Skip to main content
SpringerLink
Log in

A Certified Proof of the Cartan Fixed Point Theorems

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

Abstract

In this paper we present a machine-certified proof for the Cartan Fixed Point Theorems in the univariate case, using the HOL Light theorem prover.

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. Ahlfors, L.V.: Complex analysis, 3rd edn. In: An Introduction to the Theory of Analytic Functions of One Complex Variable, International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York (1978)

    Google Scholar 

  2. Burns, D.M., Krantz, S.G.: Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J. Am. Math. Soc. 7(3), 661–676 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cartan, H.: Sur les fonctions de deux variables complexes. Les transformations d’un domaine borné D en un domaine intérieur à D. Bull. Soc. Math. Fr. 58, 199–219 (1930)

    MathSciNet  MATH  Google Scholar 

  4. Cartan, H.: Sur les fonctions de plusieurs variables complexes. L’itération des transformations intérieures d’un domaine borné. Math. Z. 35(1), 760–773 (1932)

    Article  MathSciNet  Google Scholar 

  5. Colombo, F., Gentili, G., Sabadini, I., Struppa, D.: Extension results for slice regular functions of a quaternionic variable. Adv. Math. 222, 1793–1808 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Corbineau, P.: A declarative language for the coq proof assistant. In: Miculan, M., Scagnetto, I., Honsell, F. (eds.) TYPES. Lecture Notes in Computer Science, vol. 4941, pp. 69–84. Springer (2007)

  7. Franzoni, T., Vesentini, E.: Holomorphic Maps And Invariant Distances. Notas de Matemática [Mathematical Notes], vol. 69. North-Holland Publishing Co., Amsterdam (1980)

    Google Scholar 

  8. Fueter, R.: Die funktionentheorie der differentialgleichungen Θu = 0 und ΘΘu = 0 mit vier reellen variablen. Comment. Math. Helv. 7(1), 307–330 (1934)

    Article  MathSciNet  Google Scholar 

  9. Gentili, G., Stoppato, C.: Power series and analyticity over the quaternions (2009)

  10. Gentili, G., Stoppato, C., Struppa, D., Vlacci, F.: Recent developments for regular functions of a hypercomplex variable. In: Sabadini, I., Shapiro, M., Sommen, F. (eds.) Hypercomplex Analysis, Trends in Mathematics, pp. 165–185. Birkhauser, Basel (2009)

    Google Scholar 

  11. Gentili, G., Struppa, D.C.: A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris 342(10), 741–744 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216(1), 279–301 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gonthier, G.: Formal proof—the four-color theorem. Not. Am. Math. Soc. 55(11), 1382–1393 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Hales, T.C.: Formal proof. Not. Am. Math. Soc. 55(11), 1370–1380 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Harrison, J.: The HOL Light theorem prover. Freely available on the Internet at http://www.cl.cam.ac.uk/~jrh13/hol-light/. Accessed 9 August 2010

  16. Harrison, J.: HOL Light: a tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) Proceedings of the First International Conference on Formal Methods in Computer-Aided Design (FMCAD’96). Lecture Notes in Computer Science, vol. 1166, pp. 265–269. Springer, Verlag (1996)

  17. Harrison, J.: Proof style. In: Giménex, E., Pausin-Mohring, C. (eds.) Types for Proofs and Programs: International Workshop TYPES’96. Lecture Notes in Computer Science, vol. 1512, pp. 154–172. Springer, Verlag, Aussois, France (1996)

    Google Scholar 

  18. Harrison, J.: A HOL theory of Euclidean space. In: Hurd, J., Melham, T. (eds.) Theorem Proving in Higher Order Logics, 18th International Conference, TPHOLs 2005. Lecture Notes in Computer Science, vol. 3603, Springer, Verlag, Oxford, UK (2005)

    Google Scholar 

  19. Harrison, J.: Formalizing basic complex analysis. In: Matuszewski, R., Zalewska, A. (eds.) From Insight to Proof: Festschrift in Honour of Andrzej Trybulec. Studies in Logic, Grammar and Rhetoric, vol. 10(23), pp. 151–165. University of Białystok (2007)

  20. Harrison, J.: Formal proof—theory and practice. Not. Am. Math. Soc. 55(11), 1395–1406 (2008)

    MATH  Google Scholar 

  21. Kaup, W.: Reelle transformationsgruppen und invariante Metriken auf komplexen Räumen. Invent. Math 3, 43–70 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  22. Krantz, S.G.: Function Theory of Several Complex Variables, 2nd edn. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA (1992)

    MATH  Google Scholar 

  23. Rudin, W.: Function Theory in the Unit Ball of \(C\sp{n}\). Grundlehren der Mathematischen Wissenschaften, vol. 241. Springer, Verlag (1980)

  24. Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85(2), 199–224 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vesentini, E.: Capitoli scelti della teoria delle funzioni olomorfe. Quaderni dell’Unione Matematica Italiana. Oderisi, Gubbio (1984)

    Google Scholar 

  26. Wenzel, M.M., Technische Universität München.: Isabelle/isar—a versatile environment for human-readable formal proof documents (2002)

  27. Wiedijk, F.: The de bruijn factor (2000)

  28. Wiedijk, F.: Mizar light for hol light. In: Theorem Proving in Higher Order Logics: TPHOLs 2001. LNCS vol. 2152, pp. 378–393 (2001)

  29. Wiedijk, F.: Formal proof—getting started. Not. Am. Math. Soc. 55(11), 1408–1417 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/a, 50134, Firenze, Italy

    Gianni Ciolli, Graziano Gentili & Marco Maggesi

Authors
  1. Gianni Ciolli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Graziano Gentili
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marco Maggesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Maggesi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ciolli, G., Gentili, G. & Maggesi, M. A Certified Proof of the Cartan Fixed Point Theorems. J Autom Reasoning 47, 319–336 (2011). https://doi.org/10.1007/s10817-010-9198-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-010-9198-6

Keywords

Search

Navigation