Real Number Calculations and Theorem Proving

  • César Muñoz
  • David Lester
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3603)


Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs.


Theorem Prove Interval Arithmetic Transcendental Function Proof Assistant High Order Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abramovitz, M., Stegun, I.: Handbook of Mathematical Functions. National Bureau of Standards (1972)Google Scholar
  2. 2.
    Boutin, S.: Using reflection to build efficient and certified decision procedures. In: Ito, T., Abadi, M. (eds.) TACS 1997. LNCS, vol. 1281, pp. 515–530. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  3. 3.
    Ciaffaglione, A.: Certified Reasoning on Real Numbers and Objects in Coinductive Type Theory. PhD thesis, Università degli Studi di Udine (1993)Google Scholar
  4. 4.
    Daumas, M., Melquiond, G.: Generating formally certified bounds on values and round-off errors. In: Real Numbers and Computers, Dagstuhl, Germany, pp. 55–70 (2004)Google Scholar
  5. 5.
    Daumas, M., Melquiond, G., Muñoz, C.: Guaranteed proofs using interval arithmetic. In: Accepted for publication at the 17th IEEE Symposium on Computer Arithmetic (1995)Google Scholar
  6. 6.
    Dutertre, B.: Elements of mathematical analysis in PVS. In: von Wright, J., Harrison, J., Grundy, J. (eds.) TPHOLs 1996. LNCS, vol. 1125, pp. 141–156. Springer, Heidelberg (1996)Google Scholar
  7. 7.
    Fleuriot, J., Paulson, L.: Mechanizing nonstandard real analysis. LMS Journal of Computation and Mathematics 3, 140–190 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Gamboa, R.: Mechanically Verifying Real-Valued Algorithms in ACL2. PhD thesis, University of Texas at Austin (May 1999)Google Scholar
  9. 9.
    Gottliebsen, H.: Automated Theorem Proving for Mathematics: Real Analysis in PVS. PhD thesis, University of St Andrews (June 2001)Google Scholar
  10. 10.
    Gowland, P., Lester, D.: A survey of exact computer arithmetic. In: Blanck, J., Brattka, V., Hertling, P., Weihrauch, K. (eds.) Computability and Complexity in Analysis, CCA2000 Workshop, Swansea, Wales. Informatik Berichte, vol. 272, pp. 99–115. FernUniversität Hagen (2000)Google Scholar
  11. 11.
    Harrison, J.: Metatheory and reflection in theorem proving: A survey and critique. Technical Report CRC-053, SRI Cambridge, Millers Yard, Cambridge, UK (1995)Google Scholar
  12. 12.
    Harrison, J.: Theorem Proving with the Real Numbers. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  13. 13.
    Hickey, T., Ju, Q., van Emden, M.H.: Interval arithmetic: from principles to implementation. Journal of the ACM (2001)Google Scholar
  14. 14.
    Kearfott, R.B.: Interval computations: Introduction, uses, and resources. Euromath Bulletin 2(1), 95–112 (1996)MathSciNetGoogle Scholar
  15. 15.
    Mayero, M.: Formalisation et automatisation de preuves en analyses réelle et numérique. PhD thesis, Université Paris VI, décembre (2001)Google Scholar
  16. 16.
    Ménissier, V.: Arithmétique Exacte : Conception, Algorithmique et Performances d’une Implantation Informatique en Précision Arbitraire. PhD thesis, Université Paris VI, Paris, France (1994)Google Scholar
  17. 17.
    Muñoz, C.: PVSio reference manual – Version 2.a (2004), Available from
  18. 18.
    Muñoz, C., Carreño, V., Dowek, G., Butler, R.W.: Formal verification of conflict detection algorithms. International Journal on Software Tools for Technology Transfer 4(3), 371–380 (2003)CrossRefGoogle Scholar
  19. 19.
    Niqui, M.: Formalising Exact Arithmetic: Representations, Algorithms and Proofs. PhD thesis, Radboud University Nijmegen (1994)Google Scholar
  20. 20.
    Owre, S., Rushby, J.M., Shankar, N.: PVS: A prototype verification system. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 748–752. Springer, Heidelberg (1992)Google Scholar
  21. 21.
    Pour-El, M., Richards, J.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)Google Scholar
  22. 22.
    Robinson, R.M.: Review of Peter, R., Rekursive Funktionen. The Journal of Symbolic Logic 16, 280–282 (1951)CrossRefGoogle Scholar
  23. 23.
    Shankar, N.: Efficiently executing PVS. Project report, Computer Science Laboratory, SRI International, Menlo Park (1999), Available at
  24. 24.
    Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. The Journal of Symbolic Logic 14(3), 145–158 (1949)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 42(2), 230–265 (1936)zbMATHGoogle Scholar
  26. 26.
    von Henke, F.W., Pfab, S., Pfeifer, H., Rueß, H.: Case Studies in Meta-Level Theorem Proving. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 461–478. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  27. 27.
    Yakovlev, A.: Classification approach to programming of localizational (interval) computations. Interval Computations 1(3), 61–84 (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • César Muñoz
    • 1
  • David Lester
    • 2
  1. 1.National Institute of AerospaceHamptonUSA
  2. 2.University of ManchesterManchesterUK

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