Identity function.

id x = x

const x is a unary function which evaluates tox 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 off.

>>>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 asmap ($ 0) xs, orzipWith ($) fs xs.

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

& 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 f is the least fixed point of the functionf, i.e. the least definedx such thatf 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 5120

This uses the fact that Haskell’slet introduces recursive bindings. We can rewrite this definition usingfix,

>>>fix (\rec n -> if n <= 1 then 1 else n * rec (n-1)) 5120

Instead of making a recursive call, we introduce a dummy parameterrec; when used withinfix, this parameter then refers tofix' argument, hence the recursion is reintroduced.

on b u x y runs the binary functionbon the results of applying unary functionu to two argumentsx andy. 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)