Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Functional programming, design and analysis of algorithms / 1 / a / Solution

The right fold replaces the two list constructors by its two arguments; the left fold instead updates an accumulator while traversing the list.

foldr :: (a -> b -> b) -> b -> [a] -> b
foldr c z []     = z
foldr c z (x:xs) = c x (foldr c z xs)

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

Thus foldr is the natural list fold: it preserves the constructor arrangement while replacing [] by z and (:) by c. In particular, foldr (:) [] = id. More generally, the fold factorisation criterion says that the defining equations characterize the corresponding structural recursion. Association and lazy evaluation are separate: the accumulator in this definition of left fold is not automatically evaluated at each step.^[ref]