Theory of Computing Systems

, Volume 43, Issue 3–4, pp 583–602 | Cite as

Realizability interpretation of proofs in constructive analysis

Article
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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.

Keywords

Realizability Continuous inverse function Intermediate value theorem Program extraction Non-computational quantifier 
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References

  1. 1.
    Andersson, P.: Exact real arithmetic with automatic error estimates in a computer algebra system. Master’s thesis, Mathematics Department, Uppsala University (2001) Google Scholar
  2. 2.
    Berger, J.: Exact calculation of inverse functions. Math. Log. Q. 51(2), 201–205 (2005) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berger, U.: Uniform Heyting arithmetic. Ann. Pure Appl. Log. 133, 125–148 (2005) MATHCrossRefGoogle Scholar
  4. 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) CrossRefGoogle Scholar
  5. 5.
    Berger, U., Eberl, M., Schwichtenberg, H.: Term rewriting for normalization by evaluation. Inf. Comput. 183, 19–42 (2003) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967) MATHGoogle Scholar
  7. 7.
    Cruz-Filipe, L.: Constructive real analysis: a type-theoretical formalization and applications. Ph.D. Thesis, Nijmegen University (2004) Google Scholar
  8. 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) CrossRefGoogle Scholar
  9. 9.
    Gödel, K.: Über eine bisher noch nicht benützte Erweiterung des finiten Standpunkts. Dialectica 12, 280–287 (1958) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hurewicz, W.: Lectures on Ordinary Differential Equations. MIT Press, Cambridge (1958) MATHGoogle Scholar
  11. 11.
    Kohlenbach, U.: Real growth in standard parts of analysis. Habilitationsschrift, Fachbereich Mathematik, Universität Frankfurt am Main (1995) Google Scholar
  12. 12.
    Kohlenbach, U.: Proof theory and computational analysis. Electron. Notes Theor. Comput. Sci. 13 (1998), 34 p. Google Scholar
  13. 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. 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) CrossRefGoogle Scholar
  15. 15.
    Mandelkern, M.: Continuity of monotone functions. Pac. J. Math. 99(2), 413–418 (1982) MATHMathSciNetGoogle Scholar
  16. 16.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966) MATHGoogle Scholar
  17. 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. 18.
    Troelstra, A.S. (ed.): Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Math., vol. 344. Springer, Berlin (1973) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität MünchenMunichGermany

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