Certified Exact Transcendental Real Number Computation in Coq

  • Russell O’Connor
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5170)

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.

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References

  1. 1.
    Bishop, E., Bridges, D.: Constructive Analysis. Grundlehren der mathematischen Wissenschaften, vol. 279. Springer, Heidelberg (1985)MATHGoogle Scholar
  2. 2.
    Cruz-Filipe, L.: Constructive Real Analysis: a Type-Theoretical Formalization and Applications. PhD thesis, University of Nijmegen (April 2004)Google Scholar
  3. 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)CrossRefGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Cruz-Filipe, L., Letouzey, P.: A large-scale experiment in executing extracted programs. Electr. Notes Theor. Comput. Sci. 151(1), 75–91 (2006)CrossRefMATHGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 7.
    Gonthier, G.: A computer-checked proof of the four colour theorem. Technical report, Microsoft Research Cambridge (2005)Google Scholar
  8. 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. 9.
    Harrison, J.: Theorem Proving with the Real Numbers. Springer, Heidelberg (1998)CrossRefMATHGoogle Scholar
  10. 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. 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. 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)CrossRefGoogle Scholar
  13. 13.
    Niqui, M., Wiedijk, F.: The “Many Digits” friendly competition (2005), http://www.cs.ru.nl/~milad/manydigits
  14. 14.
    O’Connor, R.: A monadic, functional implementation of real numbers. Mathematical. Structures in Comp. Sci. 17(1), 129–159 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Odlyzko, A.M., te Riele, H.J.J.: Disproof of the Mertens conjecture. J. Reine Angew. Math. 357, 138–160 (1985)MathSciNetMATHGoogle Scholar
  16. 16.
    Richman, F.: Real numbers and other completions. Math. Log. Q. 54(1), 98–108 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    The Coq Development Team. The Coq Proof Assistant Reference Manual – Version V8.0 (April 2004), http://coq.inria.fr
  18. 18.
    Weisstein, E.W.: Machin-like formulas. From MathWorld–A Wolfram Web Resource (January 2004), http://mathworld.wolfram.com/Machin-LikeFormulas.html
  19. 19.
    Williams, R.: Arctangent formulas for PI (December 2002), http://www.cacr.caltech.edu/~roy/upi/pi.formulas.html

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Russell O’Connor
    • 1
  1. 1.Institute for Computing and Information Science Faculty of ScienceRadboud University NijmegenThe Netherlands

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