NumericPrelude.Base

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

WARNING: This function is partial, and should only be used if you are sure that the indexing will not fail. Otherwise, use !?.

WARNING: This function takes linear time in the index.

Examples

>>> ['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

($) is the function application operator.

Applying ($) to a function f and an argument x gives the same result as applying f to x directly. The definition is akin to this:

($) :: (a -> b) -> a -> b
($) f x = f x

This is id specialized from a -> a to (a -> b) -> (a -> b) which by the associativity of (->) is the same as (a -> b) -> a -> b.

On the face of it, this may appear pointless! But it's actually one of the most useful and important operators in Haskell.

The order of operations is very different between ($) and normal function application. Normal function application has precedence 10 - higher than any operator - and associates to the left. So these two definitions are equivalent:

expr = min 5 1 + 5
expr = ((min 5) 1) + 5

($) has precedence 0 (the lowest) and associates to the right, so these are equivalent:

expr = min 5 $ 1 + 5
expr = (min 5) (1 + 5)

Examples

A common use cases of ($) is to avoid parentheses in complex expressions.

For example, instead of using nested parentheses in the following Haskell function:

-- | Sum numbers in a string: strSum "100  5 -7" == 98
strSum :: String -> Int
strSum s = sum (mapMaybe readMaybe (words s))

we can deploy the function application operator:

-- | Sum numbers in a string: strSum "100  5 -7" == 98
strSum :: String -> Int
strSum s = sum $ mapMaybe readMaybe $ words s

($) is also used as a section (a partially applied operator), in order to indicate that we wish to apply some yet-unspecified function to a given value. For example, to apply the argument 5 to a list of functions:

applyFive :: [Int]
applyFive = map ($ 5) [(+1), (2^)]
>>> [6, 32]

Technical Remark (Representation Polymorphism)

($) is fully representation-polymorphic. This allows it to also be used with arguments of unlifted and even unboxed kinds, such as unboxed integers:

fastMod :: Int -> Int -> Int
fastMod (I# x) (I# m) = I# $ remInt# x m

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.

Boolean "and", lazy in the second argument

(++) appends 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.

Performance considerations

This function takes linear time in the number of elements of the first list. Thus it is better to associate repeated applications of (++) to the right (which is the default behaviour): xs ++ (ys ++ zs) or simply xs ++ ys ++ zs, but not (xs ++ ys) ++ zs. For the same reason GHC.Internal.Data.List.concat = GHC.Internal.Data.List.foldr (++) [] has linear performance, while GHC.Internal.Data.List.foldl (++) [] is prone to quadratic slowdown

Examples

>>> [1, 2, 3] ++ [4, 5, 6][1,2,3,4,5,6]

>>> [] ++ [1, 2, 3][1,2,3]

>>> [3, 2, 1] ++ [][3,2,1]

Right to left function composition.

prop> (f . g) x = f (g x)

prop> f . id = f = id . f

Examples

>>> map ((*2) . length) [[], [0, 1, 2], [0]][0,6,2]

>>> foldr (.) id [(+1), (*3), (^3)] 225

>>> let (...) = (.).(.) in ((*2)...(+)) 5 1030

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

as >>= f == f =<< as

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.

The character type Char represents Unicode codespace and its elements are code points as in definitions D9 and D10 of the Unicode Standard.

Character literals in Haskell are single-quoted: 'Q', 'Я' or 'Ω'. To represent a single quote itself use '\'', and to represent a backslash use '\\'. The full grammar can be found in the section 2.6 of the Haskell 2010 Language Report.

To specify a character by its code point one can use decimal, hexadecimal or octal notation: '\65', '\x41' and '\o101' are all alternative forms of 'A'. The largest code point is '\x10ffff'.

There is a special escape syntax for ASCII control characters:

Data.Char provides utilities to work with Char.

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

The type Either String Int is the type of values which can be either a String 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"

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 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.

The Haskell Report defines no laws for Eq. However, instances are encouraged to follow these properties:

Reflexivity

x == x

=

True

Symmetry

x == y

=

y == x

Transitivity

if

x == y && y == z

=

True

, then

x == z

=

True

Extensionality

if

x == y

=

True

and

f

is a function whose return type is an instance of

Eq

, then

f x == f y

=

True

Negation

x /= y

=

not (x == y)

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.

A type f is a Functor if it provides a function fmap which, given any types a and b lets you apply any function from (a -> b) to turn an f a into an f b, preserving the structure of f. Furthermore f needs to adhere to the following:

Identity

fmap id == id

Composition

fmap (f . g) == fmap f . fmap g

Note, that the second law follows from the free theorem of the type fmap and the first law, so you need only check that the former condition holds. See these articles by School of Haskell or David Luposchainsky for an explanation.