Copyright | Nils Anders Danielsson 2006 Alexander Berntsen 2014 |
---|---|

License | BSD-style (see the LICENSE file in the distribution) |

Maintainer | libraries@haskell.org |

Stability | stable |

Portability | portable |

Safe Haskell | Safe |

Language | Haskell2010 |

- Prelude re-exports
- Other combinators

Simple combinators working solely on and with functions.

# Prelude re-exports

Identity function.

id x = x

This function might seem useless at first glance, but it can be very useful in a higher order context.

#### Examples

`>>>`

3`length $ filter id [True, True, False, True]`

`>>>`

Just 3`Just (Just 3) >>= id`

`>>>`

1000`foldr id 0 [(^3), (*5), (+2)]`

`const x y`

always evaluates to `x`

, ignoring its second argument.

const x = \_ -> x

This function might seem useless at first glance, but it can be very useful in a higher order context.

#### Examples

`>>>`

42`const 42 "hello"`

`>>>`

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

(.) :: (b -> c) -> (a -> b) -> a -> c infixr 9 Source #

Right to left function composition.

(f . g) x = f (g x)

f . id = f = id . f

#### Examples

`>>>`

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

`>>>`

25`foldr (.) id [(+1), (*3), (^3)] 2`

`>>>`

30`let (...) = (.).(.) in ((*2)...(+)) 5 10`

flip :: (a -> b -> c) -> b -> a -> c Source #

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

f`f`

.

flip f x y = f y x

flip . flip = id

#### Examples

`>>>`

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

`>>>`

"6"`let (.>) = flip (.) in (+1) .> show $ 5`

($) :: (a -> b) -> a -> b infixr 0 Source #

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

specialized from `id`

`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

# Other combinators

(&) :: a -> (a -> b) -> b infixl 1 Source #

`&`

is a reverse application operator. This provides notational
convenience. Its precedence is one higher than that of the forward
application operator `$`

, which allows `&`

to be nested in `$`

.

This is a version of

, where `flip`

`id`

`id`

is specialized from `a -> a`

to `(a -> b) -> (a -> b)`

which by the associativity of `(->)`

is `(a -> b) -> a -> b`

.
flipping this yields `a -> (a -> b) -> b`

which is the type signature of `&`

#### Examples

`>>>`

"6"`5 & (+1) & show`

`>>>`

3.1406380562059946`sqrt $ [1 / n^2 | n <- [1..1000]] & sum & (*6)`

*Since: base-4.8.0.0*

is the least fixed point of the function `fix`

f`f`

,
i.e. the least defined `x`

such that `f x = x`

.

When `f`

is strict, this means that because, by the definition of strictness,
`f ⊥ = ⊥`

and such the least defined fixed point of any strict function is `⊥`

.

#### Examples

We can write the factorial function using direct recursion as

`>>>`

120`let fac n = if n <= 1 then 1 else n * fac (n-1) in fac 5`

This uses the fact that Haskell’s `let`

introduces recursive bindings. We can
rewrite this definition using `fix`

,

Instead of making a recursive call, we introduce a dummy parameter `rec`

;
when used within `fix`

, this parameter then refers to `fix`

’s argument, hence
the recursion is reintroduced.

`>>>`

120`fix (\rec n -> if n <= 1 then 1 else n * rec (n-1)) 5`

Using `fix`

, we can implement versions of `repeat`

as

and `fix`

`.`

`(:)`

`cycle`

as `fix`

`.`

`(++)`

`>>>`

[0,0,0,0,0,0,0,0,0,0]`take 10 $ fix (0:)`

`>>>`

[1,1,2,3,5,8,13,21,34,55]`map (fix (\rec n -> if n < 2 then n else rec (n - 1) + rec (n - 2))) [1..10]`

#### Implementation Details

on :: (b -> b -> c) -> (a -> b) -> a -> a -> c infixl 0 Source #

runs the binary function `on`

b u x y`b`

*on* the results of applying
unary function `u`

to two arguments `x`

and `y`

. From the opposite
perspective, it transforms two inputs and combines the outputs.

(op ``on`

` f) x y = f x ``op`

` f y

#### Examples

`>>>`

[[],[0],[0,1],[0,1,2]]`sortBy (compare `on` length) [[0, 1, 2], [0, 1], [], [0]]`

`>>>`

4`((+) `on` length) [1, 2, 3] [-1]`

`>>>`

(4,6)`((,) `on` (*2)) 2 3`

#### Algebraic properties

applyWhen :: Bool -> (a -> a) -> a -> a Source #

`applyWhen`

applies a function to a value if a condition is true,
otherwise, it returns the value unchanged.

It is equivalent to

.`flip`

(`bool`

`id`

)

#### Examples

`>>>`

[2,2,6,4,10,6,14,8,18,10]`map (\x -> applyWhen (odd x) (*2) x) [1..10]`

`>>>`

["Hi!","Thi...","Hop...",":D"]`map (\x -> applyWhen (length x > 6) ((++ "...") . take 3) x) ["Hi!", "This is amazing", "Hope you're doing well today!", ":D"]`

#### Algebraic properties

*Since: base-4.18.0.0*