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 $d$.

中文翻译：

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正式化多面体的面格

面在多面体的组合和计算方面起着核心作用。在本文中，我们展示了证明助手Coq中多面体的第一个形式化面。这建立在图书馆形式化的基础上，该图书馆提供了多面体的基本构造和操作，包括投影，凸包和线性地图下的图像。此外，我们设计了一种特殊的机制，可以根据证明的上下文自动引入多面体或面的适当表示形式。我们通过建立面的一些最重要的组合特性来证明这种方法的可用性，即它们构成了在区间子格下封闭的一系列渐变的原子性和涂层性格。