Proof, that lists could be modelled as functions.

ListAsFunctions.hs

{-# LANGUAGE RankNTypes #-}

type List a = 
    forall b. 
        (a -> b -> b) -> (b -> b)

nil :: List a
nil _ a = a

infixr 5 ><
(><) :: a -> List a -> List a
(x >< xs) f def =
    x `f` xs f def

dump :: Show a => List a -> String
dump list = 
    list attach "nil"

  where
    attach x acc = show x ++ " >< " ++ acc

main = print $ dump $ 1 >< 2 >< 3 >< 4 >< 5 >< nil