Journal of Automated Reasoning

, Volume 45, Issue 2, pp 189–212 | Cite as

Linear Quantifier Elimination



This paper presents verified quantifier elimination procedures for dense linear orders (two of them novel), for real and for integer linear arithmetic. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formulae themselves (by reflection). The formalization of the different theories is highly modular.


Quantifier elimination Linear arithmetic Verification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ballarin, C.: Interpretation of locales in Isabelle: theories and proof contexts. In: Borwein, J., Farmer, W. (eds.) Mathematical Knowledge Management. LNCS, vol. 4108, pp. 31–43. Springer, Heidelberg (2006)Google Scholar
  2. 2.
    Boyer, R.S., Moore, J.S.: Metafunctions: proving them correct and using them efficiently as new proof procedures. In: Boyer, R., Moore, J. (eds.) The Correctness Problem in Computer Science, pp. 103–184. Academic, New York (1981)Google Scholar
  3. 3.
    Chaieb, A.: Verifying mixed real-integer quantifier elimination. In: Furbach, U., Shankar, N. (eds.) Automated Reasoning (IJCAR 2006). LNCS, vol. 4130, pp. 528–540. Springer, Heidelberg (2006)Google Scholar
  4. 4.
    Chaieb, A., Nipkow, T.: Verifying and reflecting quantifier elimination for Presburger arithmetic. In: Stutcliffe, G., Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2005). LNCS, vol. 3835, pp. 367–380. Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Cooper, D.C.: Theorem proving in arithmetic without multiplication. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence, vol. 7, pp. 91–100. Edinburgh University Press, Edinburgh (1972)Google Scholar
  6. 6.
    Enderton, H.: A Mathematical Introduction to Logic. Academic, New York(1972)MATHGoogle Scholar
  7. 7.
    Farkas, J.: Theorie der einfachen Ungleichungen. J. Reine Angew. Math. 124, 1–27 (1902)Google Scholar
  8. 8.
    Ferrante, J., Rackoff, C.: A decision procedure for the first order theory of real addition with order. SIAM J. Comput. 4, 69–76 (1975)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fourier, J.B.J.: Solution d’une question particulière du calcul des inégalités. In: Darboux, G. (ed.) Joseph Fourier—Œuvres Complétes, vol. 2, pp. 317–328. Gauthier-Villars, Paris (1888–1890)Google Scholar
  10. 10.
    Gonthier, G.: Formal proof—the four-colour theorem. Not. Am. Math. Soc. 55, 1382–1393 (2008)MATHMathSciNetGoogle Scholar
  11. 11.
    Haftmann, F., Wenzel, M.: Constructive type classes in Isabelle. In: Altenkirch, Th., McBride, C. (eds.) Types for Proofs and Programs (TYPES 2006). LNCS, vol. 4502, pp. 160–174. Springer, Heidelberg (2007)Google Scholar
  12. 12.
    Harrison, J.: Complex quantifier elimination in HOL. In: Boulton, R., Jackson, P. (eds.) TPHOLs 2001: Supplemental Proceedings, pp. 159–174. Division of Informatics, University of Edinburgh, Edinburgh (2001)Google Scholar
  13. 13.
    Harrison, J.: Handbook of Practical Logic and Automated Reasoning. Cambridge University Press, Cambridge (2009)MATHCrossRefGoogle Scholar
  14. 14.
    Langford, C.H.: Some theorems on deducibility. Ann. Math. (2nd Series) 28, 16–40 (1927)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Loos, R., Weispfenning, V.: Applying linear quantifier elimination. Comput. J. 36, 450–462 (1993)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mahboubi, A.: Contributions à la certification des calculs sur ℝ: théorie, preuves, programmation. Ph.D. thesis, Université de Nice (2006)Google Scholar
  17. 17.
    McLaughlin, S., Harrison, J.: A proof-producing decision procedure for real arithmetic. In: Nieuwenhuis, R. (ed.) Automated Deduction—CADE-20. LNCS, vol. 3632, pp. 295–314. Springer, Heidelberg (2005)Google Scholar
  18. 18.
    Monniaux, D.: A quantifier elimination algorithm for linear real arithmetic. In: Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2008). LNCS, vol. 5330, pp. 243–257. Springer, Heidelberg (2008)Google Scholar
  19. 19.
    Motzkin, T.: Beiträge zur Theorie der Linearen Ungleichungen. PhD thesis, Universität Basel (1936)Google Scholar
  20. 20.
    Nipkow, T.: Linear quantifier elimination. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) Automated Reasoning (IJCAR 2008). LNCS, vol. 5195, pp. 18–33. Springer, Heidelberg (2008)Google Scholar
  21. 21.
    Nipkow, T.: Reflecting quantifier elimination for linear arithmetic. In: Grumberg, O., Nipkow, T., Pfaller, C. (eds.) Formal Logical Methods for System Security and Correctness, pp. 245–266. IOS, Amsterdam (2008)Google Scholar
  22. 22.
    Nipkow, T., Paulson, L., Wenzel, M.: Isabelle/HOL—A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)Google Scholar
  23. 23.
    Norrish, M.: Complete integer decision procedures as derived rules in HOL. In: Basin, D., Wolff, B. (eds.) Theorem Proving in Higher Order Logics, TPHOLs 2003. LNCS, vol. 2758, pp. 71–86. Springer, Heidelberg (2003)Google Scholar
  24. 24.
    Obua, S.: Proving bounds for real linear programs in Isabelle/HOL. In: Hurd, J. (ed.) Theorem Proving in Higher Order Logics (TPHOLs 2005). LNCS, vol. 3603, pp. 227–244. Springer, Heidelberg (2005)Google Scholar
  25. 25.
    Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du I Congrès de Mathématiciens des Pays Slaves, pp. 92–101 (1929)Google Scholar
  26. 26.
    Weispfenning, V.: The complexity of linear problems in fields. J. Symbol. Comput. 5, 3–27 (1988)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Institut für InformatikTechnische Universität MünchenMunichGermany

Personalised recommendations