Advertisement

TYPES 2000: Types for Proofs and Programs pp 79-95

# Constructive Reals in Coq: Axioms and Categoricity

• Herman Geuvers
• Milad Niqui
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2277)

## Abstract

We describe a construction of the real numbers carried out in the Coq proof assistant. The basis is a set of axioms for the constructive real numbers as used in the FTA (Fundamental Theorem of Algebra) project, carried out at Nijmegen University. The aim of this work is to show that these axioms can be satisffied, by constructing a model for them. Apart from that, we show the robustness of the set of axioms for constructive real numbers, by proving (in Coq) that any two models of it are isomorphic. Finally, we show that our axioms are equivalent to the set of axioms for constructive reals introduced by Bridges in [2]. The construction of the reals is done in the ‘classical way’: first the rational numbers are built and they are shown to be a (constructive) ordered field and then the constructive real numbers are introduced as the usual Cauchy completion of the rational numbers.

## Keywords

Rational Number Cauchy Sequence Proof Assistant Classical Axiom Constructive Version
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.

## Preview

Unable to display preview. Download preview PDF.

## References

1. [1]
E. Bishop and D. Bridges. Constructive Analysis. Number 279 in Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 1985.Google Scholar
2. [2]
D. Bridges. Constructive mathematics: a foundation for computable analysis. Theoretical Computer Science, 219:95–109, 1999.
3. [3]
D. Bridges and S. Reeves. Constructive mathematics in theory and programming practice. Philosophia Mathematica, 7, 1999.Google Scholar
4. [4]
J. Chirimar and D. Howe. Implementing constructive real analysis. In J.P. Myers and M.J. O’Donnel, editors, Constructivity in Computer Science, number 613 in LNCS, pages 165–178, 1992.
5. [5]
A. Ciaffaglione and P. Di Gianantonio. A coinductive approach to real numbers. In Th. Coquand, P. Dybjer, B. Nordström, and J. Smith, editors, Types 1999 Workshop, Lökeberg, Sweden, number 1956 in LNCS, pages 114–130, 2000.Google Scholar
6. [6]
A. Ciaffaglione and P. Di Gianantonio. A tour with constructive real numbers. In Types 2000 Workshop, Durham, UK, 2001. This Volume.Google Scholar
7. [7]
D. Delahaye and M. Mayero. Field: une procédure de décision pour les nombres réels en Coq. In Proceedings of JFLA 2001. INRIA, 2001.Google Scholar
8. [8]
B. Barras et al. The Coq Proof Assistant Reference Manual, Version 7.1. INRIA, http://coq.inria.fr/doc/main.html, sep 2001.
9. [9]
H. Geuvers, R. Pollack, F. Wiedijk, and J. Zwanenburg. The algebraic hierarchy of the FTA project. In Calculemus 2001 Proc., pages 13–27, Siena, Italy, 2001.Google Scholar
10. [10]
H. Geuvers, F. Wiedijk, J. Zwanenburg, R. Pollack, M. Niqui, and H. Barendregt. FTA project. http://www.cs.kun.nl/gi/projects/fta/, nov 2000.
11. [11]
J. Harrison. Theorem Proving with the Real Numbers. Distinguished dissertations. Springer, London, 1998.Google Scholar
12. [12]
C. Jones. Completing the rationals and metric spaces in LEGO. In G. Huet and G. Plotkin, editors, Logical Environments, pages 297–316. CUP, 1993.Google Scholar
13. [13]
A. Troelstra and D. van Dalen. Constructivism in Mathematics, vol I, volume 121 of Studies in Logic and The Foundation of Math. North Holland, 1988. 342 pp.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2002

## Authors and Affiliations

• Herman Geuvers
• 1
• Milad Niqui
• 1
1. 1.Department of Computer ScienceUniversity of NijmegenThe Netherlands