foldr, applied to a binary operator, a starting value (typically the right-identity of the operator), and a list, reduces the list using the binary operator, from right to left:
> foldr f z [x1, x2, ..., xn] == x1 `f` (x2 `f` ... (xn `f` z)...)
foldr1 is a variant of foldr that has no starting value argument, and thus must be applied to non-empty lists.
Fold over the elements of a structure, associating to the right, but strictly.
Monadic fold over the elements of a structure, associating to the right, i.e. from right to left.
The unfoldr function is a `dual' to foldr: while foldr reduces a list to a summary value, unfoldr builds a list from a seed value. The function takes the element and returns Nothing if it is done producing the list or returns Just (a,b), in which case, a is a prepended to the list and b is used as the next element in a recursive call. For example,
> iterate f == unfoldr (\x -> Just (x, f x))
In some cases, unfoldr can undo a foldr operation:
> unfoldr f' (foldr f z xs) == xs
if the following holds:
> f' (f x y) = Just (x,y)
> f' z = Nothing
A simple use of unfoldr:
> unfoldr (\b -> if b == 0 then Nothing else Just (b, b-1)) 10