Skip to main content
SpringerLink
Log in

Realizability interpretation of proofs in constructive analysis

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language—a version of Gödel’s T—evaluation is reasonably efficient.

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. Andersson, P.: Exact real arithmetic with automatic error estimates in a computer algebra system. Master’s thesis, Mathematics Department, Uppsala University (2001)

  2. Berger, J.: Exact calculation of inverse functions. Math. Log. Q. 51(2), 201–205 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berger, U.: Uniform Heyting arithmetic. Ann. Pure Appl. Log. 133, 125–148 (2005)

    Article  MATH  Google Scholar 

  4. Berger, U.: Program extraction from normalization proofs. In: Bezem, M., Groote, J. (eds.) Typed Lambda Calculi and Applications. Lecture Notes in Comput. Sci., vol. 664, pp. 91–106. Springer, Berlin (1993)

    Chapter  Google Scholar 

  5. Berger, U., Eberl, M., Schwichtenberg, H.: Term rewriting for normalization by evaluation. Inf. Comput. 183, 19–42 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  7. Cruz-Filipe, L.: Constructive real analysis: a type-theoretical formalization and applications. Ph.D. Thesis, Nijmegen University (2004)

  8. Geuvers, H., Wiedijk, F., Zwanenburg, J.: A constructive proof of the fundamental theorem of algebra without using the rationals. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) Proc. Types 2000. Lecture Notes in Comput. Sci., vol. 2277, pp. 96–111. Springer, Berlin (2000)

    Chapter  Google Scholar 

  9. Gödel, K.: Über eine bisher noch nicht benützte Erweiterung des finiten Standpunkts. Dialectica 12, 280–287 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hurewicz, W.: Lectures on Ordinary Differential Equations. MIT Press, Cambridge (1958)

    MATH  Google Scholar 

  11. Kohlenbach, U.: Real growth in standard parts of analysis. Habilitationsschrift, Fachbereich Mathematik, Universität Frankfurt am Main (1995)

  12. Kohlenbach, U.: Proof theory and computational analysis. Electron. Notes Theor. Comput. Sci. 13 (1998), 34 p.

  13. Kreisel, G.: Interpretation of analysis by means of constructive functionals of finite types. In: Heyting, A. (ed.) Constructivity in Mathematics, pp. 101–128. North-Holland, Amsterdam (1959)

    Google Scholar 

  14. Letouzey, P.: A new extraction for Coq. In: Geuvers, H., Wiedijk, F. (eds.) Types for Proofs and Programs, Second International Workshop, TYPES 2002. Lecture Notes in Comput. Sci., vol. 2646. Springer, Berlin (2003)

    Chapter  Google Scholar 

  15. Mandelkern, M.: Continuity of monotone functions. Pac. J. Math. 99(2), 413–418 (1982)

    MATH  MathSciNet  Google Scholar 

  16. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  17. Schwichtenberg, H.: Constructive analysis with witnesses. In: Schwichtenberg, H., Spies, K. (eds.) Proof Technology and Computation. Proc. NATO Advanced Study Institute, Marktoberdorf, 2003. Series III: Computer and Systems Sciences, vol. 200, pp. 323–353. IOS Press, Utrecht (2006)

    Google Scholar 

  18. Troelstra, A.S. (ed.): Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Math., vol. 344. Springer, Berlin (1973)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität München, Theresienstr. 39, 80333, Munich, Germany

    Helmut Schwichtenberg

Authors
  1. Helmut Schwichtenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helmut Schwichtenberg.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schwichtenberg, H. Realizability interpretation of proofs in constructive analysis. Theory Comput Syst 43, 583–602 (2008). https://doi.org/10.1007/s00224-007-9027-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-007-9027-4

Keywords

Search

Navigation