Four Approaches to Automated Reasoning with Differential Algebraic Structures

  • Jesús Aransay
  • Clemens Ballarin
  • Julio Rubio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3249)


While implementing a proof for the Basic Perturbation Lemma (a central result in Homological Algebra) in the theorem prover Isabelle one faces problems such as the implementation of algebraic structures, partial functions in a logic of total functions, or the level of abstraction in formal proofs. Different approaches aiming at solving these problems will be evaluated and classified according to features such as the degree of mechanization obtained or the direct correspondence to the mathematical proofs. From this study, an environment for further developments in Homological Algebra will be proposed.


Algebraic Structure Chain Complex Theorem Prover Automate Reasoning Algebraic Topology 
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.
    Ballarin, C.: Locales and Locale Expressions in Isabelle/Isar. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 34–50. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Berghofer, S.: Program Extraction in Simply-Typed Higher Order Logic. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 21–38. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Brown, R.: The twisted Eilenberg-Zilber theorem, Celebrazioni Arch. Secolo XX, Simp. Top, pp. 34-37 (1967)Google Scholar
  4. 4.
    Calmet, J.: Some Grand Mathematical Challenges in Mechanized Mathematics. In: Hardin, T., Rioboo, R. (eds.) Calculemus 2003, pp. 137–141 (2003)Google Scholar
  5. 5.
    Dousson, X., Sergeraert, F., Siret, Y.: The Kenzo program,
  6. 6.
    Glimming, J.: Logic and Automation for Algebra of Programming, Master Thesis, Maths Institute, University of Oxford (August 2001), available at
  7. 7.
    Gugenheim, V.K.A.M.: On the chain complex of a fibration. Illinois Journal of Mathematics 16, 398–414 (1972)MATHMathSciNetGoogle Scholar
  8. 8.
    Kobayashi, H., Suzuki, H., Murao, H.: Rings and Modules in Isabelle/HOL. In: Hardin, T., Rioboo, R. (eds.) Calculemus 2003, pp. 124–129 (2003)Google Scholar
  9. 9.
    Lambán, L., Pascual, V., Rubio, J.: An object-oriented interpretation of the EAT system. Appl. Algebra Eng. Commun. Comput. 14(3), 187–215Google Scholar
  10. 10.
    Mac Lane, S.: Homology. Springer, Heidelberg (1994)Google Scholar
  11. 11.
    Naraschewski, W., Wenzel, M.: Object-Oriented Verification based on Record Subtyping in Higher-Order Logic. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 349–366. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)MATHCrossRefGoogle Scholar
  13. 13.
    Paulson, L.: Defining Functions on Equivalence Classes, Report, available at
  14. 14.
    Rubio, J., Sergeraert, F.: Constructive Algebraic Topology. Lecture Notes Summer School in Fundamental Algebraic Topology, Institut Fourier (1997)Google Scholar
  15. 15.
    Rubio, J., Sergeraert, F., Siret, Y.: EAT: Symbolic Software for Effective Homology Computation, Institut Fourier, Grenoble (1997)Google Scholar
  16. 16.
    Shih, W.: Homologie des espaces fibrés, Publications Math.ématiques de l’I.H.E.S. 13 (1962)Google Scholar
  17. 17.
    Théry, L.: Proving and computing: A certified version of the Buchberger’s algorithm. In: Kirchner, C., Kirchner, H. (eds.) CADE 1998. LNCS (LNAI), vol. 1421, p. 349. Springer, Heidelberg (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jesús Aransay
    • 1
  • Clemens Ballarin
    • 2
  • Julio Rubio
    • 1
  1. 1.Dpto. de Matemáticas y ComputaciónUniv. de La RiojaLogroñoSpain
  2. 2.Institut für InformatikTechnische Univ. MünchenGarchingGermany

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