A Decision Procedure for Geometry in Coq

Abstract

We present in this paper the development of a decision procedure for affine plane geometry in the Coq proof assistant. Among the existing decision methods, we have chosen to implement one based on the area method developed by Chou, Gao and Zhang, which provides short and "readable" proofs for geometry theorems. The idea of the method is to express the goal to be proved using three geometric quantities and eliminate points in the reverse order of their construction thanks to some elimination lemmas.

Formalizing Projective Plane Geometry in Coq

“…Frédérique Guilhot realized a large Coq development about Euclidean geometry following a presentation suitable for use in french high-school [13]. In [24,25], Julien Narboux presented the formalization and implementation in the Coq proof assistant of the area decision procedure of Chou, Gao and Zhang [5] and a formalization of foundations of Euclidean geometry based on Tarski's axiom system [33,30]. In [11], Jean Duprat proposes the formalization in Coq of an axiom system for compass and ruler geometry.…”

Automatic Verification of Regular Constructions in Dynamic Geometry Systems

“…The prover can be used in conjunction with other dynamic geometry tools. Apart from the original implementation by its authors [4,5], we are aware of another two geometry provers based on the area method: one within the Theorema project [3], and one within the system Coq (COQareaMethod) [18].…”

Mechanical Theorem Proving in Tarski’s Geometry

“…Concerning the proof of programs in the field of computational geometry we can cite the formalization of convex hulls algorithms by David Pichardie and Yves Bertot in Coq [11] and by Laura Meikle and Jacques Fleuriot in Isabelle [12] and the formalization of an image segmentation algorithm by Jean-François Dufourd [13]. In [14,15], we have presented the formalization and implementation in the Coq proof assistant of the area decision procedure of Chou, Gao and Zhang [16].…”

“…This work is under progress. We also plan to enrich our formalization to use it as a foundation for other formal Coq developments about geometry such as Frédérique Guilhot formalization of geometry as it is presented in the french curriculum [10] and our implementation in Coq of the area method of Chou, Gao and Zhang [14]. A longer-term challenge would be to perform a systematic development of geometry similar to the book of Schwabhäuser, Szmielew and Tarski but in the context of a constructive axiom system such as the axiom system of von Plato [24] which has already been formalized in the Coq proof assistant by Gilles Khan [25].…”

“…The system checks whether the proof is correct. In [14], we have described the formalization of decision procedure for geometry. This formalization, allows to use the area method to generate automatically proofs which are double checked by the Coq system.…”