An Applicative Module Calculus

The SML-like module systems are small typed languages of their own. As is, one would expect a proof of their soundness following from a proof of subject reduction. Unfortunately, the subject-reduction property and the preservation of type abstraction seem to be incompatible. As a consequence, in relevant module systems, the theoretical study of reductions is meaningless, and for instance, the question of normalization of module expressions can not even be considered. In this paper, we analyze this problem as a misunderstanding of the notion of module definition. We build a variant of the SML module system — inspired from recent works by Leroy, Harper, and Lillibridge — which enjoys the subject reduction property. Type abstraction — achieved through an explicit declaration of the signature of a module at its definition — is preserved. This was the initial motivation. Besides our system enjoys other type-theoretic properties: the calculus is strongly normalizing, there are no syntactic restrictions on module paths, it enjoys a purely applicative semantics, every module has a principal type, and type inference is decidable. Neither Leroy’s system nor Harper and Lillibridge’s system has all of them.