Formal Description of Geometrical Properties

Tuan Minh Pham 1
1 MARELLE - Mathematical, Reasoning and Software
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Synthetic geometry, sometimes also called axiomatic geometry, deals purely with geometric objects. Relying on axioms and theorems relating the basic concepts of geometry makes it possible to highlight the geometrical properties during the proofs. This thesis concentrates on geometrical properties and their description in the Coq proof assistant. The two main results of this thesis are a library of formal descriptions and a proof system extension to interact directly with geometrical objects during proofs. The first part presents our formalization of Euclidean geometry based on affine geometry. We approach notions, properties, and theorems in a style similar to what is taught in high school. This development improves on the library developed by F. Guilhot by eliminating needless axioms, giving more appropriate definitions for some geometric notions and re-formalizing their properties. The second part deals with the notion of orientation in the plane. It allows us to remove ambiguities in the presentation of geometric objects and state and solve ordered geometry problems. The third part deals with foundations. In particular, we show that the axiom systems of Hilbert and Tarski can be described on top of ours. We also show that automatic proof methods like the area method and Gr\"obner bases can be integrated. The work on the area method was performed jointly with J. Narboux. The fourth part presents a combination of the Coq formal proof tool and the Geogebra dynamic geometry tool. This is based on the java-based user-interface for Coq named Pcoq. This combination allows users to easily perform geometry reasoning as taught in high school and in a interactive manner with the support of a graphic interface. This shows how a proof system could be used in education.
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Tuan Minh Pham. Formal Description of Geometrical Properties. Logic in Computer Science [cs.LO]. Univeristé Nice Sophia Antipolis, 2011. English. ⟨tel-01112334⟩

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