A Matroid-Based Automatic Prover and Coq Proof Generator for Projective Incidence Geometry

Abstract

We present an automatic theorem prover for projective incidence geometry. This prover does not consider coordinates. Instead, it follows a combinatorial approach based on the concept of rank. This allows to deal only with sets of points and to capture relations between objects of the projective space (equality, collinearity, coplanarity, etc.) in a homogenous way. Taking advantage of the computational aspect of this approach, we automatically compute by saturation the ranks of all sets of the powerset of the points of the geometric configuration we consider. Upon completion of the saturation phase, our prover then retraces the proof process and generates the corresponding Coq code. This code is then formally checked by the Coq proof assistant, thus ensuring that the proof is actually correct. We use the prover to verify some well-known, non-trivial theorems in projective space geometry, among them: Desargues’ theorem and Dandelin–Gallucci’s theorem.