Formalizing the Face Lattice of Polyhedra

TL;DR

This paper formalizes the concept of faces of polyhedra within the Coq proof assistant, enabling rigorous proofs of key combinatorial properties and theorems related to polyhedral faces.

Contribution

It introduces the first formalization of polyhedral faces in Coq, including mechanisms for automatic representation and proofs of fundamental properties and the Balinski theorem.

Findings

Faces form graded atomistic and coatomistic lattices

Faces are closed under interval sublattices

Proved Balinski's theorem on $d$-connectedness

Abstract

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 interval sublattices. We also prove a theorem due to Balinski on the $d$-connectedness of the adjacency graph of polytopes of dimension $d$.