A case study in formalizing projective geometry in Coq: Desargues theorem

Artificial Intelligence and Symbolic Computation

2018

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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.

Section: Resultsmentioning

confidence: 99%

Formalizing Some “Small” Finite Models of Projective Geometry in Coq

Artificial Intelligence and Symbolic Computation

2018

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“…First, we will finish our case study on the possibilities of full automation in the case of finite geometries. Secondly, we are interested in a partial automation of many steps in the Desargues' theorem presented in [23,24] using mainly the submodularity. To extend this analysis, we will examine two other consequent theorems in projective incidence geometry: Dandelin-Galluci theorem [1,2] and the harmonic conjugate.…”

Section: Discussionmentioning

confidence: 99%

“…Now that we have specified the axiomatization, we will focus on proving the equivalence between these two axiom systems. In previous work [23,24], only the implication from the ranks to ≥3D projective geometry has been studied. It is sufficient to allow the formalization of Desargues theorem with the ranks.…”

Section: D Rank-based Axiom Systemmentioning

confidence: 99%

Two cryptomorphic formalizations of projective incidence geometry

2018

Ann Math Artif Intell

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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.

“…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]. The proof was performed interactively: all subsets of points involved in the proof as well as their ranks has to be determined by the user.…”

Section: Matroid Theory Applied To 3d Incidence Geometrymentioning

confidence: 99%

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

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

2021

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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.