Extending a Resolution Prover for Inequalities on Elementary Functions

  • Behzad Akbarpour
  • Lawrence C. Paulson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4790)


Experiments show that many inequalities involving exponentials and logarithms can be proved automatically by combining a resolution theorem prover with a decision procedure for the theory of real closed fields (RCF). The method should be applicable to any functions for which polynomial upper and lower bounds are known. Most bounds only hold for specific argument ranges, but resolution can automatically perform the necessary case analyses. The system consists of a superposition prover (Metis) combined with John Harrison’s RCF solver and a small amount of code to simplify literals with respect to the RCF theory.


Canonical Form Elementary Function Decision Procedure Automate Reasoning Prime Number Theorem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akbarpour, B., Paulson, L.C.: Towards automatic proofs of inequalities involving elementary functions. In: Cook, B., Sebastiani, R. (eds.) PDPAR: Pragmatics of Decision Procedures in Automated Reasoning, pp. 27–37 (2006)Google Scholar
  2. 2.
    Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem. ACM Transactions on Computational Logic (in press)Google Scholar
  3. 3.
    Avigad, J., Friedman, H.: Combining decision procedures for the reals. Logical Methods in Computer Science 2(4) (2006)Google Scholar
  4. 4.
    Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, ch. 2, pp. 19–99. Elsevier Science, Amsterdam (2001)Google Scholar
  5. 5.
    Barwise, J.: An introduction to first-order logic. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 5–46. North-Holland, Amsterdam (1977)Google Scholar
  6. 6.
    Beeson, M.: Automatic generation of a proof of the irrationality of e. JSC 32(4), 333–349 (2001)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. SIGSAM Bulletin 37(4), 97–108 (2003)zbMATHCrossRefGoogle Scholar
  8. 8.
    Clarke, E., Zhao, X.: Analytica: A theorem prover for Mathematica. Mathematica Journal 3(1), 56–71 (1993)Google Scholar
  9. 9.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. Technical Report MIP-9720, Universität Passau, D-94030, Germany (1997)Google Scholar
  10. 10.
    Flanagan, C., Joshi, R., Ou, X., Saxe, J.B.: Theorem proving using lazy proof explication. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 355–367. Springer, Heidelberg (2003)Google Scholar
  11. 11.
    Grégoire, B., Mahboubi, A.: Proving equalities in a commutative ring done right in coq. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 98–113. Springer, Heidelberg (2005)Google Scholar
  12. 12.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficient. Springer, Heidelberg (2006) First published in 1983; cited by Mclaughlin and Harrison [15] Google Scholar
  13. 13.
    Hurd, J.: Metis first order prover (2007),
  14. 14.
    McCune, W., Wos, L.: Otter: The CADE-13 competition incarnations. Journal of Automated Reasoning 18(2), 211–220 (1997)CrossRefGoogle Scholar
  15. 15.
    McLaughlin, S., Harrison, J.: A proof-producing decision procedure for real arithmetic. In: Nieuwenhuis, R. (ed.) CADE-20. LNCS (LNAI), vol. 3632, pp. 295–314. Springer, Heidelberg (2005)Google Scholar
  16. 16.
    Nipkow, T., Paulson, L.C., Wenzel, M. (eds.): Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  17. 17.
    Prevosto, V., Waldmann, U.: SPASS+T. In: Sutcliffe, G., Schmidt, R., Schulz, S.: (eds.) FLoC 2006 Workshop on Empirically Successful Computerized Reasoning, vol. 192 of CEUR Workshop Proceedings, pp. 18–33 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Behzad Akbarpour
    • 1
  • Lawrence C. Paulson
    • 1
  1. 1.Computer Laboratory, University of CambridgeEngland

Personalised recommendations