Proving Bounds for Real Linear Programs in Isabelle/HOL

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 demonstrate that via
reflection and 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.