In this talk, I will present the project Coq-Polyhedra that aims at formalizing the theory of polyhedra as well as polyhedral computations in the proof assistant Coq.
I will explain how the intuitionistic nature of the logic of a proof assistant like Coq requires to define basic properties of polyhedra in a quite different way than is usually done, by relying on a formal proof of the simplex method. I will also focus on the formalization of the faces of polyhedra, and present a mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. I will demonstrate the usability of this approach by establishing some of the most important combinatorial properties of faces, namely that they constitute a family of graded atomistic and coatomistic lattices closed under interval sublattices, as well as Balinski’s theorem on the d-connectedness of the graph of d-polytopes. Finally, I will discuss recent progress on the formal computation of the graph of a polytope directly within the proof assistant, thanks to a certified algorithm that checks a posteriori the output of Avis’ vertex enumeration library lrslib.
Joint work with Quentin Canu, Ricardo D. Katz and Pierre-Yves Strub.