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  • Cited by 2
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Daumas, M. Lester, D. and Muoz, C. 2009. Verified Real Number Calculations: A Library for Interval Arithmetic. IEEE Transactions on Computers, Vol. 58, Issue. 2, p. 226.

    Paulson, Lawrence C. 2004. Organizing Numerical Theories Using Axiomatic Type Classes. Journal of Automated Reasoning, Vol. 33, Issue. 1, p. 29.

  • LMS Journal of Computation and Mathematics, Volume 3
  • January 2000, pp. 140-190

Mechanizing Nonstandard Real Analysis

  • Jacques D. Fleuriot (a1) and Lawrence C. Paulson (a2)
  • DOI:
  • Published online: 01 February 2010

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.

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2.A. M. Ballantyne and W. W. Bledsoe , ‘Automatic proofs of theorems in analysis using nonstandard analysis’, J. ACM 24 (1977) 353374.

4.M. Beeson , ‘Using nonstandard analysis to ensure the correctness of symbolic computations’, Internal. J. Found. Comput. Sci. 6 (1995) 299338.

8.J. H. Conway , On numbers and games (Academic Press Inc. (London) Ltd, 1976).

14.John Harrison , Theorem proving with the real numbers (Springer-Verlag, 1998). Also published as Technical Report 408 of the Computer Laboratory, University of Cambridge, 1996.

16.R. F. Hoskins , Standard and nonstandard analysis, Math. Appl. (Ellis Horwood Limited, 1990).

22.D. Laugwitz , ‘Infinitely small quantities in Cauchy's textbooks’, Historia Math. 14 (1987) 258274.

26.L. C. Paulson , ‘The inductive approach to verifying cryptographic protocols’, J. Computer Security (1998) 85128.

29.A. Robinson , Non-standard analysis (North-Holland, 1980).

30.E. Schechter , Handbook of analysis and its foundations (Academic Press, 1997).

31.A. P. Simpson , ‘The Infidel is innocent’, Math. Intelligencer 12 (1990) 4351.

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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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Supplementary Materials

Fleuriot and Paulson Appendix

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863 KB