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A higher-order function for filtering elements of a list

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Let's filter a given list of numbers to only retain the odd numbers. In Language:Haskell, we would use the filter function as follows:

Prelude> filter odd [1,2,3,4,5]
[1,3,5]

That is, filter is applied to a predicate (i.e., a function to return a Boolean) and a list; filter returns the list of those elements that satisfy the predicate. The higher-order function filter can be defined as follows:

-- Define filter via pattern matching
filter :: (a -> Bool) -> [a] -> [a]
filter p [] = []
filter p (x:xs) = if p x then x : ys else ys
  where
    ys = filter p xs

For what it matters, the filter function could also be defined in terms of the fold function, thereby reusing a recursion scheme as opposed to the explicitly recursive definition above.

-- Define filter via foldr
filter' p = foldr f []
  where
    f x = if p x then (:) x else id

Further, we can also define the filter function in terms of list comprehension; in this manner we hint at the meaning of list comprehensions because, in fact, list comprehensions correspond to syntactic sugar whose elimination also involves the Filter function.

-- Define filter via list comprehension
filter'' :: (a -> Bool) -> [a] -> [a]
filter'' p xs = [ x | x <- xs, p x]

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