transformation is a function
t :: (Applicative f, Applicative g) => f a -> g a
preserving the Applicative operations, i.e.
and the identity functor Identity and composition of functors Compose
are defined as
newtype Identity a = Identity a
instance Functor Identity where
fmap f (Identity x) = Identity (f x)
instance Applicative Identity where
pure x = Identity x
Identity f <*> Identity x = Identity (f x)
newtype Compose f g a = Compose (f (g a))
instance (Functor f, Functor g) => Functor (Compose f g) where
fmap f (Compose x) = Compose (fmap (fmap f) x)
instance (Applicative f, Applicative g) => Applicative (Compose f g) where
pure x = Compose (pure (pure x))
Compose f <*> Compose x = Compose ((<*>) <$> f <*> x)(The naturality law is implied by parametricity.)
Instances are similar to Functor, e.g. given a data type
data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
a suitable instance would be
instance Traversable Tree where traverse f Empty = pure Empty traverse f (Leaf x) = Leaf <$> f x traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r
This is suitable even for abstract types, as the laws for <*>
imply a form of associativity.
The superclass instances should satisfy the following:
- In the
Functorinstance,fmapshould be equivalent to traversal with the identity applicative functor (fmapDefault). - In the
Foldableinstance,foldMapshould be equivalent to traversal with a constant applicative functor (foldMapDefault).
Methods
traverse :: Applicative f => (a -> f b) -> t a -> f (t b) Source
Map each element of a structure to an action, evaluate these actions
from left to right, and collect the results. For a version that ignores
the results see traverse_.
sequenceA :: Applicative f => t (f a) -> f (t a) Source
Evaluate each action in the structure from left to right, and
and collect the results. For a version that ignores the results
see sequenceA_.
mapM :: Monad m => (a -> m b) -> t a -> m (t b) Source
Map each element of a structure to a monadic action, evaluate
these actions from left to right, and collect the results. For
a version that ignores the results see mapM_.
sequence :: Monad m => t (m a) -> m (t a) Source
Evaluate each monadic action in the structure from left to
right, and collect the results. For a version that ignores the
results see sequence_.
Instances
| Traversable [] | |
| Traversable Maybe | |
| Traversable Last | |
| Traversable First | |
| Traversable Product | |
| Traversable Sum | |
| Traversable Dual | |
| Traversable ZipList | |
| Traversable Complex | |
| Traversable NonEmpty | |
| Traversable Option | |
| Traversable Last | |
| Traversable First | |
| Traversable Max | |
| Traversable Min | |
| Traversable Identity | |
| Traversable (Either a) | |
| Traversable ((,) a) | |
| Traversable (Proxy (TYPE Lifted)) | |
| Traversable (Arg a) | |
| Traversable (Const (TYPE Lifted) m) | |
| (Traversable f, Traversable g) => Traversable (Product (TYPE Lifted) f g) | |
| (Traversable f, Traversable g) => Traversable (Sum (TYPE Lifted) f g) | |
| (Traversable f, Traversable g) => Traversable (Compose (TYPE Lifted) (TYPE Lifted) f g) | |
Miscellaneous functions
const x is a unary function which evaluates to x for all inputs.
For instance,
>>>map (const 42) [0..3][42,42,42,42]
flip :: (a -> b -> c) -> b -> a -> c Source
flip ff.
($) :: (a -> b) -> a -> b infixr 0 Source
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 map ($ 0) xszipWith ($) fs xs
until :: (a -> Bool) -> (a -> a) -> a -> a Source
until p ff until p holds.
error :: forall v. forall a. HasCallStack => [Char] -> a Source
error stops execution and displays an error message.
errorWithoutStackTrace :: forall v. forall a. [Char] -> a Source
A variant of error that does not produce a stack trace.
Since: 4.9.0.0
undefined :: forall v. forall a. HasCallStack => a Source
The value of seq a b is bottom if a is bottom, and
otherwise equal to b. seq is usually introduced to
improve performance by avoiding unneeded laziness.
A note on evaluation order: the expression seq a b does
not guarantee that a will be evaluated before b.
The only guarantee given by seq is that the both a
and b will be evaluated before seq returns a value.
In particular, this means that b may be evaluated before
a. If you need to guarantee a specific order of evaluation,
you must use the function pseq from the "parallel" package.
($!) :: (a -> b) -> a -> b infixr 0 Source
Strict (call-by-value) application operator. It takes a function and an argument, evaluates the argument to weak head normal form (WHNF), then calls the function with that value.
List operations
map :: (a -> b) -> [a] -> [b] Source
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 Source
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] Source
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]
Return all the elements of a list except the last one. The list must be non-empty.
null :: Foldable t => t a -> Bool Source
Test whether the structure is empty. The default implementation is optimized for structures that are similar to cons-lists, because there is no general way to do better.
length :: Foldable t => t a -> Int Source
Returns the size/length of a finite structure as an Int. The
default implementation is optimized for structures that are similar to
cons-lists, because there is no general way to do better.
(!!) :: [a] -> Int -> a infixl 9 Source
List index (subscript) operator, starting from 0.
It is an instance of the more general genericIndex,
which takes an index of any integral type.
reverse xs returns the elements of xs in reverse order.
xs must be finite.
Special folds
any :: Foldable t => (a -> Bool) -> t a -> Bool Source
Determines whether any element of the structure satisfies the predicate.
all :: Foldable t => (a -> Bool) -> t a -> Bool Source
Determines whether all elements of the structure satisfy the predicate.
concat :: Foldable t => t [a] -> [a] Source
The concatenation of all the elements of a container of lists.
concatMap :: Foldable t => (a -> [b]) -> t a -> [b] Source
Map a function over all the elements of a container and concatenate the resulting lists.
Building lists
Scans
Infinite lists
iterate :: (a -> a) -> a -> [a] Source
iterate f x returns an infinite list of repeated applications
of f to x:
iterate f x == [x, f x, f (f x), ...]
replicate :: Int -> a -> [a] Source
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 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 :: Int -> [a] -> [a] Source
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 :: Int -> [a] -> [a] Source
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 ( when take n xs, drop n xs)n is not _|_
(splitAt _|_ xs = _|_).
splitAt is an instance of the more general genericSplitAt,
in which n may be of any integral type.
takeWhile :: (a -> Bool) -> [a] -> [a] Source
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]) Source
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]) Source
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
lookup :: Eq a => a -> [(a, b)] -> Maybe b Source
lookup key assocs looks up a key in an association list.
Zipping and unzipping lists
unzip :: [(a, b)] -> ([a], [b]) Source
unzip transforms a list of pairs into a list of first components
and a list of second components.
Functions on strings
lines :: String -> [String] Source
lines breaks a string up into a list of strings at newline
characters. The resulting strings do not contain newlines.
Note that after splitting the string at newline characters, the last part of the string is considered a line even if it doesn't end with a newline. For example,
lines "" == [] lines "\n" == [""] lines "one" == ["one"] lines "one\n" == ["one"] lines "one\n\n" == ["one",""] lines "one\ntwo" == ["one","two"] lines "one\ntwo\n" == ["one","two"]
Thus lines ss.
words :: String -> [String] Source
words breaks a string up into a list of words, which were delimited
by white space.
Converting to and from String
Converting to String
Conversion of values to readable Strings.
Derived instances of Show have the following properties, which
are compatible with derived instances of Read:
- The result of
showis 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
showsPrecwill produce infix applications of the constructor. - the representation will be enclosed in parentheses if the
precedence of the top-level constructor in
xis less thand(associativity is ignored). Thus, ifdis0then the result is never surrounded in parentheses; ifdis11it is always surrounded in parentheses, unless it is an atomic expression. - If the constructor is defined using record syntax, then
showwill 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 = 5Note that right-associativity of :^: is ignored. For example,
show(Leaf 1 :^: Leaf 2 :^: Leaf 3)"Leaf 1 :^: (Leaf 2 :^: Leaf 3)".
Methods
showsPrec :: Int -> a -> ShowS Source
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.
Instances
showChar :: Char -> ShowS Source
utility function converting a Char to a show function that
simply prepends the character unchanged.
showString :: String -> ShowS Source
utility function converting a String to a show function that
simply prepends the string unchanged.
Converting from String
Parsing of Strings, producing values.
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
Readinstance 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
Readwill parse only the record-syntax form, and furthermore, the fields must be given in the same order as the original declaration. - The derived
Readinstance 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 = 5Note 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 = readListPrecDefaultMethods
readsPrec :: Int -> ReadS a Source
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.
Instances
| Read Bool | |
| Read Char | |
| Read Double | |
| Read Float | |
| Read Int | |
| Read Int8 | |
| Read Int16 | |
| Read Int32 | |
| Read Int64 | |
| Read Integer | |
| Read Ordering | |
| Read Word | |
| Read Word8 | |
| Read Word16 | |
| Read Word32 | |
| Read Word64 | |
| Read () | |
| Read GeneralCategory | |
| Read Lexeme | |
| Read SomeSymbol | |
| Read SomeNat | |
| Read DecidedStrictness | |
| Read SourceStrictness | |
| Read SourceUnpackedness | |
| Read Associativity | |
| Read Fixity | |
| Read Any | |
| Read All | |
| Read IOMode | |
| Read CUIntMax | |
| Read CIntMax | |
| Read CUIntPtr | |
| Read CIntPtr | |
| Read CSUSeconds | |
| Read CUSeconds | |
| Read CTime | |
| Read CClock | |
| Read CSigAtomic | |
| Read CWchar | |
| Read CSize | |
| Read CPtrdiff | |
| Read CDouble | |
| Read CFloat | |
| Read CULLong | |
| Read CLLong | |
| Read CULong | |
| Read CLong | |
| Read CUInt | |
| Read CInt | |
| Read CUShort | |
| Read CShort | |
| Read CUChar | |
| Read CSChar | |
| Read CChar | |
| Read IntPtr | |
| Read WordPtr | |
| Read SeekMode | |
| Read NewlineMode | |
| Read Newline | |
| Read BufferMode | |
| Read ExitCode | |
| Read Fd | |
| Read CRLim | |
| Read CTcflag | |
| Read CSpeed | |
| Read CCc | |
| Read CUid | |
| Read CNlink | |
| Read CGid | |
| Read CSsize | |
| Read CPid | |
| Read COff | |
| Read CMode | |
| Read CIno | |
| Read CDev | |
| Read GCStats | |
| Read Version | |
| Read Natural | |
| Read Void | Reading a |
| Read a => Read [a] | |
| Read a => Read (Maybe a) | |
| (Integral a, Read a) => Read (Ratio a) | |
| Read (U1 p) | |
| Read p => Read (Par1 p) | |
| Read a => Read (Down a) | |
| Read a => Read (Last a) | |
| Read a => Read (First a) | |
| Read a => Read (Product a) | |
| Read a => Read (Sum a) | |
| Read a => Read (Dual a) | |
| Read a => Read (ZipList a) | |
| Read a => Read (Complex a) | |
| HasResolution a => Read (Fixed a) | |
| Read a => Read (NonEmpty a) | |
| Read a => Read (Option a) | |
| Read m => Read (WrappedMonoid m) | |
| Read a => Read (Last a) | |
| Read a => Read (First a) | |
| Read a => Read (Max a) | |
| Read a => Read (Min a) | |
| Read a => Read (Identity a) | This instance would be equivalent to the derived instances of the
|
| (Read a, Read b) => Read (Either a b) | |
| Read (f p) => Read (Rec1 f p) | |
| (Read a, Read b) => Read (a, b) | |
| Read (Proxy k s) | |
| (Read a, Read b) => Read (Arg a b) | |
| Read c => Read (K1 i c p) | |
| (Read (f p), Read (g p)) => Read ((:+:) f g p) | |
| (Read (f p), Read (g p)) => Read ((:*:) f g p) | |
| Read (f (g p)) => Read ((:.:) f g p) | |
| (Read a, Read b, Read c) => Read (a, b, c) | |
| (~) k a b => Read ((:~:) k a b) | |
| Coercible k a b => Read (Coercion k a b) | |
| Read (f a) => Read (Alt k f a) | |
| Read a => Read (Const k a b) | This instance would be equivalent to the derived instances of the
|
| Read (f p) => Read (M1 i c f p) | |
| (Read a, Read b, Read c, Read d) => Read (a, b, c, d) | |
| (Read1 f, Read1 g, Read a) => Read (Product (TYPE Lifted) f g a) | |
| (Read1 f, Read1 g, Read a) => Read (Sum (TYPE Lifted) f g a) | |
| (Read a, Read b, Read c, Read d, Read e) => Read (a, b, c, d, e) | |
| (Read1 f, Read1 g, Read a) => Read (Compose (TYPE Lifted) (TYPE Lifted) f g a) | |
| (Read a, Read b, Read c, Read d, Read e, Read f) => Read (a, b, c, d, e, f) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g) => Read (a, b, c, d, e, f, g) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h) => Read (a, b, c, d, e, f, g, h) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i) => Read (a, b, c, d, e, f, g, h, i) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j) => Read (a, b, c, d, e, f, g, h, i, j) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k) => Read (a, b, c, d, e, f, g, h, i, j, k) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k, Read l) => Read (a, b, c, d, e, f, g, h, i, j, k, l) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k, Read l, Read m) => Read (a, b, c, d, e, f, g, h, i, j, k, l, m) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k, Read l, Read m, Read n) => Read (a, b, c, d, e, f, g, h, i, j, k, l, m, n) | |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k, Read l, Read m, Read n, Read o) => Read (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) | |
read :: Read a => String -> a Source
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
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 :: TYPE Lifted -> TYPE Lifted Source
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 do-notation
or the >> and >>= operations from the Monad class.
Simple I/O operations
Output functions
print :: Show a => a -> IO () Source
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 Source
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 () Source
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 Source
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 () Source
The computation writeFile file str function writes the string str,
to the file file.
appendFile :: FilePath -> String -> IO () Source
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 Source