Boolean "and", lazy in the second argument

Boolean "or", lazy in the second argument

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.

Examples

>>> maybe False odd (Just 3)
True

>>> maybe False odd Nothing
False

>>> import Text.Read ( readMaybe )
>>> maybe 0 (*2) (readMaybe "5")
10
>>> maybe 0 (*2) (readMaybe "")
0

>>> maybe "" show (Just 5)
"5"
>>> maybe "" show Nothing
""

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").

Examples

>>> let s = Left "foo" :: Either String Int
>>> s
Left "foo"
>>> let n = Right 3 :: Either String Int
>>> n
Right 3
>>> :type s
s :: Either String Int
>>> :type n
n :: Either String Int

>>> let s = Left "foo" :: Either String Int
>>> let n = Right 3 :: Either String Int
>>> fmap (*2) s
Left "foo"
>>> fmap (*2) n
Right 6

>>> import Data.Char ( digitToInt, isDigit )
>>> :{
    let parseEither :: Char -> Either String Int
        parseEither c
          | isDigit c = Right (digitToInt c)
          | otherwise = Left "parse error"
>>> :}

>>> :{
    let parseMultiple :: Either String Int
        parseMultiple = do
          x <- parseEither '1'
          y <- parseEither '2'
          return (x + y)
>>> :}

>>> parseMultiple
Right 3

>>> :{
    let parseMultiple :: Either String Int
        parseMultiple = do
          x <- parseEither 'm'
          y <- parseEither '2'
          return (x + y)
>>> :}

>>> parseMultiple
Left "parse error"

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.

Examples

>>> let s = Left "foo" :: Either String Int
>>> let n = Right 3 :: Either String Int
>>> either length (*2) s
3
>>> either length (*2) n
6

The character type Char is an enumeration whose values represent Unicode (or equivalently ISO/IEC 10646) code points (i.e. 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.

See Data.List for operations on lists.

Extract the first component of a pair.

Extract the second component of a pair.

curry converts an uncurried function to a curried function.

Examples

>>> curry fst 1 2
1

uncurry converts a curried function to a function on pairs.

Examples

>>> uncurry (+) (1,2)
3

>>> uncurry ($) (show, 1)
"1"

>>> map (uncurry max) [(1,2), (3,4), (6,8)]
[2,4,8]

Arbitrary precision integers. In contrast with fixed-size integral types such as Int, the Integer type represents the entire infinite range of integers.

Integers are stored in a kind of sign-magnitude form, hence do not expect two's complement form when using bit operations.

If the value is small (fit into an Int), IS constructor is used. Otherwise Integer and IN constructors are used to store a BigNat representing respectively the positive or the negative value magnitude.

Invariant: Integer and IN are used iff value doesn't fit in IS

fmap is used to apply a function of type (a -> b) to a value of type f a, where f is a functor, to produce a value of type f b. Note that for any type constructor with more than one parameter (e.g., Either), only the last type parameter can be modified with fmap (e.g., b in `Either a b`).

Some type constructors with two parameters or more have a Bifunctor instance that allows both the last and the penultimate parameters to be mapped over.

Examples

>>> fmap show Nothing
Nothing
>>> fmap show (Just 3)
Just "3"

>>> fmap show (Left 17)
Left 17
>>> fmap show (Right 17)
Right "17"

>>> fmap (*2) [1,2,3]
[2,4,6]

>>> fmap even (2,2)
(2,True)

>>> fmap even ("hello", 1.0, 4)
("hello",1.0,True)

Identity function.

id x = x

const x is a unary function which evaluates to x for all inputs.

>>> const 42 "hello"
42

>>> map (const 42) [0..3]
[42,42,42,42]

Function composition.

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

>>> flip (++) "hello" "world"
"worldhello"

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.

Note that ($) is levity-polymorphic in its result type, so that foo $ True where foo :: Bool -> Int# is well-typed.

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 variant of error that does not produce a stack trace.

Since: base-4.9.0.0

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.

The value of seq a b is bottom if a is bottom, and otherwise equal to b. In other words, it evaluates the first argument a to weak head normal form (WHNF). 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.

\(\mathcal{O}(n)\). 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, ...]

>>> map (+1) [1, 2, 3]
[2,3,4]

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.

\(\mathcal{O}(n)\). 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]

>>> filter odd [1, 2, 3]
[1,3]

\(\mathcal{O}(1)\). Extract the first element of a list, which must be non-empty.

>>> head [1, 2, 3]
1
>>> head [1..]
1
>>> head []
*** Exception: Prelude.head: empty list

\(\mathcal{O}(n)\). Extract the last element of a list, which must be finite and non-empty.

>>> last [1, 2, 3]
3
>>> last [1..]
* Hangs forever *
>>> last []
*** Exception: Prelude.last: empty list

\(\mathcal{O}(1)\). Extract the elements after the head of a list, which must be non-empty.

>>> tail [1, 2, 3]
[2,3]
>>> tail [1]
[]
>>> tail []
*** Exception: Prelude.tail: empty list

\(\mathcal{O}(n)\). Return all the elements of a list except the last one. The list must be non-empty.

>>> init [1, 2, 3]
[1,2]
>>> init [1]
[]
>>> init []
*** Exception: Prelude.init: empty list

List index (subscript) operator, starting from 0. It is an instance of the more general genericIndex, which takes an index of any integral type.

>>> ['a', 'b', 'c'] !! 0
'a'
>>> ['a', 'b', 'c'] !! 2
'c'
>>> ['a', 'b', 'c'] !! 3
*** Exception: Prelude.!!: index too large
>>> ['a', 'b', 'c'] !! (-1)
*** Exception: Prelude.!!: negative index

reverse xs returns the elements of xs in reverse order. xs must be finite.

>>> reverse []
[]
>>> reverse [42]
[42]
>>> reverse [2,5,7]
[7,5,2]
>>> reverse [1..]
* Hangs forever *

\(\mathcal{O}(n)\). 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

>>> scanl (+) 0 [1..4]
[0,1,3,6,10]
>>> scanl (+) 42 []
[42]
>>> scanl (-) 100 [1..4]
[100,99,97,94,90]
>>> scanl (\reversedString nextChar -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']
["foo","afoo","bafoo","cbafoo","dcbafoo"]
>>> scanl (+) 0 [1..]
* Hangs forever *

\(\mathcal{O}(n)\). scanl1 is a variant of scanl that has no starting value argument:

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

>>> scanl1 (+) [1..4]
[1,3,6,10]
>>> scanl1 (+) []
[]
>>> scanl1 (-) [1..4]
[1,-1,-4,-8]
>>> scanl1 (&&) [True, False, True, True]
[True,False,False,False]
>>> scanl1 (||) [False, False, True, True]
[False,False,True,True]
>>> scanl1 (+) [1..]
* Hangs forever *

\(\mathcal{O}(n)\). scanr is the right-to-left dual of scanl. Note that the order of parameters on the accumulating function are reversed compared to scanl. Also note that

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

>>> scanr (+) 0 [1..4]
[10,9,7,4,0]
>>> scanr (+) 42 []
[42]
>>> scanr (-) 100 [1..4]
[98,-97,99,-96,100]
>>> scanr (\nextChar reversedString -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']
["abcdfoo","bcdfoo","cdfoo","dfoo","foo"]
>>> force $ scanr (+) 0 [1..]
*** Exception: stack overflow

\(\mathcal{O}(n)\). scanr1 is a variant of scanr that has no starting value argument.

>>> scanr1 (+) [1..4]
[10,9,7,4]
>>> scanr1 (+) []
[]
>>> scanr1 (-) [1..4]
[-2,3,-1,4]
>>> scanr1 (&&) [True, False, True, True]
[False,False,True,True]
>>> scanr1 (||) [True, True, False, False]
[True,True,False,False]
>>> force $ scanr1 (+) [1..]
*** Exception: stack overflow

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

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

Note that iterate is lazy, potentially leading to thunk build-up if the consumer doesn't force each iterate. See iterate' for a strict variant of this function.

>>> take 10 $ iterate not True
[True,False,True,False...
>>> take 10 $ iterate (+3) 42
[42,45,48,51,54,57,60,63...

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

>>> take 20 $ repeat 17
[17,17,17,17,17,17,17,17,17...

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.

>>> replicate 0 True
[]
>>> replicate (-1) True
[]
>>> replicate 4 True
[True,True,True,True]

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.

>>> cycle []
*** Exception: Prelude.cycle: empty list
>>> take 20 $ cycle [42]
[42,42,42,42,42,42,42,42,42,42...
>>> take 20 $ cycle [2, 5, 7]
[2,5,7,2,5,7,2,5,7,2,5,7...

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.

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).

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.

\(\mathcal{O}(\min(m,n))\). zip takes two lists and returns a list of corresponding pairs.

>>> zip [1, 2] ['a', 'b']
[(1,'a'),(2,'b')]

If one input list is shorter than the other, excess elements of the longer list are discarded, even if one of the lists is infinite:

>>> zip [1] ['a', 'b']
[(1,'a')]
>>> zip [1, 2] ['a']
[(1,'a')]
>>> zip [] [1..]
[]
>>> zip [1..] []
[]

zip is right-lazy:

>>> zip [] undefined
[]
>>> zip undefined []
*** Exception: Prelude.undefined
...

zip is capable of list fusion, but it is restricted to its first list argument and its resulting list.

zip3 takes three lists and returns a list of triples, analogous to zip. It is capable of list fusion, but it is restricted to its first list argument and its resulting list.

\(\mathcal{O}(\min(m,n))\). zipWith generalises zip by zipping with the function given as the first argument, instead of a tupling function.

zipWith (,) xs ys == zip xs ys
zipWith f [x1,x2,x3..] [y1,y2,y3..] == [f x1 y1, f x2 y2, f x3 y3..]

For example, zipWith (+) is applied to two lists to produce the list of corresponding sums:

>>> zipWith (+) [1, 2, 3] [4, 5, 6]
[5,7,9]

zipWith is right-lazy:

>>> let f = undefined
>>> zipWith f [] undefined
[]

zipWith is capable of list fusion, but it is restricted to its first list argument and its resulting list.

The zipWith3 function takes a function which combines three elements, as well as three lists and returns a list of the function applied to corresponding elements, analogous to zipWith. It is capable of list fusion, but it is restricted to its first list argument and its resulting list.

zipWith3 (,,) xs ys zs == zip3 xs ys zs
zipWith3 f [x1,x2,x3..] [y1,y2,y3..] [z1,z2,z3..] == [f x1 y1 z1, f x2 y2 z2, f x3 y3 z3..]

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

>>> unzip []
([],[])
>>> unzip [(1, 'a'), (2, 'b')]
([1,2],"ab")

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

>>> unzip3 []
([],[],[])
>>> unzip3 [(1, 'a', True), (2, 'b', False)]
([1,2],"ab",[True,False])

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 s contains at least as many elements as newlines in s.

words breaks a string up into a list of words, which were delimited by white space.

>>> words "Lorem ipsum\ndolor"
["Lorem","ipsum","dolor"]

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

>>> unlines ["Hello", "World", "!"]
"Hello\nWorld\n!\n"

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

>>> unwords ["Lorem", "ipsum", "dolor"]
"Lorem ipsum dolor"

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 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)".

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.

Write a character to the standard output device (same as hPutChar stdout).

Write a string to the standard output device (same as hPutStr stdout).

The same as putStr, but adds a newline character.

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]])

Read a character from the standard input device (same as hGetChar stdin).

Read a line from the standard input device (same as hGetLine stdin).

The getContents operation returns all user input as a single string, which is read lazily as it is needed (same as hGetContents stdin).

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.

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.

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.

The computation writeFile file str function writes the string str, to the file file.

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]])

The readIO function is similar to read except that it signals parse failure to the IO monad instead of terminating the program.

The readLn function combines getLine and readIO.

The Haskell 2010 type for exceptions in the IO monad. Any I/O operation may raise an IOException instead of returning a result. For a more general type of exception, including also those that arise in pure code, see Exception.

In Haskell 2010, this is an opaque type.

Raise an IOException in the IO monad.

Construct an IOException value with a string describing the error. The fail method of the IO instance of the Monad class raises a userError, thus:

instance Monad IO where
  ...
  fail s = ioError (userError s)