Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps

This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by structural or noetherian induction: Genus Theorem, Euler's Formula, constructive planarity criteria. A notion of ring of faces is inductively defined and a Jordan Curve Theorem is stated and proven for any planar hypermap

Formal study of plane Delaunay triangulation

This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe triangulations, we rely on a combinatorial hypermap specification framework we have been developing for years. We embed hypermaps in the plane by attaching coordinates to elements in a consistent way. We then describe what are legal and illegal Delaunay edges and a flipping operation which we show preserves hypermap, triangulation, and embedding invariants. To prove the termination of the algorithm, we use a generic approach expressing that any non-cyclic relation is well-founded when working on a finite set

Formalizing generalized maps in Coq

AbstractThis paper is the first half of a two-part series devoted to an exemplary formal proof of a fundamental result in the field of geometry—the theorem of classification of surfaces—which has major implications in computer graphics. We study here the specification of generalized maps, a topological combinatory model for surfaces subdivisions. We show how we developed in Coq two fundamentally distinct formalizations of generalized maps, each based on one of the standard definitions, in a single common framework, then used this specification to prove for the first time their complete equivalence