Higher-order functions allow common programming patterns to be encapsulated as functions.

## Basic Concepts #

A function that takes a function as an argument or returns a function as a result is called a higher-order function.
Because the term curried already exists for returning functions as results, the term higher-order is often just used for taking functions as arguments.

twice :: (a -> a) -> a -> a
twice f x = f (f x)

twice (*2) 3 -- 12
twice reverse [1,2,3] -- [1,2,3]

## Processing Lists #

The standard library defines a number of useful higher-order functions for processing lists.

map :: (a -> b) -> [a] -> [b]
map f xs = [ f x | x <- xs ]
----
map :: (a -> b) -> [a] -> [b]
map f []= []
map f (x:xs) = f x : map f xs
----
map (+1) [1,3,5,7] -- [2,4,6,8]

map even [1,2,3,4] -- [False,True,False,True]

map reverse ["abc","def","ghi"] -- ["cba","fed","ihg"]
----
map (map (+1)) [[1,2,3],[4,5]] -- { applying the outer map }

[map (+1) [1,2,3], map (+1) [4,5]] -- { applying the inner maps }

[[2,3,4],[5,6]]
filter :: (a -> Bool) -> [a] -> [a]
filter p xs = [ x | x <- xs, p x ]
----
filter :: (a -> Bool) -> [a] -> [a]
filter p [] = []
filter p (x : xs) | p x = x : filter p xs
| otherwise = filter p xs
----
filter even [1..10] -- [2,4,6,8,10]

filter (> 5) [1..10] -- [6,7,8,9,10]

filter (/= ' ') "abc def ghi" -- "abcdefghi"
all even [2,4,6,8] -- True
----
any odd [2,4,6,8] -- False
----
takeWhile even [2,4,6,7,8] -- [2,4,6]
----
dropWhile odd [1,3,5,6,7] -- [6,7]

## The foldr Function #

Many functions that take a list as their argument can be defined using the following pattern of recursion on lists:

f []     = v
f (x:xs) = x # f xs

The function maps the empty list to a value v, and any non-empty list to an operator # applied to the head of the list and the result of recursively processed tail.

For example:

sum []       = 0
sum (x : xs) = x + sum xs
----
product [] = 1
product (x : xs) = x * product xs
----
or [] = False
or (x : xs) = x || or xs
----
and [] = True
and (x : xs) = x && and xs

The higher-order library function foldr (fold right) encapsulates this pattern of recursion for defining functions on lists.

Fold right function assumes that the given operator associates to the right: 1+(2+(3+0)).

foldr :: (a -> b -> b) -> b -> [a] -> b
foldr f v [] = v
foldr f v (x : xs) = f x (foldr f v xs)
----
sum :: Num a => [a] -> a
sum = foldr (+) 0
----
product :: Num a => [a] -> a
product = foldr (*) 1
----
or :: [Bool] -> Bool
or = foldr (||) False
----
and :: [Bool] -> Bool
and = foldr (&&) True

It is easier to reason about foldr f v in a non-recursive way, as simply replacing each : (cons) operator in a list by the function f, and the empty list at the end by the value v.

For example, applying the function foldr (+) 0 to the list 1 : (2 : (3 : [])) gives the result 1 + (2 + (3 + 0)) in which : and [] have been replaced by + and 0.

A quick reminder: [1,2,3] and 1 : (2 : (3 : [])) are equivalent.

Many functions can be redefined with foldr:

length :: [a] -> Int
length [] = 0
length (_ : xs) = 1 + length xs
----
length [1,2,3]

1 : (2 : (3 : []))

1 + (1 + (1 + 0))

3
----
length :: [a] -> Int
length = foldr (\_ n -> 1 + n) 0

----

reverse :: [a] -> [a]
reverse [] = []
reverse (x : xs) = reverse xs ++ [x]

reverse [1,2,3]

1 : (2 : (3 : []))

(([] ++ [3]) ++ [2]) ++ [1]
----
snoc x xs = xs ++ [x] -- cons backwards

reverse :: [a] -> [a]
reverse = foldr snoc []

## The foldl Function #

Opposite of foldr, assumes that operator associates to the left: ((0+1)+2)+3.

foldl :: (a -> b -> a) -> a -> [b] -> a
foldl f v [] = v
foldl f v (x : xs) = foldl f (f v x) xs

It is useful for mapping the empty list to the accumulator value v, and any non-empty list to the result of recursively processing the tail using a new accumulator value obtained by applying an operator # to the current value and the head of the list.

f v []     = v
f v (x:xs) = f (v # x) xs

When a function can be defined using both foldr and foldl the choice of which definition is preferable is usually based on efficiency and requires considering the evaluation mechanism of Haskell.

## The Composition Operator #

The standard operator . returns the composition of two functions as a single function.

(.) :: (b -> c) -> (a -> b) -> (a -> c)
f . g = \x -> f (g x)
----
odd n = not (even n)
odd = not . even
----
twice f x = f (f x)
twice = f . f
----
id :: a -> a
id = \x -> x
-- compose a list of functions
compose :: [a -> a] -> (a -> a)
compose = foldr (.) id

Continuing later on.