Type checking with universes

Abstract

Various formulations of constructive type theories have been proposed to serve as the basis for machine-assisted proof and as a theoretical basis for studying programming languages. Many of these calculi include a cumulative hierarchy of “universes”, each a type of types closed under a collection of type-forming operations. Universes are of interest for a variety of reasons, some philosophical (predicative vs. impredicative type theories), some theoretical (limitations on the closure properties of type theories) and some practical (to achieve some of the advantages of a type of all types without sacrificing consistency.) The Generalized Calculus of Constructions (CC ω) is a formal theory of types that includes such a hierarchy of universes. Although essential to the formalization of constructive mathematics, universes are tedious to use in practice, for one is required to make specific choices of universe levels and to ensure that all choices are consistent. In this paper we study several problems associated with type checking in the presence of universes in the context of CC ω. First, we consider the basic type checking and well-typedness problems for this calculus. Second, we consider a formulation of Russell and Whitehead's “typical ambiguity” convention whereby universe levels may be elided, provided that some consistent assignment of levels leads to a correct derivation. Third, we consider the introduction of definitions to both the basic calculus and the calculus with typical ambiguity. This extension leads to a notion of “universe polymorphism” analogous to the type polymorphism of ML. Although our study is conducted for CC ω, we expect that our methods will apply to other variants of the Calculus of Constructions and to type theories such as Constable's V3.