Abstract
Linear programming is a basic mathematical technique for optimizing a linear function on a domain that is constrained by linear inequalities. We restrict ourselves to linear programs on bounded domains that involve only real variables. In the context of theorem proving, this restriction makes it possible for any given linear program to obtain certificates from external linear programming tools that help to prove arbitrarily precise bounds for the given linear program. To this end, an explicit formalization of matrices in Isabelle/HOL is presented, and how the concept of lattice-ordered rings allows for a smooth integration of matrices with the axiomatic type classes of Isabelle.
As our work is a contribution to the Flyspeck project, we argue that with the above techniques it is now possible to prove bounds for the linear programs arising in the proof of the Kepler conjecture sufficiently fast.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hales, T.C.: Some algorithms arising in the proof of the Kepler conjecture, sect. 3.1.1., arXiv:math.MG/0205209
Hales, T.C.: A Proof of the Kepler Conjecture. Annals of Mathematics (to appear)
The Flyspeck Project Fact Sheet. http://www.math.pitt.edu/~thales/flyspeck/index.html
Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Springer, Heidelberg (2002)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley & Sons, Chichester (1986)
Paulson, L.C.: Organizing Numerical Theories Using Axiomatic Type Classes. Journal of Automated Reasoning (in press)
Paulson, L.C.: Defining Functions on Equivalence Classes. ACM Transactions on Computational Logic (in press)
Two Fast Algorithms for Sparse Matrices: Multiplication and Permuted Transposition. ACM Transactions on Mathematical Software 4(3), 250–269 (1978)
Barras, B.: Programming and Computing in HOL. In: Aagaard, M.D., Harrison, J. (eds.) TPHOLs 2000 LNCS, vol. 1869, pp. 17–37. Springer, Heidelberg (2000)
Jacobs, B., Melham, T.: Translating Dependent Type Theory into Higher Order Logic. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993, vol. 664, pp. 209–229. Springer, Heidelberg (1993)
Lang, S.: Algebra. Addison-Wesley, Reading (1974)
Birkhoff, G.: Lattice Theory. AMS (1967)
Fuchs, L.: Partially ordered algebraic systems. Addison-Wesley, Reading (1963)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Obua, S. (2005). Proving Bounds for Real Linear Programs in Isabelle/HOL. In: Hurd, J., Melham, T. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2005. Lecture Notes in Computer Science, vol 3603. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11541868_15
Download citation
DOI: https://doi.org/10.1007/11541868_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28372-0
Online ISBN: 978-3-540-31820-0
eBook Packages: Computer ScienceComputer Science (R0)