Dependent Types with Subtyping and Late-Bound Overloading

Abstract

We present a calculus with dependent types, subtyping, and late-bound overloading. Besides its theoretical interest this work is motivated by several practical needs that range form the definition of logic encodings to proof specialization and reuse and to object-oriented extension of the SML module system. The theoretical study of this calculus is not straightforward. While confluence is relatively easy to prove, subject reduction is much harder. We were not able to add overloading to any existing system with dependent types and subtyping, and prove subject reduction. This is why we also define here as by-product a new subtyping system for dependent types that improves previous systems and enjoys several properties (notably the transitivity elimination property). The calculus with overloading is then obtained as a conservative extension of this new system. Another difficult point is strong normalization, which is a necessary condition to the decidability of subtyping and typing relations. The calculus with overloading is not strongly normalizing. However, we show that a reasonably useful fragment of the calculus enjoys this property and that its strong normalization implies the decidability of its subtyping and typing relations. The article is divided into two parts: the first three scetions provide a general overview of the systems and its motivations and can be read separately; the remaining sections develop the formal study.