Soundness and principal contexts for a shallow polymorphic type system based on classical logic

Abstract

In this paper we investigate how to adapt the well-known notion of ML-style polymorphism (shallow polymorphism) to a term calculus based on a Curry-Howard correspondence with classical sequent calculus, namely, the X i -calculus. We show that the intuitive approach is unsound, and pinpoint the precise nature of the problem. We define a suitably refined type system, and prove its soundness. We then define a notion of principal contexts for the type system, and provide an algorithm to compute these, which is proved to be sound and complete with respect to the type system. In the process, we formalise and prove correctness of generic unification, which generalises Robinson's unification to shallow-polymorphic types.

Monadic translation of classical sequent calculus

“…† The following analysis can also be applied to Church-style systems by defining each translation of terms as a mapping from terms with their type derivations. On the other hand, Curry-style formulations do not appear to be suitable since there is some evidence that Curry-style cbv polymorphic calculi with control operators are unsound(Harper and Lillibridge 1993;Fujita 1999;Summers 2011). ‡ The rules RIntro2 and Intro2 are superficially the same rule, but they range over different systems of types and terms.…”