Constructing List Homomorphisms from Proofs

The well-known third list homomorphism theorem states that if a function h is both an instance of foldr and foldl , it is a list homomorphism. Plenty of previous works devoted to constructing list homomorphisms, however, overlook the fact that proving h is both a foldr and a foldl is often the hardest part which, once done, already provides a useful hint about what the resulting list homomorphism could be. In this paper we propose a new approach: to construct a possible candidate of the associative operator and, at the same time, to transform a proof that h is both a foldr and a foldl to a proof that h is a list homomorphism. The effort constructing the proof is thus not wasted, and the resulting program is guaranteed to be correct.