Application operator. This operator is redundant, since ordinary application (f x) means the same as (f $ x). However, $ has low, right-associative binding precedence, so it sometimes allows parentheses to be omitted; for example:

     f $ g $ h x  =  f (g (h x))

It is also useful in higher-order situations, such as map ($ 0) xs, or zipWith ($) fs xs.

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

Boolean "and"

Boolean "or"

morphism composition

Boolean "not"

otherwise is defined as the value True. It helps to make guards more readable. eg.

  f x | x < 0     = ...
      | otherwise = ...

Extract the first component of a pair.

Extract the second component of a pair.

the identity morphism

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.

Case analysis for the Either type. If the value is Left a, apply the first function to a; if it is Right b, apply the second function to b.

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

Constant function.

error stops execution and displays an error message.

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.

unzip transforms a list of pairs into a list of first components and a list of second components.

zipWith generalises zip by zipping with the function given as the first argument, instead of a tupling function. For example, zipWith (+) is applied to two lists to produce the list of corresponding sums.

or returns the disjunction of a Boolean list. For the result to be False, the list must be finite; True, however, results from a True value at a finite index of a finite or infinite list.

Write a string to stdout, followed by a newline.

elem is the list membership predicate, usually written in infix form, e.g., x `elem` xs. For the result to be False, the list must be finite; True, however, results from an element equal to x found at a finite index of a finite or infinite list.

uncurry converts a curried function to a function on pairs.

curry converts an uncurried function to a curried function.

Swap the components of a pair.

until p f yields the result of applying f until p holds.

asTypeOf is a type-restricted version of const. It is usually used as an infix operator, and its typing forces its first argument (which is usually overloaded) to have the same type as the second.

A special case of error. It is expected that compilers will recognize this and insert error messages which are more appropriate to the context in which undefined appears.

Evaluates its first argument to head normal form, and then returns its second argument as the result.

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.

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.

Minimal complete definition: either == or /=.

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 succ maxBound and pred minBound should result in a runtime error.
• fromEnum and toEnum should give a runtime error if the result value is not representable in the result type. For example, toEnum 7 :: Bool is an error.
• 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

Conversion of values to readable Strings.

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,

• show (Leaf 1 :^: Leaf 2 :^: Leaf 3) produces the string "Leaf 1 :^: (Leaf 2 :^: Leaf 3)".

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.

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.

Same as >>=, but with the arguments interchanged.

Basic numeric class.

Minimal complete definition: all except negate or (-)

Integral numbers, supporting integer division.

Minimal complete definition: quotRem and toInteger

Fractional numbers, supporting real division.

Minimal complete definition: fromRational and (recip or (/))

Trigonometric and hyperbolic functions and related functions.

Minimal complete definition: pi, exp, log, sin, cos, sinh, cosh, asin, acos, atan, asinh, acosh and atanh

Extracting components of fractions.

Minimal complete definition: properFraction

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

Minimal complete definition: all except exponent, significand, scaleFloat and atan2

The Maybe type encapsulates an optional value. A value of type Maybe a either contains a value of type a (represented as Just a), or it is empty (represented as 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.

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).

A value of type IO a is a computation which, when performed, does some I/O before returning a value of type 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.

The Either type represents values with two possibilities: a value of type Either a b is either Left a or 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").

A space-efficient representation of a Word8 vector, supporting many efficient operations. A ByteString contains 8-bit characters only.

Instances of Eq, Ord, Read, Show, Data, Typeable

A space efficient, packed, unboxed Unicode text type.

A Map from keys k to values a.

A map from keys to values. A map cannot contain duplicate keys; each key can map to at most one value.

A set of values a.

A set of values. A set cannot contain duplicate values.

Boxed vectors, supporting efficient slicing.

The class of types that can be converted to a hash value.

Minimal implementation: hash or hashWithSalt.

A Word is an unsigned integral type, with the same size as Int.

8-bit unsigned integer type

32-bit unsigned integer type

64-bit unsigned integer type

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.

32-bit signed integer type

64-bit signed integer type

Arbitrary-precision integers.

Arbitrary-precision rational numbers, represented as a ratio of two Integer values. A rational number may be constructed using the % operator.

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.

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.

raise a number to a non-negative integral power

raise a number to an integral power

the same as flip (-).

Because - is treated specially in the Haskell grammar, (- e) is not a section, but an application of prefix negation. However, (subtract exp) is equivalent to the disallowed section.

general coercion from integral types

general coercion to fractional types

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 = foldr mappend mempty

The method names refer to the monoid of lists under concatenation, but there are many other instances.

Minimal complete definition: mempty and mappend.

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.

Send the first component of the input through the argument arrow, and copy the rest unchanged to the output.

A mirror image of first.

The default definition may be overridden with a more efficient version if desired.

Split the input between the two argument arrows and combine their output. Note that this is in general not a functor.

The default definition may be overridden with a more efficient version if desired.

Fanout: send the input to both argument arrows and combine their output.

The default definition may be overridden with a more efficient version if desired.

The mapMaybe function is a version of map which can throw out elements. In particular, the functional argument returns something of type Maybe b. If this is Nothing, no element is added on to the result list. If it just Just b, then b is included in the result list.

The catMaybes function takes a list of Maybes and returns a list of all the Just values.

The fromMaybe function takes a default value and and Maybe value. If the Maybe is Nothing, it returns the default values; otherwise, it returns the value contained in the Maybe.

Partitions a list of Either into two lists All the Left elements are extracted, in order, to the first component of the output. Similarly the Right elements are extracted to the second component of the output.

(*) `on` f = \x y -> f x * f y.

Typical usage: sortBy (compare `on` fst).

Algebraic properties:

• (*) `on` id = (*) (if (*) ∉ {⊥, const ⊥})

• ((*) `on` f) `on` g = (*) `on` (f . g)

• flip on f . flip on g = flip on (g . f)

 comparing p x y = compare (p x) (p y)

Useful combinator for use in conjunction with the xxxBy family of functions from Data.List, for example:

   ... sortBy (comparing fst) ...

A functor with application, providing operations to

• embed pure expressions (pure), and
• sequence computations and combine their results (<*>).

A minimal complete definition must include implementations of these functions satisfying the following laws:

identity

pure id <*> v = v

composition

pure (.) <*> u <*> v <*> w = u <*> (v <*> w)

homomorphism

pure f <*> pure x = pure (f x)

interchange

u <*> pure y = pure ($ y) <*> u

The other methods have the following default definitions, which may be overridden with equivalent specialized implementations:

      u *> v = pure (const id) <*> u <*> v
      u <* v = pure const <*> u <*> v

As a consequence of these laws, the Functor instance for f will satisfy

      fmap f x = pure f <*> x

If f is also a Monad, it should satisfy pure = return and (<*>) = ap (which implies that pure and <*> satisfy the applicative functor laws).

An infix synonym for fmap.

Left-to-right Kleisli composition of monads.

Lift a computation from the argument monad to the constructed monad.

Monads in which IO computations may be embedded. Any monad built by applying a sequence of monad transformers to the IO monad will be an instance of this class.

Instances should satisfy the following laws, which state that liftIO is a transformer of monads:

• liftIO . return = return

• liftIO (m >>= f) = liftIO m >>= (liftIO . f)

Lift a computation from the IO monad.

Any type that you wish to throw or catch as an exception must be an instance of the Exception class. The simplest case is a new exception type directly below the root:

 data MyException = ThisException | ThatException
     deriving (Show, Typeable)

 instance Exception MyException

The default method definitions in the Exception class do what we need in this case. You can now throw and catch ThisException and ThatException as exceptions:

*Main> throw ThisException `catch` \e -> putStrLn ("Caught " ++ show (e :: MyException))
Caught ThisException

In more complicated examples, you may wish to define a whole hierarchy of exceptions:

 ---------------------------------------------------------------------
 -- Make the root exception type for all the exceptions in a compiler

 data SomeCompilerException = forall e . Exception e => SomeCompilerException e
     deriving Typeable

 instance Show SomeCompilerException where
     show (SomeCompilerException e) = show e

 instance Exception SomeCompilerException

 compilerExceptionToException :: Exception e => e -> SomeException
 compilerExceptionToException = toException . SomeCompilerException

 compilerExceptionFromException :: Exception e => SomeException -> Maybe e
 compilerExceptionFromException x = do
     SomeCompilerException a <- fromException x
     cast a

 ---------------------------------------------------------------------
 -- Make a subhierarchy for exceptions in the frontend of the compiler

 data SomeFrontendException = forall e . Exception e => SomeFrontendException e
     deriving Typeable

 instance Show SomeFrontendException where
     show (SomeFrontendException e) = show e

 instance Exception SomeFrontendException where
     toException = compilerExceptionToException
     fromException = compilerExceptionFromException

 frontendExceptionToException :: Exception e => e -> SomeException
 frontendExceptionToException = toException . SomeFrontendException

 frontendExceptionFromException :: Exception e => SomeException -> Maybe e
 frontendExceptionFromException x = do
     SomeFrontendException a <- fromException x
     cast a

 ---------------------------------------------------------------------
 -- Make an exception type for a particular frontend compiler exception

 data MismatchedParentheses = MismatchedParentheses
     deriving (Typeable, Show)

 instance Exception MismatchedParentheses where
     toException   = frontendExceptionToException
     fromException = frontendExceptionFromException

We can now catch a MismatchedParentheses exception as MismatchedParentheses, SomeFrontendException or SomeCompilerException, but not other types, e.g. IOException:

*Main> throw MismatchedParentheses catch e -> putStrLn ("Caught " ++ show (e :: MismatchedParentheses))
Caught MismatchedParentheses
*Main> throw MismatchedParentheses catch e -> putStrLn ("Caught " ++ show (e :: SomeFrontendException))
Caught MismatchedParentheses
*Main> throw MismatchedParentheses catch e -> putStrLn ("Caught " ++ show (e :: SomeCompilerException))
Caught MismatchedParentheses
*Main> throw MismatchedParentheses catch e -> putStrLn ("Caught " ++ show (e :: IOException))
*** Exception: MismatchedParentheses

The SomeException type is the root of the exception type hierarchy. When an exception of type e is thrown, behind the scenes it is encapsulated in a SomeException.

A variant of throw that can only be used within the IO monad.

Although throwIO has a type that is an instance of the type of throw, the two functions are subtly different:

 throw e   `seq` x  ===> throw e
 throwIO e `seq` x  ===> x

The first example will cause the exception e to be raised, whereas the second one won't. In fact, throwIO will only cause an exception to be raised when it is used within the IO monad. The throwIO variant should be used in preference to throw to raise an exception within the IO monad because it guarantees ordering with respect to other IO operations, whereas throw does not.

An alias for append.

An alias for addExtension.

Get whether a FilePath’s last extension is the predicate.

Retrieve a FilePath’s basename component.

 basename "foo/bar.txt" == "bar"

Retrieve a FilePath’s filename component.

 filename "foo/bar.txt" == "bar.txt"

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]])