Journal of Automated Reasoning

, Volume 40, Issue 4, pp 271–292 | Cite as

A Mechanized Proof of the Basic Perturbation Lemma



We present a complete mechanized proof of the result in homological algebra known as basic perturbation lemma. The proof has been carried out in the proof assistant Isabelle, more concretely, in the implementation of higher-order logic (HOL) available in the system. We report on the difficulties found when dealing with abstract algebra in HOL, and also on the ongoing stages of our project to give a certified version of some of the algorithms present in the Kenzo symbolic computation system.


Homological algebra Isabelle Basic perturbation lemma 


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  1. 1.
    Aransay, J.: Mechanized reasoning in homological algebra. Ph.D. Thesis, Universidad de La Rioja. (2006)
  2. 2.
    Aransay, J.: A formalized proof of the basic perturbation lemma in Isabelle/HOL. (2007)
  3. 3.
    Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem. ACM Trans. Comput. Log. (2008)Google Scholar
  4. 4.
    Ballarin, C.: Locales and locale expressions in Isabelle/Isar. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003, 3rd International Workshop on Types for Proofs and Programs, Torino, Italy, May 2003. Lecture Notes in Computer Science, vol. 3085, pp. 34–50. Springer (2004)Google Scholar
  5. 5.
    Ballarin, C.: Interpretation of locales in Isabelle: managing dependencies between locales. Technical Report TUM-I0607, Technische Universität München. (2006a)
  6. 6.
    Ballarin, C.: Interpretation of locales in Isabelle: theories and proof contexts. In: Borwein, J.M., Farmer, W.M. (eds.) MKM 2006, 5th International Conference on Mathematical Knowledge Management, Wokingham, UK, August 2006. Lecture Notes in Artificial Intelligence, vol. 4108, pp. 31–43. Springer (2006b)Google Scholar
  7. 7.
    Ballarin, C., Kammüller, F., Paulson, L.: The Isabelle/HOL algebra library. (2005)
  8. 8.
    Barnes, D., Lambe, L.: Fixed point approach to homological perturbation theory. Proc. Am. Math. Soc. 112(3), 881–892 (1991)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Bauer, G., Wenzel, M.: Computer-assisted mathematics at work (the Hahn–Banach theorem in Isabelle/Isar). In: Coquand, T., Dybjer, P., Nordström, B., Smith, J. (eds.) TYPES’99, Types for Proofs and Programs International Workshop, Lökeberg, Sweden, June 1999. Lecture Notes in Computer Science, vol. 1956, pp. 61–76. Springer (2000)Google Scholar
  10. 10.
    Berghofer, S.: Program extraction in simply-typed higher order logic. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002, 2nd International Workshop on Types for Proofs and Programs, Berg en Dal, The Netherlands, April 2002. Lecture Notes in Computer Science, vol. 2646, pp. 21–38. Springer (2003a)Google Scholar
  11. 11.
    Berghofer, S.: Proofs, programs and executable specifications in higher order logic. Ph.D. Thesis, Technische Universität München (2003b)Google Scholar
  12. 12.
    Berghofer, S.: Answer to Tom Ridge. Available at the mail list, February 18. (2005)
  13. 13.
    Bertot, Y., Castéran, P.: Interactive theorem proving and program development. In: Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science, vol. 25. Springer (2004)Google Scholar
  14. 14.
    Calmet, J.: Some grand mathematical challenges in mechanized mathematics. In: Hardin, T., Rioboo, R. (eds.) Calculemus 2003, 11th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, Rome, Italy, September 2003, pp. 137–141. Aracne Editrice S.R.L. (2003)Google Scholar
  15. 15.
    Coquand, T., Lombardi, H.: A logical approach to abstract algebra. Math. Struct. Comput. Sci. 16(5), 885–900 (2006)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Coquand, T., Spiwack, A.: Towards constructive homological algebra in type theory. In: Kauers, R.M.M., Kerber, M., Windsteiger, W. (eds.) 14th Symposium, Calculemus 2007, 6th International Conference, MKM 2007, Hagenberg, Austria, June 2007. Lecture Notes in Computer Science, vol. 4573, pp. 40–54. Springer (2007)Google Scholar
  17. 17.
    Dousson, X., Sergeraert, F., Siret, Y.: The Kenzo program. (1999)
  18. 18.
    Gugenheim, V.K.A.M.: On the chain complex of a fibration. Ill. J. Math. 16(3), 398–414 (1972)MathSciNetMATHGoogle Scholar
  19. 19.
    Jacobson, N.: Basic Algebra 2, 2nd edn. W.H. Freeman and Company (1989)Google Scholar
  20. 20.
    Johnstone, P.T.: Notes on Logic and Set Theory. Cambridge University Press (1987)Google Scholar
  21. 21.
    Kammüller, F., Paulson, L.C.: A formal proof of Sylow’s theorem – an experiment in abstract algebra with Isabelle/HOL. J. Autom. Reason. 23(3), 235–264 (1999)CrossRefMATHGoogle Scholar
  22. 22.
    Kobayashi, H., Suzuki, H., Ono, Y.: Formalization of Hensel’s lemma. In: Hurd, J., Smith, E., Darbari, A. (eds.) Theorem Proving in Higher Order Logics: Emerging Trends Proceedings. Oxford University Computing Laboratory (2005)Google Scholar
  23. 23.
    Mac Lane, S.: Homology. Springer (1963)Google Scholar
  24. 24.
    Markov, A.A.: On constructive mathematics. Am. Math. Soc. Transl. 2(98), 1–9 (1971)Google Scholar
  25. 25.
    Müller, O., Slind, K.: Treating partiality in a logic of total functions. Comput. J. 40(10), 640–652 (1997)CrossRefGoogle Scholar
  26. 26.
    Naraschewski, W., Wenzel, M.: Object-oriented verification based on record subtyping in higher-order logic. In: Grundy, J., Newey, M. (eds.) TPHOLs’98, 11th International Conference on Theorem Proving in Higher Order Logics, Canberra, Australia, September 1998. Lecture Notes in Computer Science. vol. 1479, pp. 349–366. Springer (1998)Google Scholar
  27. 27.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: a proof assistant for higher order logic. Lecture Notes in Computer Science, vol. 2283. Springer (2002)Google Scholar
  28. 28.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle’s logics: HOL. (2005)
  29. 29.
    Paulson, L.C.: The foundation of a generic theorem prover. J. Autom. Reason. 5(3), 363–397 (1989)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Paulson, L.C.: A formulation of the simple theory of types (for Isabelle). In: Martin-Löf, P., Mints, G. (eds.) COLOG-88, International Conference on Computer Logic, Tallinn, USSR, December 1988. Lecture Notes in Computer Science, vol. 417, pp. 246–274. Springer (1990a)Google Scholar
  31. 31.
    Paulson, L.C.: Isabelle: the next 700 theorem provers. In: Odifreddi, P. (ed.) Logic and Computer Science, pp. 361–386. Academic Press (1990b)Google Scholar
  32. 32.
    Rubio, J., Sergeraert, F.: Constructive algebraic topology. Lecture Notes Summer School in Fundamental Algebraic Topology, Institut Fourier. (1997)
  33. 33.
    Rubio, J., Sergeraert, F.: Constructive algebraic topology. Bulletin des Sciences Mathématiques 126(5), 389–412 (2002)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Thompson, S.: Type Theory and Functional Programming. Addison-Wesley (1991)Google Scholar
  35. 35.
    Troelstra, A., van Dalen, D.: Constructivism in Mathematics, vol. 2. Studies in Logic and the Foundations of Mathematics, vol. 123. North-Holland Press (1988)Google Scholar
  36. 36.
    Wenzel, M.: Isabelle/Isar – a versatile environment for human-readable formal proof documents. Ph.D. Thesis, Technische Universität München (2002)Google Scholar

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© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Dpto. de Matemáticas y ComputaciónUniversidad de La RiojaLogroñoSpain
  2. 2.Institut für InformatikUniversität InnsbruckInnsbruckAustria

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