Framework for Constructive Geometry (Based on the Area Method

2006

In this technical report we present a rational reconstruction of the area method (developed by Chou, Gao and Zhang) for automated theorem proving for Euclidean geometry. Our rational reconstruction covers all relevant lemmas proved in full details and also full details of required algebraic reasoning (missing from the papers introducint the area method). We also present our implementation of this algorithm, made within the program GCLC. The area method main idea is to express the hypothesis of a theorem using a set of constructive statements each of then introducing a new point, and to express the conclusion by a polynomial in a some geometry quantities, without any relation to a given system of coordinates. The proof is then developed by eliminating, in reverse order, the point introduced before, using for that purpose a set of lemmas. After eliminating all the introduced points the polynomial is just an equality between two rational expression in independent variables. Hence if they are equal the statement is true, otherwise it is false. The proofs generated by the prover (developed as a part of GCLC) are generally short and readable. The program can prove many non-trivial theorems in a very efficient way.

Journal of Automated Reasoning

The area method for Euclidean constructive geometry was proposed by Chou, Gao and Zhang in the early 1990’s. The method can efficiently prove many non-trivial geometry theorems and is one of the most interesting and most successful methods for automated theorem proving in geometry. The method produces proofs that are often very concise and human-readable. In this paper, we provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted in the original presentations. We also give a variant of Chou, Gao and Zhang’s axiom system. Based on this axiom system, we proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there are only three implementations more. Although the basic idea of the method is simple, implementing it is a very challenging task because of a number of details that has to be dealt with. With the description of the method given in this paper, implementing the method should be still complex, but a straightforward task. In the paper we describe all these implementations and also some of their applications.

Journal of Automated Reasoning, 2004

J. Autom. Reason., 2012

The area method for Euclidean constructive geometry was pro posed by Chou, Gao and Zhang in the early 1990’s. The method can efficiently p rove many non-trivial geometry theorems and is one of the most interesting and most su cce sful methods for automated theorem proving in geometry. The method produces proo fs that are often very concise and human-readable. In this paper, we provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted i n the original presentations. We also give a variant of Chou, Gao and Zhang’s axiom system. Bas ed on this axiom system, we proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there are only three implementations more . Although the basic idea of the method is simple, implementing it is a very challenging task because of a number of de...

… Science and Its Applications-ICCSA 2011, 2011

In this article, we present the development of a library of formal proofs for theorem proving in plane geometry in a pedagogical context. We use the Coq proof assistant . This library includes the basic geometric notions to state theorems and provides a database of theorems to construct interactive proofs more easily. It is an extension of the library of F. Guilhot for interactive theorem proving at the level of high-school geometry [7], where we eliminate redundant axioms and give formalizations for the geometric concepts using a vector approach. We also enrich this library by offering an automated deduction method which can be used as a complement to interactive proof. For that purpose, we integrate the formalization of the area method [3] which was developed by J. Narboux in Coq .

Journal of Symbolic Computation

This paper describes the formalization of the arithmetization of Euclidean plane geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of the book by Schwabhäuser Szmielew and Tarski Metamathematische Methoden in der Geometrie. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a "back-translation" from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method. Moreover, we solve a challenge proposed by Beeson: we prove that, given two points, an equilateral triangle based on these two points can be constructed in Euclidean Hilbert planes. Finally, we derive the axioms for another automated deduction method: the area method.

Automated Deduction in Geometry, 2013

We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations. But this does not produce computer-checkable first-order proofs in geometry. We might try to produce such proofs directly, or we might try to develop a "back-translation" from algebra to geometry, following Descartes but with computer in hand. This paper discusses the relations between the two approaches, the attempts that have been made, and the obstacles remaining. On the theoretical side we give a new first-order theory of "vector geometry", suitable for formalizing geometry and algebra and the relations between them. On the practical side we report on some experiments in automated deduction in these areas. 1 A very good piece of work towards formalizing Euclid is [1], but because it mixes computations (decision procedures) with first-order proofs, it does not furnish a counterexample to the statement in the text.

Collins' Cylindrical Algebraic Decomposition method (CAD) can be used to prove geometry theorems that involve order relation (that is, the algebraic form consists of polynomial equalities and inequalities). Unfortunately only very simple geometric statements can be proved this way, as the method is very time consuming. To overcome the slowness of Collins' CAD method for complicated polynomials we propose a method (section 4) that combines the area method for computing geometric quantities and the CAD method. We present an implementation of this method as part of the Geometry Prover in the frame of the Theorema project.

Electronic Proceedings in Theoretical Computer Science, 2021

Mathematical proof is undoubtedly the cornerstone of mathematics. The emergence, in the last years, of computing and reasoning tools, in particular automated geometry theorem provers, has enriched our experience with mathematics immensely. To avoid disparate efforts, the Open Geometry Prover Community Project aims at the integration of the different efforts for the development of geometry automated theorem provers, under a common "umbrella". In this article the necessary steps to such integration are specified and the current implementation of some of those steps is described.

2009

The area method for Euclidean constructive geometry was proposed by Chou et al. in early 1990's. The method produces human-readable proofs and can efficiently prove many nontrivial theorems. It can be considered as one of the most interesting and most successful methods in geometry theorem proving and probably the most successful in the domain of automated production of readable proofs. In this research report, we focus on the rigorous proofs of all the lemmas of the area method. This text is meant as a support text for the article, The Area Method: a Recapitulation,