Toward the use of a proof assistant to teach mathematics

International audienceProof is a crucial aspect of mathematics and must have a prominent role in the education. Dynamic Geometry Software (D.G.S.) and Computer Algebra Software (C.A.S) are widely used in a pedagogical context. These tools are used to explore, visualize, calculate, find counter examples, conjectures, or check facts, but most of them can not be used to build a proof in itself. But there are software whose sole purpose is to help the user produce proofs : the proof assistants. We believe that proof assistants are now mature enough to be adapted to the education. After giving a quick overview of what a proof assistant is, we will discuss the possible advantages of using it in the education. Finally we report on the ongoing work to ease the use of a proof assistant in the classroom

Proof-checking Euclid

We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski's formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48 propositions, we proved 235 theorems. The extras were partly "Book Zero", preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid's gaps, and partly just variants of Euclid's propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid's logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq.Comment: 53 page

Les assistants de preuve, ou comment avoir confiance en ses démonstrations.

National audiencePrésentation de vulgarisation sur les assistants de preuve et la correspondance de Curry-Howar

A formalization of diagrammatic proofs in abstract rewriting

Diagrams are in common use in the rewriting community. In this paper, we present a formalization of this kind of diagrams. We give a formal definition for the diagrams used to state properties. We propose inference rules to formalize the reasoning depicted by some well known diagrammatic proofs : a transitivity property of some abstract rewriting systems and the Newman's lemma. We show that the system proposed is both correct and complete for a class of formulas called coherent logic

Mechanical Theorem Proving in Tarski's geometry.

International audienceThis paper describes the mechanization of the proofs of the first height chapters of Schwabäuser, Szmielew and Tarski's book: Metamathematische Methoden in der Geometrie. The goal of this development is to provide foundations for other formalizations of geometry and implementations of decision procedures. We compare the mechanized proofs with the informal proofs. We also compare this piece of formalization with the previous work done about Hilbert's Gründlagen der Geometrie. We analyze the differences between the two axiom systems from the formalization point of view

Herbrand's theorem and non-Euclidean geometry

We use Herbrand's theorem to give a new proof that Euclid's parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.Comment: 12 pages, 5 figure

A Synthetic Proof of Pappus’ Theorem in Tarski’s Geometry

In this paper, we report on the formalization of a synthetic proof of Pappus' theorem. We provide two versions of the theorem: the first one is proved in neutral geometry (without assuming the parallel postulate), the second (usual) version is proved in Euclidean geometry. The proof that we formalize is the one presented by Hilbert in The Foundations of Geometry which has been detailed by Schwabhäuser , Szmielew and Tarski in part I of Metamathematische Methoden in der Geometrie. We highlight the steps which are still missing in this later version. The proofs are checked formally using the Coq proof assistant. Our proofs are based on Tarski's axiom system for geometry without any continuity axiom. This theorem is an important milestone toward obtaining the arithmetization of geometry which will allow us to provide a connection between analytic and synthetic geometry

Towards a Certified Version of the Encyclopedia of Triangle Centers

Triangle centers such as the center of gravity, the circumcenter, the orthocenter are well studied by geometers. Recently, under the guidance of Clark Kimberling, an electronic encyclopedia of triangle centers (ETC) has been developed, it contains more than 7000 centers and many properties of these points. In this paper, we describe how we created a certified version of ETC such that some of the properties described come along with a computer checked proof of its validity

A Coq-based Library for Interactive and Automated Theorem Proving in Plane Geometry

International audienceIn 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, 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 which was developed by J. Narboux in Coq