Two cryptomorphic formalizations of projective incidence geometry

Abstract

Incidence geometry is a well-established theory which captures the very basic properties of all geometries in terms of points belonging to lines, planes, etc. Moreover, projective incidence geometry leads to a simple framework where many properties can be studied. In this article, we consider two very different but complementary mathematical approaches formalizing this theory within the Coq proof assistant. The first one consists of the usual and synthetic geometric axiom system often encountered in the literature. The second one is more original and relies on combinatorial aspects through the notion of rank which is based on the matroid structure of incidence geometry. This paper mainly contributes to the field by proving the equivalence between these two approaches in both 2D and 3D. This result allows us to study the further automation of many proofs of projective geometry theorems. We give an overview of techniques that will be heavily used in the equivalence proof and are generic enough to be reused later in yet-to-bewritten proofs. Finally, we discuss the possibilities of future automation that can be envisaged using the rank notion.

Mechanization of Incidence Projective Geometry in Higher Dimensions, a Combinatorial Approach

“…The solver is also able to yield a Coq proof from the history of the computation of the rank function. Besides, the proof of equivalence between the geometric and the algebraic approaches has been proved by our team in a previous work ( [14] and [6]). This allows us to consider mechanizing theorems expressed in geometric terms, the constraints expressed through the rank function being in a way, the "assembling language" of the prover.…”

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

“…Using the Coq proof assistant, we formally showed in [6] that this set of axioms together with the matroid axioms (A1) to (A3) is equivalent to the usual synthetic axiom system presented in Sect. We successfully applied the approach based on ranks to prove a 2D version of Desargues property in a 3D setting [16].…”

Integrating an Automated Prover for Projective Geometry as a New Tactic in the Coq Proof Assistant

“…The above two approaches are shown to be equivalent [5] and the combinatorial one can be successfully used to automatically prove some emblematic theorems of 3D projective incidence geometry [4]. Among them we can cite Desargues' theorem and Dandelin-Gallucci's theorem [6].…”