Pierre Boutry

National Institute for Research in Computer Science and Control | INRIA · MARELLE - Mathematical, Reasoning and Software Research Team

In this paper we focus on the formalization of the proof of equivalence between different versions of Euclid's 5 th postulate. This postulate is of historical importance because for centuries many mathematicians believed that this statement was rather a theorem which could be derived from the first four of Euclid's postulates and history is rich of...

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 g...

Using small scale automation to improve both accessibility and readability of formal proofs in geometry. Francisco Botana and Pedro Quaresma. Abstract. This paper describes some techniques to help building formal proofs in geometry and at the same time improving readability. Rather than trying to completely automate the proving process we provide s...

This paper describes the formalization and implementation of a reflexive tactic for automated generation of proofs of incidence to an affine variety. Incidence proofs occur frequently in formal proofs of geometric statements. Nevertheless they are most of the time omitted in pen-and-paper proofs since they do not contribute to the understanding of...

We describe formalization of the Poincaré disc model of hyperbolic geometry within the Isabelle/HOL proof assistant. The model is defined within the complex projective line CP1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgre...

In this thesis, we investigate how a proof assistant can be used to study the foundations of geometry. We start by focusing on ways to axiomatize Euclidean geometry and their relationship to each other. Then, we expose a new proof that Euclid's parallel postulate is not derivable from the other axioms of first-order Euclidean geometry.This leads us...

In this paper we focus on the formalization of the proofs of equivalence between different versions of Euclid’s 5th postulate. Our study is performed in the context of Tarski’s neutral geometry, or equivalently in Hilbert’s geometry defined by the first three groups of axioms, and uses an intuitionistic logic, assuming excluded-middle only for poin...

This paper describes the formalization of the arithmetization of Euclidean plane geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of the book by Schwabhäuser,...

This paper describes the formalization of the arithmetization of Euclidean geometry in the Coq proof assistant. As a basis for this work, Tarski’s system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of Metamathematische Methoden in d...