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

Abstract

Mechanizing proofs of geometric theorems in 3D is significantly more challenging than in 2D. As a first noteworthy case study, we consider an iconic theorem of 3D geometry: Dandelin-Gallucci's theorem. We work in the very simple but powerful framework of projective incidence geometry, where only incidence relationships are considered. We study and compare two new and very different approaches to prove this theorem. First, we propose a new proof based on the well-known Wu's method. Second, we use an original method based on matroid theory to generate a proof script which is then checked by the Coq proof assistant. For each method, we point out which parts of the proof we manage to carry out automatically and which parts are more difficult to automate and require human interaction. We hope these first developments will lead to formally proving more 3D theorems automatically and that it will be used to formally verify some key properties of computational geometry algorithms in 3D. CCS CONCEPTS• Theory of computation → Automated reasoning; • Computing methodologies → Theorem proving algorithms.

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

“…This theorem was proved interactively using Coq a few years ago [14]. It proceeds as follows: we prove the property in the case where the 2 triangles are not coplanar 4 (as shown in Fig. 3).…”

“…We only give a brief description here, the interested reader can consult David's thesis [5] (in French), or another brief introduction in [4]. The algorithm is based on a graph whose nodes are labeled by a set, a lower bound and an upper bound for the value of the rank function for this set.…”

Towards Automatic Transformations of Coq Proof Scripts

“…Even though our framework can be used to implement other proof script transformations, this one is of special interest to us. Indeed, we recently designed a prover for projective incidence geometry [3,12] which relies on the concept of rank to carry out proofs of geometric theorems such as Desargues or Dandelin-Gallucci automatically. This prover produces a trace (a large Coq proof script containing several statements and their proofs).…”

“…We recently developed a new way [3,12], based on ranks, to automatically prove statements in projective incidence geometry. Our approach works well but produces proof scripts which are very large and often feature several auxiliary lemmas.…”

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]. The automated prover, named Bip for matroid Based Incidence Prover, is designed to prove equality between ranks of various sets of points.…”