Formalizing the Face Lattice of Polyhedra

RICARDO D. KATZ; XAVIER ALLAMIGEON; PIERRE YVES STRUB

Paris

Conferencia; 10th International Joint Conference on Automated Reasoning (IJCAR 2020); 2020

Université Paris 13

Faces play a central role in the combinatorial and computational aspects of polyhedra. In this paper, we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library providing the basic constructions and operations over polyhedra, including projections, convex hulls and images under linear maps. Moreover, we design a special mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. We 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 sublattices.