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Suppose a predicate accepts some $u$ and rejects some $v$. Dropping its initial accepted list prefix maps both the empty list and the singleton $u$ to the empty list, but maps the extensions by $v$ to different lists. A direct list fold cannot do this: its combining function receives the same head and the same folded-tail summary in both cases. This is an instance of the fold factorisation criterion. An auxiliary list fold retaining more information followed by projection is not ruled out.

The structural recursion that replaces an empty list by a chosen value and a cons constructor by a combining function applied to the head and folded tail. A direct list fold is strict in the list spine at its outermost constructor, even when the combining function does not demand the folded tail. Its algebraic recognition rule is the fold factorisation criterion.

A list whose spine or elements may contain bottom values. The displayed value has a known finite spine prefix and an undefined tail. An entirely undefined list exposes no outer constructor, while [] exposes a terminating empty list. These distinctions matter to strict functions and to the fold factorisation criterion.

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]

Let $F=\mathrm{foldl}(\ominus )0$, using the specified operation $u\ominus v=v-u$. Direct evaluation gives $F([])=0$, $F([0])=0$, $F([1])=1$, and $F([1,0])=-1$.

Suppose $F=\mathrm{foldr}cb$ with the same integer result type. The empty list forces $b=0$. The two singleton calculations force $c00=0$ and $c10=1$. But thencontrary to $F([1,0])=-1$. No such c,b exist. This is the information restriction in the fold factorisation criterion: equal folded tails must remain indistinguishable after the same head is attached. A larger, function-valued result type can retain the missing information, as the continuation representation of a left fold does below.

In general, no direct list fold with result type [a] can implement dropWhile p. Choose total values $u,v$ for which $p(u)=\mathrm{True}$ and $p(v)=\mathrm{False}$. The results on [] and [u] are both []. A hypothetical implementation foldr h z would therefore receive identical folded-tail values when extending either input by the same head $v$. Yet the required outputs are [v] and [v,u], respectively. This contradicts the fold factorisation criterion: the combining function cannot distinguish information already discarded by its tail summary.

The claim concerns a direct same-result-type list fold. A list fold returning extra information, followed by a projection, is possible: a pair can retain both the original tail and the dropped list suffix. Also, special predicates can have direct list folds: a constantly false predicate gives the identity; a constantly true predicate can use foldr (\_ r -> r) [], which correctly diverges when it must traverse an infinite spine. Those special cases do not supply an implementation for an arbitrary predicate.

The list fold criterion is: there must be a value z and a definable combining function c such thatThe last equality says that all information needed about the tail factors through f xs; the combining function receives no other access to xs. These conditions give f = foldr c z.

Necessity follows from the defining equations of foldr, including its case distinction on the input list. Sufficiency follows by structural induction for finite lists and finite partial-list approximations. For infinite or partial lazy lists, the usual continuous semantics of these definable functions extends equality from the approximations to their limit. The strictness condition matters: agreement only on finite total lists does not establish agreement at $\perp$. This is the fold factorisation criterion.

tails is not a direct list fold. Every foldr c z is strict in its list argument, but tails undefined = undefined : undefined has a defined outer constructor. Applying null distinguishes this result from $\perp$.

ptails is not a direct list fold either. On a type with distinct values u and v,A fold step given the same new head x and these equal folded tails must return the same result. However,Thus the tail information needed by the next step does not factor through ptails; this violates the fold factorisation criterion even on finite total lists.

ntails is a direct list fold:When the tail is nonempty, its first nonempty suffix of a list is the tail itself; otherwise it is recovered as []. Hence original (ntailsFold xs) = xs. The inner case distinction is delayed beneath constructors, so the definition also preserves the required behaviour on partial and infinite lazy lists. Building a suffix head does not first demand the whole folded tail.

inits is not a direct list fold: inits undefined = [] : undefined has a defined outer constructor, while foldr c z undefined is $\perp$.

Both pinits and ninits are direct list folds:For pinits, the combining function is exactly the recursive step above. For ninits, expand inits xs = [] : ninits xs and useNeither step needs to inspect yss before producing its first outer constructor. Thus the definitions satisfy the fold factorisation criterion on partial as well as total lazy lists.