base-4.12.0.0: Basic libraries

Data.Function

Contents

Description

Simple combinators working solely on and with functions.

Synopsis
• id :: a -> a
• const :: a -> b -> a
• (.) :: (b -> c) -> (a -> b) -> a -> c
• flip :: (a -> b -> c) -> b -> a -> c
• ($) :: forall r a (b :: TYPE r). (a -> b) -> a -> b • (&) :: a -> (a -> b) -> b • fix :: (a -> a) -> a • on :: (b -> b -> c) -> (a -> b) -> a -> a -> c # Prelude re-exports id :: a -> a Source # Identity function. id x = x const :: a -> b -> a Source # const x is a unary function which evaluates to x for all inputs. >>> const 42 "hello" 42  >>> map (const 42) [0..3] [42,42,42,42]  (.) :: (b -> c) -> (a -> b) -> a -> c infixr 9 Source # Function composition. flip :: (a -> b -> c) -> b -> a -> c Source # flip f takes its (first) two arguments in the reverse order of f. >>> flip (++) "hello" "world" "worldhello"  ($) :: forall r a (b :: TYPE r). (a -> b) -> a -> b infixr 0 Source #

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

# 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 $.

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


Since: 4.8.0.0

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

fix f is the least fixed point of the function f, i.e. the least defined x such that f x = x.

For example, 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,

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


Instead of making a recursive call, we introduce a dummy parameter rec; when used within fix, this parameter then refers to fix' argument, hence the recursion is reintroduced.

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

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.

((+) on f) x y = f x + f y

Typical usage: sortBy (compare on fst).

Algebraic properties:

• (*) on id = (*) -- (if (*) ∉ {⊥, const ⊥})
• ((*) on f) on g = (*) on (f . g)
• flip on f . flip on g = flip on (g . f)