Theory of linear and integer programming
Theory of linear and integer programming
Two Fast Algorithms for Sparse Matrices: Multiplication and Permuted Transposition
ACM Transactions on Mathematical Software (TOMS)
Translating Dependent Type Theory into Higher Order Logic
TLCA '93 Proceedings of the International Conference on Typed Lambda Calculi and Applications
Programming and Computing in HOL
TPHOLs '00 Proceedings of the 13th International Conference on Theorem Proving in Higher Order Logics
Organizing Numerical Theories Using Axiomatic Type Classes
Journal of Automated Reasoning
Defining functions on equivalence classes
ACM Transactions on Computational Logic (TOCL)
Isabelle/HOL: a proof assistant for higher-order logic
Isabelle/HOL: a proof assistant for higher-order logic
IJCAR '08 Proceedings of the 4th international joint conference on Automated Reasoning
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
Flyspeck II: the basic linear programs
Annals of Mathematics and Artificial Intelligence
Fast reflexive arithmetic tactics the linear case and beyond
TYPES'06 Proceedings of the 2006 international conference on Types for proofs and programs
Journal of Automated Reasoning
Formally verified conditions for regularity of interval matrices
AISC'10/MKM'10/Calculemus'10 Proceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics
Point-free, set-free concrete linear algebra
ITP'11 Proceedings of the Second international conference on Interactive theorem proving
Partizan games in Isabelle/HOLZF
ICTAC'06 Proceedings of the Third international conference on Theoretical Aspects of Computing
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Formal global optimisation with taylor models
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Checking conservativity of overloaded definitions in higher-order logic
RTA'06 Proceedings of the 17th international conference on Term Rewriting and Applications
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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.