Yun-Yan Chi and Shin-Cheng Mu. In the 9th Asian Symposium on Programming Languages and Systems (APLAS 2011), LNCS 7078, pages 74-88.
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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.