Abstract
In this paper, we investigate the possibility to mechanize the proof of some real complex mathematical theorems in B [1]. For this, we propose a little structure language which allows one to encode mathematical structures and their accompanying theorems. A little tool is also proposed, which translates this language into B, so that Atelier B, the tool associated with B, can be used to prove the theorems. As an illustrative example, we eventually (mechanically) prove the Theorem of Zermelo [6] stating that any set can be well-ordered. The present study constitutes a complete reshaping of an earlier (1993) unpublished work (referenced in [4]) done by two of the authors, where the classical theorems of Haussdorf and Zorn were also proved.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J.R. Abrial. The B-Book:Assigning Programs to Meanings. Cambridge University Press (1996).
N. Bourbaki. Théorie des ensembles. Hermann, Paris, 1970.
S. Owre, N. Shankar, J. M. Rushby, and D. W. J. Stringer-Calvert PVS language reference version 2.3. Technical report, SRI International, September 1999.
L. C. Paulson and K. Grabczewski. Mechanizing set theory: Cardinal arithmetic and the axiom of choice. Journal of Automated Reasoning, 17(3):291–323, December 1996.
Markus Wenzel. Using axiomatic type classes in Isabelle. part of the Isabelle distribution. Technical report, TU München, February 2001.
E. Zermelo. Neuer Beweis für die Möglichkeit einer Wohlordnung. Mathematische Annalen, 65:107–128, 1908.
A. J. M. van Gasteren. On the Shape of Mathematical Arguments, volume 445 of Lecture Notes in Computer Science. Springer-Verlag, 1990.
Clear Sy. Atelier B (version 3.6). 2001.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Abrial, JR., Cansell, D., Laffitte, G. (2002). “Higher-Order” Mathematics in B. In: Bert, D., Bowen, J.P., Henson, M.C., Robinson, K. (eds) ZB 2002:Formal Specification and Development in Z and B. ZB 2002. Lecture Notes in Computer Science, vol 2272. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45648-1_19
Download citation
DOI: https://doi.org/10.1007/3-540-45648-1_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43166-4
Online ISBN: 978-3-540-45648-3
eBook Packages: Springer Book Archive