Diophantus' 20th Problem and Fermat's Last Theorem for n=4: Formalization of Fermat's Proofs in the Coq Proof Assistant

We present the proof of Diophantus' 20th problem (book VI of Diophantus' Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used the "wonderful" method (ipse dixit Fermat) of infinite descent. This method, which is, historically, the first use of induction, consists in producing smaller and smaller non-negative integer solutions assuming that one exists; this naturally leads to a reductio ad absurdum reasoning because we are bounded by zero. We describe the formalization of this proof which has been carried out in the Coq proof assistant. Moreover, as a direct and no less historical application, we also provide the proof (by Fermat) of Fermat's last theorem for n=4, as well as the corresponding formalization made in Coq.Comment: 16 page

The Three Gap Theorem : Specification and Proof in Coq

Projet COQWe present a specification and a proof in Coq of the three gap theorem, or initially Steinhaus conjecture whose result is the following: let N be the distribution of points placed consecutively around a circle by an angle of $\alpha{}$; then the points partition the circle into gaps of at most three different lengths. We start by making an axiomatization of the real numbers in Coq in order to use them in the development. Thereafter, we define all the mathematical tools necessary and some lemmas used in the proof. Finally we state the theorem

Coloured Petri Net Refinement Specification and Correctness Proof with Coq

In this work, we address the formalisation of symmetric nets, a subclass of coloured Petri nets, refinement in COQ. We first provide a formalisation of the net models, and of their type refinement in COQ. Then the COQ proof assistant is used to prove the refinement correctness lemma. An example adapted from a protocol example illustrates our work

Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program

Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring specific accuracy issues due to their massive use of floating-point computations. Yet, it is uncommon to guarantee their correctness. Indeed, we had to extend existing methods and tools for proving the correct behavior of programs to verify an existing numerical analysis program. This C program implements the second-order centered finite difference explicit scheme for solving the 1D wave equation. In fact, we have gone much further as we have mechanically verified the convergence of the numerical scheme in order to get a complete formal proof covering all aspects from partial differential equations to actual numerical results. To the best of our knowledge, this is the first time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with arXiv:1112.179

Certified, Efficient and Sharp Univariate Taylor Models in COQ

International audienceWe present a formalisation, within the COQ proof assistant, of univariate Taylor models. This formalisation being executable, we get a generic library whose correctness has been formally proved and with which one can effectively compute rigorous and sharp approximations of univariate functions composed of usual functions such as 1/x, sqrt(x), exp(x), sin(x) among others. In this paper, we present the key parts of the formalisation and we evaluate the quality of our certified library on a set of examples

A Coq Formalization of Lebesgue Induction Principle and Tonelli's Theorem

International audienceLebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus, its formalization in a proof assistant is to be designed to fit different goals and projects. Once the Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is Tonelli's theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. This article describes the formal definition and proof in Coq of product sigma-algebras, product measures and their uniqueness, the construction of iterated integrals, up to Tonelli's theorem. We also advertise the Lebesgue induction principle provided by an inductive type for nonnegative measurable functions

Une formalisation en Coq de l'intégrale de Lebesgue des fonctions positives

Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs involving the use of integration, directly or indirectly. By its capability to extend the (Riemann) integral to a wide class of irregular functions, and to functions defined on more general spaces than the real line, the Lebesgue integral is perfectly suited for use in mathematical fields such as probability theory, numerical mathematics, and real analysis. In this article, we present the Coq formalization of $\sigma$-algebras, measures, simple functions, and integration of nonnegative measurable functions, up to the full formal proofs of the Beppo Levi (monotone convergence) theorem and Fatou's lemma. More than a plain formalization of the known literature, we present several design choices made to balance the harmony between mathematical readability and usability of Coq theorems. These results are a first milestone toward the formalization of $L^p$~spaces such as Banach spaces.Le calcul intégral, tout comme le calcul différentiel, est un outil fondamental utilisé largement dans de nombreux domaines scientifiques. La formalisation de la notion mathématique d'intégrale et de ses propriétés dans un assistant de preuve aide à donner la plus grande confiance sur la correction de programmes numériques utilisant l'intégration, directement ou indirectement. De part sa capacité à étendre l'intégrale (de Riemann) à une large classe de fonctions irrégulières, et à des fonctions définies sur des espaces plus généraux que la droite réelle, l'intégrale de Lebesgue est considérée comme parfaitement adaptée aux domaines mathématiques tels que la théorie des probabilités, les mathématiques numériques et l'analyse réelle. Dans cet article, nous présentons la formalisation en Coq des tribus (ou $\sigma$-algèbres), des mesures, des fonctions étagées et de l'intégrale des fonctions mesurables positives, jusqu'aux preuves formelles complètes du théorème de convergence monotone de Beppo Levi et du lemme de Fatou. Plus qu'une simple formalisation de la littérature connue, nous présentons plusieurs choix de design menés pour équilibrer l'harmonie entre la lisibilité mathématique et l'ergonomie des théorèmes Coq. Ces résultats sont un premier jalon vers la formalisation des espaces~$L^p$ comme espaces de Banach

Wave Equation Numerical Resolution: a Comprehensive Mechanized Proof of a C Program

We formally prove correct a C program that implements a numerical scheme for the resolution of the one-dimensional acoustic wave equation. Such an implementation introduces errors at several levels: the numerical scheme introduces method errors, and floating-point computations lead to round-off errors. We annotate this C program to specify both method error and round-off error. We use Frama-C to generate theorems that guarantee the soundness of the code. We discharge these theorems using SMT solvers, Gappa, and Coq. This involves a large Coq development to prove the adequacy of the C program to the numerical scheme and to bound errors. To our knowledge, this is the first time such a numerical analysis program is fully machine-checked.Comment: No. RR-7826 (2011

The Three Gap Theorem (Steinhauss Conjecture)

We deal with the distribution of N points placed consecutively around the circle by a fixed angle of a. From the proof of Tony van Ravenstein, we propose a detailed proof of the Steinhaus conjecture whose result is the following: the N points partition the circle into gaps of at most three different lengths. We study the mathematical notions required for the proof of this theorem revealed during a formal proof carried out in Coq