Automatic Deduction in Dynamic Geometry using Sage

We present a symbolic tool that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction. The main prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction. In one worksheet, diagrams constructed with the open source dynamic geometry system GeoGebra are accepted. In this worksheet, Groebner bases are used to either compute the equation of a geometric locus in the case of a locus construction or to determine the truth of a general geometric statement included in the GeoGebra construction as a boolean variable. In the second worksheet, locus constructions coded using the common file format for dynamic geometry developed by the Intergeo project are accepted for computation. The prototype and several examples are provided for testing. Moreover, a third Sage worksheet is presented in which a novel algorithm to eliminate extraneous parts in symbolically computed loci has been implemented. The algorithm, based on a recent work on the Groebner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Detailed examples are discussed.Comment: In Proceedings THedu'11, arXiv:1202.453

Integrating DGSs and GATPs in an Adaptative and Collaborative Blended-Learning Web-Environment

The area of geometry with its very strong and appealing visual contents and its also strong and appealing connection between the visual content and its formal specification, is an area where computational tools can enhance, in a significant way, the learning environments. The dynamic geometry software systems (DGSs) can be used to explore the visual contents of geometry. This already mature tools allows an easy construction of geometric figures build from free objects and elementary constructions. The geometric automated theorem provers (GATPs) allows formal deductive reasoning about geometric constructions, extending the reasoning via concrete instances in a given model to formal deductive reasoning in a geometric theory. An adaptative and collaborative blended-learning environment where the DGS and GATP features could be fully explored would be, in our opinion a very rich and challenging learning environment for teachers and students. In this text we will describe the Web Geometry Laboratory a Web environment incorporating a DGS and a repository of geometric problems, that can be used in a synchronous and asynchronous fashion and with some adaptative and collaborative features. As future work we want to enhance the adaptative and collaborative aspects of the environment and also to incorporate a GATP, constructing a dynamic and individualised learning environment for geometry.Comment: In Proceedings THedu'11, arXiv:1202.453

Proof Certificates for Algebra and their Application to Automatic Geometry Theorem Proving

Post-proceedings of ADG 2008 (Automated Deduction in Geometry)International audienceIntegrating decision procedures in proof assistants in a safe way is a major challenge. In this paper, we describe how, starting from Hilbert's Nullstellensatz theorem, we combine a modified version of Buchberger's algorithm and some reflexive techniques to get an effective procedure that automatically produces formal proofs of theorems in geometry. The method is implemented in the Coq system but, since our specialised version of Buchberger's algorithm outputs explicit proof certificates, it could be easily adapted to other proof assistants

Formalising Geometric Axioms for Minkowski Spacetime and Without-Loss-of-Generality Theorems

This contribution reports on the continued formalisation of an axiomatic system for Minkowski spacetime (as used in the study of Special Relativity) which is closer in spirit to Hilbert's axiomatic approach to Euclidean geometry than to the vector space approach employed by Minkowski. We present a brief overview of the axioms as well as of a formalisation of theorems relating to linear order. Proofs and excerpts of Isabelle/Isar scripts are discussed, with a focus on the use of symmetry and reasoning "without loss of generality".Comment: In Proceedings ADG 2021, arXiv:2112.1477

A Parametric Approach to 3D Dynamic Geometry

Dynamic geometry systems are computer applications allowing the exact on-screen drawing of geometric diagrams and their interactive manipulation by mouse dragging. Whereas there exists an extensive list of 2D dynamic geometry environments, the number of 3D systems is reduced. Most of them, both in 2D and 3D, share a common approach, using numerical data to manage geometric knowledge and elementary methods to compute derived objects. This paper deals with a parametric approach for automatic management of 3D Euclidean constructions. An open source library, implementing the core functions in a 3D dynamic geometry system, is described here. The library deals with constructions by using symbolic parameters, thus enabling a full algebraic knowledge about objects such as loci and envelopes. This parametric approach is also a prerequisite for performing automatic proof. Basic functions are defined for symbolically checking the truth of statements. Using recent results from the theory of parametric polynomial systems solving, the bottleneck in the automatic determination of geometric loci and envelopes is solved. As far as we know, there is no comparable library in the 3D case, except the paramGeo3D library (designed for computing equations of simple 3D geometric objects, which, however, lacks specific functions for finding loci and envelopes)