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A Coq-Based Library for Interactive and Automated Theorem Proving in Plane Geometry

  • Tuan-Minh Pham
  • Yves Bertot
  • Julien Narboux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6785)

Abstract

In this article, we present the development of a library of formal proofs for theorem proving in plane geometry in a pedagogical context. We use the Coq proof assistant [4]. This library includes the basic geometric notions to state theorems and provides a database of theorems to construct interactive proofs more easily. It is an extension of the library of F. Guilhot for interactive theorem proving at the level of high-school geometry [7], where we eliminate redundant axioms and give formalizations for the geometric concepts using a vector approach. We also enrich this library by offering an automated deduction method which can be used as a complement to interactive proof. For that purpose, we integrate the formalization of the area method [3] which was developed by J. Narboux in Coq [12, 10].

Keywords

formalization automation geometry Coq teaching 

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References

  1. 1.
    Berger, M., Pansu, P., Berry, J.P., Saint-Raymond, X.: Problems in Geometry. Springer, Heidelberg (1984)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bertot, Y., Guilhot, F., Pottier, L.: Visualizing Geometrical Statements with GeoView. Electronic Notes in Theoretical Computer Science 103, 49–65 (2003)CrossRefGoogle Scholar
  3. 3.
    Chou, S.C., Gao, X.S., Zhang, J.Z.: Machine Proofs in Geometry. World Scientific, Singapore (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Coq development team: The Coq Proof Assistant Reference Manual, Version 8.3. TypiCal Project (2010), http://coq.inria.fr
  5. 5.
    Dehlinger, C., Dufourd, J.-F., Schreck, P.: Higher-Order Intuitionistic Formalization and Proofs in Hilbert’s Elementary Geometry. In: Richter-Gebert, J., Wang, D. (eds.) ADG 2000. LNCS (LNAI), vol. 2061, pp. 306–324. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Geogebra development team: Introduction to GeoGebra, http://www.geogebra.org/book/intro-en/
  7. 7.
    Guilhot, F.: Formalisation en Coq et visualisation d’un cours de géométrie pour le lycée. TSI 24, 1113–1138 (2005) (in french)CrossRefGoogle Scholar
  8. 8.
    Hilbert, D.: Les fondements de la géométrie. In: Gabay, J. (ed.) Edition critique avec introduction et compléments préparée par Paul Rossier. Dunod, Paris (1971)Google Scholar
  9. 9.
    Janicić, P.: Geometry construction language. Journal of Automated Reasoning 44, 3–24 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Janicic, P., Narboux, J., Quaresma, P.: The Area Method: a Recapitulation. Journal of Automated Reasoning (2010)Google Scholar
  11. 11.
    Meikle, L., Fleuriot, J.: Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In: Theorem Proving in Higher Order Logics, pp. 319–334 (2003)Google Scholar
  12. 12.
    Narboux, J.: A Decision Procedure for Geometry in Coq. In: Slind, K., Bunker, A., Gopalakrishnan, G.C. (eds.) TPHOLs 2004. LNCS, vol. 3223, pp. 225–240. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Narboux, J.: Toward the use of a proof assistant to teach mathematics. In: Proceedings of the 7th International Conference on Technology in Mathematics Teaching, ICTMT 7 (2005)Google Scholar
  14. 14.
    Narboux, J.: A graphical user interface for formal proofs in geometry. J. Autom. Reasoning 39(2), 161–180 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Narboux, J.: Mechanical Theorem Proving in Tarski’s Geometry. In: Botana, F., Recio, T. (eds.) ADG 2006. LNCS (LNAI), vol. 4869, pp. 139–156. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Pham, T.-M., Bertot, Y.: A combination of a dynamic geometry software with a proof assistant for interactive formal proofs. In: UITP Workshops 2010 (2010)Google Scholar
  17. 17.
    Scott, P., Fleuriot, J.: Idle time discovery in geometry theorem proving. In: Proceedings of ADG 2010 (2010)Google Scholar
  18. 18.
    Tarski, A.: What is Elementary Geometry?. In: Henkin, L., Suppes, P., Tarski, A. (eds.) The axiomatic Method, with special reference to Geometry and Physics, pp. 16–29. North-Holland, Amsterdam (1959)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Tuan-Minh Pham
    • 1
  • Yves Bertot
    • 1
  • Julien Narboux
    • 2
  1. 1.INRIA Sophia AntipolisFrance
  2. 2.LSIIT, Université de Strasbourg, CNRSFrance

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