Boolean "and"

Boolean "or"

Boolean "not"

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

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

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 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.

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").

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.

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 String is a list of characters. String constants in Haskell are values of type String.

Extract the first component of a pair.

Extract the second component of a pair.

curry converts an uncurried function to a curried function.

uncurry converts a curried function to a function on pairs.

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 /=.

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.

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

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.

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.

Arbitrary-precision integers.

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.

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

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 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.

gcd x y is the non-negative factor of both x and y of which every common factor of x and y is also a factor; for example gcd 4 2 = 2, gcd (-4) 6 = 2, gcd 0 4 = 4. gcd 0 0 = 0. (That is, the common divisor that is "greatest" in the divisibility preordering.)

Note: Since for signed fixed-width integer types, abs minBound < 0, the result may be negative if one of the arguments is minBound (and necessarily is if the other is 0 or minBound) for such types.

lcm x y is the smallest positive integer that both x and y divide.

raise a number to a non-negative integral power

raise a number to an integral power

general coercion from integral types

general coercion to fractional types

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.

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.

mapM f is equivalent to sequence . map f.

mapM_ f is equivalent to sequence_ . map f.

Evaluate each action in the sequence from left to right, and collect the results.

Evaluate each action in the sequence from left to right, and ignore the results.

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

Identity function.

Constant function.

Function composition.

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

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.

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.

error stops execution and displays an error message.

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.

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

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, ...]

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, 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]

Extract the first element of a list, which must be non-empty.

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

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

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

Test whether a list is 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.

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.

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 is a variant of foldl that has no starting value argument, and thus must be applied to non-empty lists.

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 is a variant of foldr that has no starting value argument, and thus must be applied to non-empty lists.

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

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.

Applied to a predicate and a list, any determines if any element of the list satisfies the predicate. For the result to be False, the list must be finite; True, however, results from a True value for the predicate applied to an element at a finite index of a finite or infinite list.

Applied to a predicate and a list, all determines if all elements of the list satisfy the predicate. For the result to be True, the list must be finite; False, however, results from a False value for the predicate applied to an element at a finite index of a finite or infinite list.

The sum function computes the sum of a finite list of numbers.

The product function computes the product of a finite list of numbers.

Concatenate a list of lists.

Map a function over a list and concatenate the results.

maximum returns the maximum value from a list, which must be non-empty, finite, and of an ordered type. It is a special case of maximumBy, which allows the programmer to supply their own comparison function.

minimum returns the minimum value from a list, which must be non-empty, finite, and of an ordered type. It is a special case of minimumBy, which allows the programmer to supply their own comparison function.

scanl is similar to foldl, but returns a list of successive reduced values from the left:

scanl f z [x1, x2, ...] == [z, z `f` x1, (z `f` x1) `f` x2, ...]

Note that

last (scanl f z xs) == foldl f z xs.

scanl1 is a variant of scanl that has no starting value argument:

scanl1 f [x1, x2, ...] == [x1, x1 `f` x2, ...]

scanr is the right-to-left dual of scanl. Note that

head (scanr f z xs) == foldr f z xs.

scanr1 is a variant of scanr that has no starting value argument.

iterate f x returns an infinite list of repeated applications of f to x:

iterate f x == [x, f x, f (f x), ...]

repeat x is an infinite list, with x the value of every element.

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.

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 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) when n is not _|_ (splitAt _|_ xs = _|_). splitAt is an instance of the more general genericSplitAt, in which n may be of any integral type.

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] == []

dropWhile p xs returns the suffix remaining after takeWhile p xs:

dropWhile (< 3) [1,2,3,4,5,1,2,3] == [3,4,5,1,2,3]
dropWhile (< 9) [1,2,3] == []
dropWhile (< 0) [1,2,3] == [1,2,3]

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])

span p xs is equivalent to (takeWhile p xs, dropWhile p xs)

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],[])

break p is equivalent to span (not . p).

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.

notElem is the negation of elem.

lookup key assocs looks up a key in an association list.

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 takes three lists and returns a list of triples, analogous to zip.

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.

The zipWith3 function takes a function which combines three elements, as well as three lists and returns a list of their point-wise combination, analogous to zipWith.

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

The unzip3 function takes a list of triples and returns three lists, analogous to unzip.

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.

unlines is an inverse operation to lines. It joins lines, after appending a terminating newline to each.

unwords is an inverse operation to words. It joins words with separating spaces.

The shows functions return a function that prepends the output String to an existing String. This allows constant-time concatenation of results using function composition.

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)".