## Xavier Allamigeon ; Ricardo D. Katz ; Pierre-Yves Strub - Formalizing the Face Lattice of Polyhedra

lmcs:7436 - Logical Methods in Computer Science, May 18, 2022, Volume 18, Issue 2 - https://doi.org/10.46298/lmcs-18(2:10)2022
Formalizing the Face Lattice of Polyhedra

Authors: Xavier Allamigeon ; Ricardo D. Katz ; Pierre-Yves Strub

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 $d$.

Volume: Volume 18, Issue 2
Published on: May 18, 2022
Accepted on: March 7, 2022
Submitted on: May 3, 2021
Keywords: Computer Science - Logic in Computer Science,Mathematics - Combinatorics,Mathematics - Optimization and Control