map f xs is the list obtained by applying f to each element of xs, i.e.,
> map f [x1, x2, ..., xn] == [f x1, f x2, ..., f xn]
> map f [x1, x2, ...] == [f x1, f x2, ...]

An infix synonym for fmap.

Promote a function to a monad.

A variant of <*> with the arguments reversed.

iterate f x returns an infinite list of repeated applications of f to x:
> iterate f x == [x, f x, f (f x), ...]

dropWhile p xs returns the suffix remaining after takeWhile p xs:
> dropWhile (< 3) [1,2,3,4,5,1,2,3] == [3,4,5,1,2,3]
> dropWhile (< 9) [1,2,3] == []
> dropWhile (< 0) [1,2,3] == [1,2,3]

filter, applied to a predicate and a list, returns the list of those elements that satisfy the predicate; i.e.,
> filter p xs = [ x | x <- xs, p x]

takeWhile, applied to a predicate p and a list xs, returns the longest prefix (possibly empty) of xs of elements that satisfy p:
> takeWhile (< 3) [1,2,3,4,1,2,3,4] == [1,2]
> takeWhile (< 9) [1,2,3] == [1,2,3]
> takeWhile (< 0) [1,2,3] == []

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 Data.List.zipWith ($) fs xs.

Strict (call-by-value) application, defined in terms of seq.

Applied to a predicate and a list, all determines if all elements of the list satisfy the predicate. For the result to be True, the list must be finite; False, however, results from a False value for the predicate applied to an element at a finite index of a finite or infinite list.

Applied to a predicate and a list, any determines if any element of the list satisfies the predicate. For the result to be False, the list must be finite; True, however, results from a True value for the predicate applied to an element at a finite index of a finite or infinite list.

In many situations, the liftM operations can be replaced by uses of ap, which promotes function application.
> return f `ap` x1 `ap` ... `ap` xn
is equivalent to
> liftMn f x1 x2 ... xn

The sortWith function sorts a list of elements using the user supplied function to project something out of each element

This function may be used as a value for Data.Foldable.foldMap in a Foldable instance.

Direct MonadPlus equivalent of filter filter = (mfilter:: (a -> Bool) -> [a] -> [a] applicable to any MonadPlus, for example mfilter odd (Just 1) == Just 1 mfilter odd (Just 2) == Nothing

Determines whether all elements of the structure satisfy the predicate.

Determines whether any element of the structure satisfies the predicate.

The groupWith function uses the user supplied function which projects an element out of every list element in order to to first sort the input list and then to form groups by equality on these projected elements

until p f yields the result of applying f until p holds.

Function composition.

(*) `on` f = \x y -> f x * f y.
Typical usage: Data.List.sortBy (compare `on` fst).
Algebraic properties:
* (*) `on` id = (*) (if (*) {¥, const ¥})
* (*) `on` f) `on` g = (*) `on` (f .
* on f . flip on g = flip on (g .
>

> comparing p x y = compare (p x) (p y)
Useful combinator for use in conjunction with the xxxBy family of functions from Data.List, for example:
> ... sortBy (comparing fst) ...

The maybe function takes a default value, a function, and a Maybe value. If the Maybe value is Nothing, the function returns the default value. Otherwise, it applies the function to the value inside the Just and returns the result.

This function maps one exception into another as proposed in the paper "A semantics for imprecise exceptions".

This function maps one exception into another as proposed in the paper "A semantics for imprecise exceptions".