Relude.Function

relude-1.2.2.0: Safe, performant, user-friendly and lightweight Haskell Standard Library

Copyright | (c) 2016 Stephen Diehl (c) 2016-2018 Serokell (c) 2018-2023 Kowainik
License | MIT
Maintainer | Kowainik <xrom.xkov@gmail.com>
Stability | Stable
Portability | Portable
Safe Haskell | Safe
Language | Haskell2010

Relude.Function

Contents

- Reexports
  - Data.Function reexports
  - Control.Category reexports
  - Control.Arrow reexports
- Combinators

Description

This module reexports very basic and primitive functions and function combinators.

Synopsis

- (.) :: (b -> c) -> (a -> b) -> a -> c
- ($) :: (a -> b) -> a -> b
- (&) :: a -> (a -> b) -> b
- id :: a -> a
- const :: a -> b -> a
- flip :: (a -> b -> c) -> b -> a -> c
- fix :: (a -> a) -> a
- on :: (b -> b -> c) -> (a -> b) -> a -> a -> c
- (>>>) :: forall {k} cat (a :: k) (b :: k) (c :: k). Category cat => cat a b -> cat b c -> cat a c
- (<<<) :: forall {k} cat (b :: k) (c :: k) (a :: k). Category cat => cat b c -> cat a b -> cat a c
- (&&&) :: Arrow a => a b c -> a b c' -> a b (c, c')
- identity :: a -> a

# Reexports

## Data.Function reexports

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

Right to left function composition.

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

    f . id = f = id . f

#### Examples

Expand

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

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

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

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

``($)`` 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

Expand

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)

Expand

`($)` 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

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

`&` 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 ``flip` `id``, where `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

Expand

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

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

@since base-4.8.0.0

id :: a -> a #

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

Expand

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

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

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

const :: a -> b -> a #

`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

Expand

    >>> const 42 "hello"
    42

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

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

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

    flip f x y = f y x

    flip . flip = id

#### Examples

Expand

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

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

fix :: (a -> a) -> a #

``fix` f` is the least fixed point of the function `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

Expand

We can write the factorial function using direct recursion as

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

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.

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

Using `fix`, we can implement versions of `repeat` as ``fix` `.` `(:)`` and `cycle` as ``fix` `.` `(++)``

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

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

#### Implementation Details

Expand

The current implementation of `fix` uses structural sharing

    fix f = let x = f x in x

A more straightforward but non-sharing version would look like

    fix f = f (fix f)

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

``on` b u x y` runs the binary function `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

Expand

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

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

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

#### Algebraic properties

Expand

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

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

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

## Control.Category reexports

(>>>) :: forall {k} cat (a :: k) (b :: k) (c :: k). Category cat => cat a b -> cat b c -> cat a c infixr 1 #

Left-to-right composition

(<<<) :: forall {k} cat (b :: k) (c :: k) (a :: k). Category cat => cat b c -> cat a b -> cat a c infixr 1 #

Right-to-left composition

## Control.Arrow reexports

(&&&) :: Arrow a => a b c -> a b c' -> a b (c, c') infixr 3 #

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.

# Combinators

identity :: a -> a #

Renamed version of `id`.

    >>> identity 10
    10

    >>> fmap identity [1,2,3]
    [1,2,3]