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

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

Anscombe's Account of Voluntary Action in "Intention"

Podria semblar que el llibre d'Anscombe Intention considera que el concepte d'allò voluntari té una importància filosòfica secundària. No obstant això, aquesta impressió és errònia. Deriva d'un malentès de la filosofia de l'acció d'Anscombe en general i d'Intention en particular. L'argument principal d'aquest article és que, per entendre l'abast i la naturalesa de la contribució d'Intention a la comprensió d'allò voluntari, hem d'arribar a un enteniment no només de l'exposició positiva que avança el llibre partint del seus mètodes, sinó també de la naturalesa dels problemes que, sobre la base d'aquests mateixos mètodes, defuig deliberadament perquè impliquen consideracions relatives a l'ètica. Aquest article està dividit en set seccions. La secció introductòria exposa el fet que a Intention es relega el concepte de 'voluntari' a la perifèria de la filosofia de l'acció. La segona secció situa el §49 considerant Intention en conjunt i mostra per què una explicació sistemàtica d'allò voluntari es posterga fins a una etapa tan tardana de la recerca. A continuació, es procedeix a comentar la secció §49 amb l'objectiu de desplegar i defensar les diverses idees sobre el tema d'allò voluntari que s'hi apleguen sistemàticament en el context de la distinció fonamental entre allò intencional i allò voluntari. Les seccions 3-6, que constitueixen la major part de l'article, estan dedicades respectivament als quatre epígrafs sota els quals Anscombe aprehèn successivament la distinció entre allò intencional i allò voluntari en el §49. Finalment, a la darrera secció s'intenta posar en relleu la unitat subjacent de la teoria d'allò voluntari que figura en el §49, així com el caràcter deliberat de les limitacions que acompanyen aquesta explicació.It might seem that Anscombe's book Intention dismisses the concept of the voluntary as of secondary philosophical significance. However, this impression is misconceived and stems from a misunderstanding of Anscombe's philosophy of action in general and the contribution of Intention in particular. The main contention of this essay is that to understand the scope and nature of the contribution of Intention to an understanding of the voluntary we must come to terms with not only the positive account that the book advances on the basis of its methods but also the nature of the problems that it deliberately leaves out, based on these same methods, on the grounds that they involve considerations pertaining to ethics. This essay is divided into seven sections. The introductory section expounds the charge that Intention relegates the concept of the voluntary into the periphery of the philosophy of action. The next section places §49 within Intention as a whole. It seeks to explain why a systematic account of the voluntary is deferred until such a late stage in the inquiry. I then proceed to give a commentary of section §49 with the aim of unpacking and defending the various insights that are there systematically brought together against the background of the pivotal distinction between the intentional and the voluntary. Sections 3 to 6, which constitute the main bulk of this essay, are respectively devoted to the four headings under which Anscombe successively apprehends the distinction between the intentional and the voluntary in §49. Finally, in the last section, I try to bring out the underlying unity of the account of the voluntary given in §49 as well as the deliberate nature of the limitations in this account.Podría parecer que el libro de Anscombe Intention considera que el concepto de lo voluntario tiene una importancia filosófica secundaria. Sin embargo, esta impresión es errónea. Se deriva de un malentendido de la filosofía de la acción de Anscombe en general y de Intention en particular. El principal argumento de este artículo es que, para entender el alcance y la naturaleza de la contribución de Intention a la comprensión de lo voluntario, debemos llegar a un entendimiento no solo de la exposición positiva que avanza el libro partiendo de sus métodos, sino también de la naturaleza de los problemas que, sobre la base de esos mismos métodos, rehúye deliberadamente porque implican consideraciones relativas a la ética. Este artículo está dividido en siete secciones. La sección introductoria expone el hecho de que en Intention se relega el concepto de 'voluntario' a la periferia de la filosofía de la acción. La segunda sección sitúa el §49 considerando Intention en su conjunto y muestra por qué una explicación sistemática de lo voluntario se posterga hasta una etapa tan tardía de la investigación. A continuación, se procede a comentar la sección §49 con el objetivo de desplegar y defender las diversas ideas sobre el tema de lo voluntario que se reúnen sistemáticamente allí en el contexto de la distinción fundamental entre lo intencional y lo voluntario. Las secciones 3-6, que constituyen la mayor parte del artículo, están dedicadas respectivamente a los cuatro epígrafes bajo los cuales Anscombe aprehende sucesivamente la distinción entre lo intencional y lo voluntario en el §49. Finalmente, en la última sección se intenta poner de relieve la unidad subyacente de la teoría de lo voluntario que figura en el §49, así como el carácter deliberado de las limitaciones que acompañan a esta explicación

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

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

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A Graphical User Interface for Formal Proofs in Geometry.

International audienceWe present in this paper the design of a graphical user interface to deal with proofs in geometry. The software developed combines three tools: a dynamic geometry software to explore, measure and invent conjectures, an automatic theorem prover to check facts and an interactive proof system (Coq) to mechanically check proofs built interactively by the user

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