Mechanizing Nonstandard Real Analysis

This paper first describes the construction and use of the hyperreals in the theorem-prover Isabelle within the framework of higher-order logic (HOL). The theory, which includes infinitesimals and infinite numbers, is based on the hyperreal number system developed by Abraham Robinson in his nonstandard analysis (NSA). The construction of the hyperreal number system has been carried out strictly through the use of definitions to ensure that the foundations of NSA in Isabelle are sound. Mechanizing the construction has required that various number systems including the rationals and the reals be built up first. Moreover, to construct the hyperreals from the reals has required developing a theory of filters and ultrafilters and proving Zorn's lemma, an equivalent form of the axiom of choice. This paper also describes the use of the new types of numbers and new relations on them to formalize familiar concepts from analysis. The current work provides both standard and nonstandard definitions for the various notions, and proves their equivalence in each case. To achieve this aim, systematic methods, through which sets and functions are extended to the hyperreals, are developed in the framework. The merits of the nonstandard approach with respect to the practice of analysis and mechanical theorem-proving are highlighted throughout the exposition.