Geoproof, a user interface for formal proofs in geometry

2005

This paper outlines Geometry Explorer, a prototype system that allows users to create Euclidean geometry constructions using a dynamic geometry interface, specify conjectures about them and then use a full-angle method prover to automatically produce diagram independent, human-readable proofs to theorems. Our system can then automatically generate novel diagrammatic proofs of the forward-chaining and backward-chaining reasoning used by the geometry theorem prover, as well as visualise multiple proofs to single theorems. We discuss the features of our system, how they were implemented and the issues encountered when trying to create diagrammatic full-angle method proofs.

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.

… 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 .

2006

The axiomatic presentation of geometry fills the gap between formal logic and our spatial intuition. The study of geometry is, and will always be, very important for a mathematical practitioner. GCLCprover, an automatic theorem prover (ATP) integrated with dynamic geometry software (DGS) gives its user a tool to bridge his/her spatial intuition with formal, Euclidean geometry proofs. GeoThms, a system consisting of the mentioned programs and a database geoDB, provides a framework for exploring geometrical knowledge. A GeoThms user can browse through a list of available geometric problems, their statements, illustrations, and proofs. He/she can also interactively produce new geometrical constructions, theorems, and proofs and add new results to the existing ones. GeoThms framework provides an environment suitable for new ways of studying and teaching geometry at different levels. GeoThms also provides a system for storing mathematical knowledge (in a explicit, declarative form) — not only theorem statements, but also their (automatically generated) proofs and corresponding illustrations.

Journal of Automated Reasoning, 2007

Rapport Technique

C OQ is a Proof Assistant for a Logical Framework known as the Calculus of Induc- tive Constructions. It allows the interactive construction of formal proofs, and also the manipulation of functional programs consistently with ...

Theorem Proving in Higher Order Logics, 2004

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.

2003

Formalizing mathematical proofs has as aim to represent arbitrary mathematical notions and proofs on a computer in order to construct a database of certified results useful to learn and develop the subject. At present it is mathematically not appealing to construct formal proofs. To make formalzing more mathematician-friendly one should have a good interface for proofs, definitions and computations. The proof-assistant Mizar does have a good interface for proofs, but not for making computations. Other assistants, like Coq based on type theory, does have a good interface for computations, but not for proofs. This paper sketches ways in which proofs are represented in a mathematical way. Although the underlying formalized statements come from the system Coq, this is not essential. Mainly the paper has as aim to convince implementers of mathematical assistants to make systems in such a way that formalizing proofs becomes natural. Much further developed is the work on Isar providing a mathematical proof language for the assistant Isabelle. The approach in this paper is to approximate a proof language by writing proof-sketches, a notion by Wiedijk, with the aim that they should eventually be verifiable by a proof-checker. Nederpelt [2002] has a different approach: there the emphasis is on the ease of providing formalizations of mathematical definitions. 1 <www.cs.kun.nl/~freek/aut> 2 <www.mizar.org> 3 <www.cs.cornell.edu/Info/Projects/NuPrl/nuprl.html> 4 <www.cl.cam.ac.uk/Research/HVG/Isabelle> 5 <archive.comlab.ox.ac.uk/formal-methods/hol.html> 6 <pauillac.inria.fr/coq> 7 <isabelle.in.tum.de/Isar>

Electronic Proceedings in Theoretical Computer Science

The geometry automated theorem proving area distinguishes itself by a large number of specific methods and implementations, different approaches (synthetic, algebraic, semi-synthetic) and different goals and applications (from research in the area of artificial intelligence to applications in education). Apart from the usual measures of efficiency (e.g. CPU time), the possibility of visual and/or readable proofs is also an expected output against which the geometry automated theorem provers (GATP) should be measured. The implementation of a competition between GATP would allow to create a test bench for GATP developers to improve the existing ones and to propose new ones. It would also allow to establish a ranking for GATP that could be used by "clients" (e.g. developers of educational e-learning systems) to choose the best implementation for a given intended use.

Lecture Notes in Computer Science

This paper introduces Proof General Kit, a framework for software components tailored to interactive proof development. The goal of the framework is to enable flexible environments for managing formal proofs across their life-cycle: creation, maintenance and exploitation. The framework connects together different kinds of component, exchanging messages using a common communication infrastructure and protocol called PGIP. The main channel connects provers to displays. Provers are the back-end interactive proof engines and displays are components for interacting with the user, allowing browsing or editing of proofs. At the core of the framework is a broker middleware component which manages proof-in-progress and mediates between components.