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Certified Exact Transcendental Real Number Computation in Coq

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Book cover Theorem Proving in Higher Order Logics (TPHOLs 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5170))

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Abstract

Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete metric spaces in the Coq proof assistant and used this library to build a constructive real number implementation including elementary real number functions and proofs of correctness. Using this library, I have created a tactic that automatically proves strict inequalities over closed elementary real number expressions by computation.

This document has been produced using \({\rm\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}_{\textsc{macs}}\)(see http://www.texmacs.org)

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References

  1. Bishop, E., Bridges, D.: Constructive Analysis. Grundlehren der mathematischen Wissenschaften, vol. 279. Springer, Heidelberg (1985)

    MATH  Google Scholar 

  2. Cruz-Filipe, L.: Constructive Real Analysis: a Type-Theoretical Formalization and Applications. PhD thesis, University of Nijmegen (April 2004)

    Google Scholar 

  3. Cruz-Filipe, L., Geuvers, H., Wiedijk, F.: C-CoRN: the constructive Coq repository at Nijmegen. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) MKM 2004. LNCS, vol. 3119, pp. 88–103. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  4. Cruz-Filipe, L., Spitters, B.: Program extraction from large proof developments. In: Basin, D., Wolff, B. (eds.) TPHOLs 2003. LNCS, vol. 2758, pp. 205–220. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  5. Cruz-Filipe, L., Letouzey, P.: A large-scale experiment in executing extracted programs. Electr. Notes Theor. Comput. Sci. 151(1), 75–91 (2006)

    Article  MATH  Google Scholar 

  6. Geuvers, H., Niqui, M.: Constructive reals in Coq: Axioms and categoricity. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 79–95. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. Gonthier, G.: A computer-checked proof of the four colour theorem. Technical report, Microsoft Research Cambridge (2005)

    Google Scholar 

  8. Hales, T.C.: A computer verification of the Kepler conjecture. In: Proceedings of the International Congress of Mathematicians, Beijing, vol. III, pp. 795–804. Higher Ed. Press (2002)

    Google Scholar 

  9. Harrison, J.: Theorem Proving with the Real Numbers. Springer, Heidelberg (1998)

    Book  MATH  Google Scholar 

  10. Jones, C.: Completing the rationals and metric spaces in LEGO. In: The second annual Workshop on Logical environments, New York, NY, USA, pp. 297–316. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  11. Julien, N.: Certified exact real arithmetic using co-induction in arbitrary integer base. In: Functional and Logic Programming Symposium (FLOPS). LNCS, Springer, Heidelberg (2008)

    Google Scholar 

  12. Muñoz, C., Lester, D.: Real number calculations and theorem proving. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 195–210. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  13. Niqui, M., Wiedijk, F.: The “Many Digits” friendly competition (2005), http://www.cs.ru.nl/~milad/manydigits

  14. O’Connor, R.: A monadic, functional implementation of real numbers. Mathematical. Structures in Comp. Sci. 17(1), 129–159 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Odlyzko, A.M., te Riele, H.J.J.: Disproof of the Mertens conjecture. J. Reine Angew. Math. 357, 138–160 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Richman, F.: Real numbers and other completions. Math. Log. Q. 54(1), 98–108 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. The Coq Development Team. The Coq Proof Assistant Reference Manual – Version V8.0 (April 2004), http://coq.inria.fr

  18. Weisstein, E.W.: Machin-like formulas. From MathWorld–A Wolfram Web Resource (January 2004), http://mathworld.wolfram.com/Machin-LikeFormulas.html

  19. Williams, R.: Arctangent formulas for PI (December 2002), http://www.cacr.caltech.edu/~roy/upi/pi.formulas.html

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Author information

Authors and Affiliations

  1. Institute for Computing and Information Science Faculty of Science, Radboud University Nijmegen, The Netherlands

    Russell O’Connor

Authors
  1. Russell O’Connor
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Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, Concordia University, 1455 de Maisonneuve Blvd. W, H3G 1M8, Montreal, Quebec, Canada

    Otmane Ait Mohamed

  2. National Institute of Aerospace, 100 Exploration Way, VA 23666, Hampton, USA

    César Muñoz

  3. Dept. of Electrical & Computer Engineering, Concordia University, 1455 de Maisonette W, H3G 1M8, Montreal, Quebec, Canada

    Sofiène Tahar

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© 2008 Springer-Verlag Berlin Heidelberg

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O’Connor, R. (2008). Certified Exact Transcendental Real Number Computation in Coq. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2008. Lecture Notes in Computer Science, vol 5170. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71067-7_21

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  • DOI: https://doi.org/10.1007/978-3-540-71067-7_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-71065-3

  • Online ISBN: 978-3-540-71067-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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