Formalizing Some " Small " Finite Models of Projective Geometry in Coq

David Braun (1) , Nicolas Magaud (1) , Pascal Schreck (1)

1 ICube - Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie

We study two different descriptions of incidence projective geometry: a synthetic, mathematics-oriented one and a more practical, computation-oriented one, based on the combinatorial concept of rank of a set of points. Using both axiom systems, we prove that some specific finite planes (resp. spaces) verify the axioms of projective plane (resp. space) geometry and Desargues' property. It requires using repeated case analysis on all variables of some finite inductive data-types and leads to numerous (sub-)goals in the Coq proof assistant. We thus investigate to what extend Coq can deal with such a combinatorial explosion in the number of cases to handle. We propose some easy-to-implement but relevant proof optimizations which, combined together, lead to an efficient way to deal with such large proofs.

Mots clés: Coq; proof automation; combinatorial explosion; finite inductive types; projective geometry; finite geometry; Desargues' property

Domaines: Intelligence artificielle [cs.AI]; Calcul formel [cs.SC]

David Braun, Nicolas Magaud, Pascal Schreck. Formalizing Some " Small " Finite Models of Projective Geometry in Coq. Proceedings of the 13th International Conference on Artificial Intelligence and Symbolic Computation (AISC'2018), Sep 2018, Suzhou, China. ⟨hal-01835493⟩