Polyhedra genus theorem and Euler formula: A hypermap-formalized intuitionistic proof

Authors:
Jean-François Dufourd
Affiliations:
Université de Strasbourg UFR de Mathématique et dInformatique, Laboratoire des Sciences de lImage, de lInformatique et de la Télédétection (LSIIT, UMR CNRS-ULP 7005), P ...
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

This article presents formalized intuitionistic proofs for the polyhedra genus theorem, the Euler formula and a sufficient condition of planarity. They are based on a hypermap model for polyhedra and on formal specifications in the Calculus of Inductive Constructions. First, a type of free maps is inductively defined from three atomic constructors. Next, a hierarchy of types defined by invariants, with operations constrained by preconditions, is built on the free maps: hypermaps, orientated combinatorial maps and a central notion of quasi-hypermaps. Besides, the proofs of their properties are established until the genus theorem and the Euler formula, mainly using a simple induction principle based on the free map term algebra. Finally, a constructive sufficient condition for polyhedra to be planar is set and proved. The whole process is assisted by the interactive Coq proof system.