A brief overview is presented of MetiTarski [4], an automatic theorem prover for real-valued special functions: ln , \(\exp\), sin, cos, etc. MetiTarski operates through a unique interaction between decision procedures and resolution theorem proving. Its history is briefly outlined, along with current projects. A simple collision avoidance example is presented.


Hybrid System Decision Procedure Collision Avoidance Theorem Prover Automate Reasoning 
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.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Wiley (1972)Google Scholar
  2. 2.
    Akbarpour, B., Paulson, L.C.: Extending a Resolution Prover for Inequalities on Elementary Functions. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 47–61. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Akbarpour, B., Paulson, L.C.: MetiTarski: An Automatic Prover for the Elementary Functions. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) AISC 2008, Calculemus 2008, and MKM 2008. LNCS (LNAI), vol. 5144, pp. 217–231. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Akbarpour, B., Paulson, L.C.: MetiTarski: An automatic theorem prover for real-valued special functions. Journal of Automated Reasoning 44(3), 175–205 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Akbarpour, B., Paulson, L.C.: Applications of MetiTarski in the Verification of Control and Hybrid Systems. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 1–15. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem. ACM Transactions on Computational Logic 9(1) (2007)Google Scholar
  8. 8.
    Beeson, M.: Automatic generation of a proof of the irrationality of e. JSC 32(4), 333–349 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Blanchette, J.C., Böhme, S., Paulson, L.C.: Extending Sledgehammer with SMT Solvers. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 116–130. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Bledsoe, W.W.: Non-resolution theorem proving. Artificial Intelligence 9, 1–35 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bridge, J., Paulson, L.C.: Case splitting in an automatic theorem prover for real-valued special functions. Journal of Automated Reasoning (in press, 2012),
  12. 12.
    Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. SIGSAM Bulletin 37(4), 97–108 (2003)zbMATHCrossRefGoogle Scholar
  13. 13.
    Bullen, P.S.: A Dictionary of Inequalities. Longman (1998)Google Scholar
  14. 14.
    Clarke, E., Zhao, X.: Analytica: A theorem prover for Mathematica. Mathematica Journal 3(1), 56–71 (1993)Google Scholar
  15. 15.
    Cuyt, A., Petersen, V., Verdonk, B., Waadeland, H., Jones, W.B.: Handbook of Continued Fractions for Special Functions. Springer (2008)Google Scholar
  16. 16.
    Daumas, M., Muñoz, C., Lester, D.: Verified real number calculations: A library for integer arithmetic. IEEE Trans. Computers 58(2), 226–237 (2009)CrossRefGoogle Scholar
  17. 17.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symbolic Comp. 5, 29–35 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    de Moura, L., Bjørner, N.S.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Denman, W., Akbarpour, B., Tahar, S., Zaki, M., Paulson, L.C.: Formal verification of analog designs using MetiTarski. In: Biere, A., Pixley, C. (eds.) Formal Methods in Computer Aided Design, pp. 93–100. IEEE (2009)Google Scholar
  20. 20.
    van den Dries, L.: Alfred Tarski’s elimination theory for real closed fields. The Journal of Symbolic Logic 53(1), 7–19 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hardy, R.: Formal Methods for Control Engineering: A Validated Decision Procedure for Nichols Plot Analysis. PhD thesis, University of St Andrews (2006)Google Scholar
  22. 22.
    Harrison, J.: Verifying Nonlinear Real Formulas Via Sums of Squares. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 102–118. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  23. 23.
    Hurd, J.: First-order proof tactics in higher-order logic theorem provers. In: Archer, M., Di Vito, B., Muñoz, C. (eds.) Design and Application of Strategies/Tactics in Higher Order Logics, NASA/CP-2003-212448 in NASA Technical Reports, pp. 56–68 (September 2003)Google Scholar
  24. 24.
    Hurd, J.: Metis first order prover (2007), Website at,
  25. 25.
    Jovanovic̀, D., de Moura, L.: Solving Non-linear Arithmetic. Technical Report MSR-TR-2012-20, Microsoft Research (2012) (accepted to IJCAR 2012)Google Scholar
  26. 26.
    Lassez, J.-L., Maher, M.J.: On Fourier’s algorithm for linear arithmetic constraints. Journal of Automated Reasoning 9(3), 373–379 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    McLaughlin, S., Harrison, J.: A Proof-Producing Decision Procedure for Real Arithmetic. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 295–314. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  28. 28.
    Mitrinović, D.S., Vasić, P.M.: Analytic Inequalities. Springer (1970)Google Scholar
  29. 29.
    Muñoz, C., Lester, D.R.: 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
  30. 30.
    Passmore, G.O., Paulson, L.C., de Moura, L.: Real algebraic strategies for MetiTarski proofs. In: Jeuring, J. (ed.) Conferences on Intelligent Computer Mathematics, CICM 2012. Springer (in press, 2012)Google Scholar
  31. 31.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Logic Computation 20(1), 309–352 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Platzer, A.: Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer (2010)Google Scholar
  33. 33.
    Platzer, A., Quesel, J.-D.: KeYmaera: A Hybrid Theorem Prover for Hybrid Systems (System Description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 171–178. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  34. 34.
    Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers. ACM Trans. Comput. Logic 7(4), 723–748 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ratschan, S., She, Z.: Benchmarks for safety verification of hybrid systems (2008),

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lawrence C. Paulson
    • 1
  1. 1.Computer LaboratoryUniversity of CambridgeEngland

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