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Proving with ACL2 the Correctness of Simplicial Sets in the Kenzo System

  • Jónathan Heras
  • Vico Pascual
  • Julio Rubio
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6564)

Abstract

Kenzo is a Common Lisp system devoted to Algebraic Topology. Although Kenzo uses higher-order functional programming intensively, we show in this paper how the theorem prover ACL2 can be used to prove the correctness of first order fragments of Kenzo. More concretely, we report on the verification in ACL2 of the implementation of simplicial sets. By means of a generic instantiation mechanism, we achieve the reduction of the proving effort for each family of simplicial sets, letting ACL2 automate the routine parts of the proofs.

Keywords

Simplicial Complex Common Lisp Moore Space Binary List Geometric Simplex 
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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References

  1. 1.
    Andrés, M., Lambán, L., Rubio, J., Ruiz-Reina, J.L.: Formalizing Simplicial Topology in ACL2. In: Proceedings ACL2 Workshop 2007, pp. 34–39. University of Austin (2007)Google Scholar
  2. 2.
    Aransay, J., Ballarin, C., Rubio, J.: A mechanized proof of the Basic Perturbation Lemma. Journal of Automated Reasoning 40(4), 271–292 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aransay, J., Ballarin, C., Rubio, J.: Generating certified code from formal proofs: a case study in homological algebra. Formal Aspects of Computing 22(2), 193–213 (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dousson, X., Sergeraert, F., Siret, Y.: The Kenzo program. Institut Fourier, Grenoble (1998), http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo Google Scholar
  5. 5.
    Heras, J.: ACL2 verification of Kenzo simplicial sets (2010), http://www.unirioja.es/cu/joheras/ss-tool.html
  6. 6.
    Heras, J., Pascual, V., Rubio, J.: Using Open Mathematical Documents to interface Computer Algebra and Proof Assistant systems. In: Carette, J., Dixon, L., Coen, C.S., Watt, S.M. (eds.) MKM 2009, Held as Part of CICM 2009. LNCS, vol. 5625, pp. 467–473. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Kaufmann, M., Moore, J.S.: ACL2 Home Page, http://www.cs.utexas.edu/users/moore/acl2/
  8. 8.
    Lambán, L., Pascual, V., Rubio, J.: An object-oriented interpretation of the EAT system. Applicable Algebra in Engineering, Communication and Computing 14(3), 187–215 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Martín-Mateos, F.J., Alonso, J.A., Hidalgo, M.J., Ruiz-Reina, J.L.: A Generic Instantiation Tool and a Case Study: A Generic Multiset Theory. In: Proceedings of the Third ACL2 Workshop, pp. 188–203. University of Grenoble, France (2002)Google Scholar
  10. 10.
    Martín-Mateos, F.J., Rubio, J., Ruiz-Reina, J.L.: ACL2 verification of simplicial degeneracy programs in the kenzo system. In: Carette, J., Dixon, L., Coen, C.S., Watt, S.M. (eds.) MKM 2009, Held as Part of CICM 2009. LNCS, vol. 5625, pp. 106–121. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    May, J.P.: Simplicial objects in Algebraic Topology. Van Nostrand Mathematical Studies, vol. 11 (1967)Google Scholar
  12. 12.
    Rubio, J., Sergeraert, F., Siret, Y.: EAT: Symbolic Software for Effective Homology Computation. Institut Fourier, Grenoble (1990), http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/#Eat Google Scholar
  13. 13.
    Sergeraert, F.: The computability problem in Algebraic Topology. Advances in Mathematics 104, 1–29 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jónathan Heras
    • 1
  • Vico Pascual
    • 1
  • Julio Rubio
    • 1
  1. 1.Departamento de Matemáticas y ComputaciónUniversidad de La RiojaLogroñoSpain

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