  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 $d$.
