Mechanizing Nonstandard Real Analysis

Abstract

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.

Organizing Numerical Theories Using Axiomatic Type Classes

“…The hypernatural numbers, which are less well known, extend the natural numbers with infinite values; they form an ordered semiring, and are therefore much better behaved than type natinf. All three "hyper" types were defined from the standard types (using ultrafilters) by Fleuriot [2]. Most users will not require non-standard analysis, but it is significant that type classes can cope with so many different types.…”

The Poincaré-Bendixson theorem in Isabelle/HOL

“…Some other porting efforts by Paulson are briefly mentioned in a different context [1]. The library for analysis in Isabelle/HOL, HOL-Analysis, has its origins in a formalization of nonstandard real analysis by Fleuriot and Paulson [10], and a formalization of multivariate analysis by Harrison [13] in HOL Light. Hölzl et al [15] describe how the formalization was developed further to profit from Isabelle/HOL's type class system and center the treatment of limits around filters.…”

“…Our formalization builds on the existing libraries for analysis and ordinary differential equations [10,13,15,18,19,21] in Isabelle/HOL and the Archive of Formal Proofs. This section recalls relevant concepts from these libraries.…”

Verified Real Number Calculations: A Library for Interval Arithmetic

“…The arctangent function is heavily used in aeronautic applications as it is fundamental to many Geodesic formulas 5 . One common implementation technique uses an approximation of the arctangent on the interval x = [−1/30, 1/30] after argument reduction [21].…”

“…The PVS specification of this problem for some values of i is presented in Listing 4. All the lemmas 5 See, for example, Ed William's Aviation Formulary at http://williams.best.vwh.net/avform.htm. are automatically discharged by the instint strategy with different splitting and taylor's expansion degrees.…”