Skip to main content
SpringerLink
Log in

On the Mechanization of the Proof of Hessenberg’s Theorem in Coherent Logic

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

We propose to combine interactive proof construction with proof automation for a fragment of first-order logic called Coherent Logic (CL). CL allows enough existential quantification to make Skolemization unnecessary. Moreover, CL has a constructive proof system based on forward reasoning, which is easy to automate and where standardized proof objects can easily be obtained. We have implemented in Prolog a CL prover which generates Coq proof scripts. We test our approach with a case study: Hessenberg’s Theorem, which states that in elementary projective plane geometry Pappus’ Axiom implies Desargues’ Axiom. Our CL prover makes it possible to automate large parts of the proof, in particular taking care of the large number of degenerate cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bezem, M., Coquand, T.: Newman’s lemma—a case study in proof automation and geometric logic. In: Păun, G., Rozenberg, G., Salomaa, A. (eds.) Current Trends in Theoretical Computer Science, vol. 2, pp. 267–282. World Scientific, Singapore (2004)

    Google Scholar 

  2. Bezem, M., Coquand, T.: Automating coherent logic. In: Sutcliffe, G., Voronkov, A. (eds.) Proceedings LPAR-12. Lecture Notes in Computer Science, vol. 3825, pp. 246–260. Springer-Verlag, Berlin (2005)

    Google Scholar 

  3. Bezem, M., Hendriks, D.: Web page including CL tool, input files, Coq files. http://www.cs.vu.nl/~diem/research/ht/.

  4. Bezem, M., Hendriks, D., de Nivelle, H.: Automated proof construction in type theory using resolution. J. Autom. Reason. 29(3), 253–275 (2002)

    Article  MATH  Google Scholar 

  5. Bonichon, R., Delahaye, D., Doligez, D.: Zenon: an extensible automated theorem prover producing checkable proofs. In: Dershowitz, N., Voronkov, A. (eds.) Proceedings LPAR-14. Lecture Notes in Computer Science, vol. 4790, pp. 151–165. Springer-Verlag, Berlin (2007)

    Google Scholar 

  6. Coxeter, H.S.M.: The Real Projective Plane, 2nd edn. Cambridge University Press, Cambridge (1955)

    MATH  Google Scholar 

  7. Cronheim, A.: A proof of Hessenberg’s theorem. Proc. AMS 4(2), 219–221 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  8. de Nivelle, H.: Translation of resolution proofs into short first-order proofs without choice axioms. Inf. Comput. 199(1), 24–54 (2005) [Special Issue on the 19th International Conference on Automated Deduction (CADE-19)]

    Article  MATH  Google Scholar 

  9. de Nivelle, H., Meng, J.: Geometric resolution: a proof procedure based on finite model search. In: Harrison, J., Furbach, U., Shankar, N. (eds.) International Joint Conference on Automated Reasoning 2006, p. 15. Springer, Seattle (2006)

    Google Scholar 

  10. Harper, R., Honsell, F., Plotkin, G.D.: A framework for defining logics. J. ACM, 40(1), 143–184 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hessenberg, G.: Beweis des Desarguesschen Satzes aus dem Pascalschen. Math. Ann. 61, 161–172 (1905)

    Article  MathSciNet  Google Scholar 

  12. Johnstone, P.: Sketches of an Elephant: A Topos Theory Compendium, vol. 2. Oxford University Press, London (2002)

    Google Scholar 

  13. Kusak, E., Leończuk, W.: Hessenberg theorem. Formaliz. Math. 2(2), 217–219 (1991)

    Google Scholar 

  14. Nederpelt, R.P., Geuvers, J.H., de Vrijer, R.C. (eds.): Selected Papers on Automath. North-Holland, Amsterdam (1994)

    MATH  Google Scholar 

  15. Paulson, L.C.: Isabelle: A Generic Theorem Prover (with a contribution by T. Nipkow). Lecture Notes in Computer Science, vol.  828. Springer, Berlin (1994)

    Google Scholar 

  16. Paulson, L.C.: A generic tableau prover and its integration with Isabelle. J. Univers. Comput. Sci. 5(3), 73–87 (1999)

    MATH  MathSciNet  Google Scholar 

  17. Robinson, J.A., Voronkov, A. (eds.): Handbook of Automated Reasoning (in 2 volumes). Elsevier and MIT Press, Amsterdam (2001)

    Google Scholar 

  18. Skolem, Th.: Logisch-Kombinatorische Untersuchungen über die Erfüllbarkeit und Beweisbarkeit Mathematischen Sätze nebst einem Theoreme über Dichte Mengen, Skrifter I, 4, 1–36, Det Norske Videnskaps-Akademi, 1920. In: Fenstad, J.E. (ed.) Selected Works in Logic, pp. 103–136. Universitetsforlaget, Oslo (1970)

  19. Sutcliffe, G., Suttner, C.: The TPTP problem library: CNF release v1.2.1. J. Autom. Reason. 21(2), 177–203 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sutcliffe, G., Suttner, C.: The state of CASC. AI Commun. 19(1), 35–48 (2006)

    MATH  MathSciNet  Google Scholar 

  21. The Coq development team. The Coq Proof Assistant Reference Manual, version 8.1, 2006. http://coq.inria.fr/.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Bergen, P.O. Box 7800, 5020, Bergen, Norway

    Marc Bezem

  2. Department of Computer Science, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081, HV Amsterdam, The Netherlands

    Dimitri Hendriks

Authors
  1. Marc Bezem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitri Hendriks
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Bezem.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bezem, M., Hendriks, D. On the Mechanization of the Proof of Hessenberg’s Theorem in Coherent Logic. J Autom Reasoning 40, 61–85 (2008). https://doi.org/10.1007/s10817-007-9086-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-007-9086-x

Keywords

Search

Navigation