Implementing Constructive Real Analysis: Preliminary Report

In this paper we present the results of an investigation into the use of the Nuprl proof development system to implement higher constructive mathematics. As a first step in exploring the issues involved, we have developed a basis for formalizing substantial parts of real analysis. More specifically, we have: developed type-theoretic representations of concepts from Bishop's treatment of constructive mathematics that allow reasonably direct formalizations; used Nuprl's facility for sound extension of its inference system to implement automated reasoners for analysis; and tested these ideas in a formalization of rational and real arithmetic and of a proof of the completeness theorem for the reals (every Cauchy sequence converges).