Question

I have a question why this function cannot work with “:” but works with || and why “:” is replaced by ||
I mean why cannot write

contains xs (y:ys) = prefix xs (y:ys) : contains xs ys

 contains :: String -> String -> Bool
 contains [] _ = True
 contains _ [] = False
 contains xs (y:ys) = prefix xs (y:ys) || contains xs ys

Answer 1

Key idea

Use : to build up a list answer, || or && to build up a True/False answer, and + or * to build up a numerical answer. Think about what you’re combining – it doesn’t have to be : all the time.

Detail

I think you mean to ask why you can’t write

contains xs (y:ys) = prefix xs (y:ys)  :  contains xs ys

: is for gluing an element to the front of a list, so it has type a -> [a] -> [a]. This means you put a single thing on the left and a list on the right. You often use that in writing recursive functions, for example, you could define

increase :: [Int] -> [Int]
increase [] = []
increase (x:xs) = x + 5 : increase xs

Here x + 5 is an Int and increase xs is a [Int].

Let’s look at your function. Its type is contains :: String -> String -> Bool so it takes a String, then a String, then gives you a Bool. Straight after the = you get prefix xs (y:ys). Now prefix has the same type: prefix :: String -> String -> Bool, so now we’ve given it xs and (y:ys) it’s giving us a Bool.

At the end of the line we’ve got contains xs ys which is a Bool too, so if we put : between them we’ve got something a bit like

False : True

but we’re not allowed to do that because True isn’t a list. (Remember (:) :: a -> [a] -> [a].) You can do False:[True] (which would be [False,True], but that’s not what you want anyway.

You need to check whether the xs is there – either at the front of (y:ys) or later on. The symbol for or is ||. So the code says

xs is in (y:ys) if it’s at the front of it (prefix xs (y:ys)) or it’s somewhere in ys (contains xs ys).

With the : (put-in-front-of symbol) in it it would say

xs is in (y:ys) if it’s at the front of it (prefix xs (y:ys)), put that answer in front of these answers – it’s somewhere in ys (contains xs ys).

which doesn’t make sense. That’s why you need or (||) not in-front-of (:).