Repenser la bibliothèque réelle de Coq : vers une formalisation de l'analyse classique mieux adaptée

Real analysis is pervasive to many applications, if only because it is a suitable tool for modeling physical or socio-economical systems. As such, its support is warranted in proof assistants, so that the users have a way to formally verify mathematical theorems and correctness of critical systems. The Coq system comes with an axiomatization of standard real numbers and a library of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results. For instance, the definitions of integrals and derivatives are based on dependent types, which make them cumbersome to use in practice. This thesis first describes various state-of-the-art libraries available in proof assistants. To palliate the inadequacies of the Coq standard library, we have designed a user-friendly formalization of real analysis: Coquelicot. An easier way of writing formulas and theorem statements is achieved by relying on total functions in place of dependent types for limits, derivatives, integrals, power series, and so on. To help with the proof process, the library comes with a comprehensive set of theorems that cover not only these notions, but also some extensions such as parametric integrals and asymptotic behaviors. Moreover, an algebraic hierarchy makes it possible to apply some of the theorems in a more generic setting, such as complex numbers or matrices. Coquelicot is a conservative extension of the classical analysis of Coq's standard library and we provide correspondence theorems between the two formalizations. We have exercised the library on several use cases: in an exam at university entry level, for the definitions and properties of Bessel functions, and for the solution of the one-dimensional wave equation.L'analyse réelle a de nombreuses applications car c'est un outil approprié pour modéliser de nombreux phénomènes physiques et socio-économiques. En tant que tel, sa formalisation dans des systèmes de preuve formelle est justifié pour permettre aux utilisateurs de vérifier formellement des théorèmes mathématiques et l'exactitude de systèmes critiques. La bibliothèque standard de Coq dispose d'une axiomatisation des nombres réels et d'une bibliothèque de théorèmes d'analyse réelle. Malheureusement, cette bibliothèque souffre de nombreuses lacunes. Par exemple, les définitions des intégrales et des dérivées sont basées sur les types dépendants, ce qui les rend difficiles à utiliser dans la pratique. Cette thèse décrit d'abord l'état de l'art des différentes bibliothèques d'analyse réelle disponibles dans les assistants de preuve. Pour pallier les insuffisances de la bibliothèque standard de Coq, nous avons conçu une bibliothèque facile à utiliser : Coquelicot. Une façon plus facile d'écrire les formules et les théorèmes a été mise en place en utilisant des fonctions totales à la place des types dépendants pour écrire les limites, dérivées, intégrales et séries entières. Pour faciliter l'utilisation, la bibliothèque dispose d'un ensemble complet de théorèmes couvrant ces notions, mais aussi quelques extensions comme les intégrales à paramètres et les comportements asymptotiques. En plus, une hiérarchie algébrique permet d'appliquer certains théorèmes dans un cadre plus générique comme les nombres complexes pour les matrices. Coquelicot est une extension conservative de l'analyse classique de la bibliothèque standard de Coq et nous avons démontré les théorèmes de correspondance entre les deux formalisations. Nous avons testé la bibliothèque sur plusieurs cas d'utilisation : sur une épreuve du Baccalauréat, pour les définitions et les propriétés des fonctions de Bessel ainsi que pour la solution de l'équation des ondes en dimension 1

Différentiabilité et intégrabilité en Coq. Application à la formule de d'Alembert

National audienceLa bibliothèque standard de Coq contient de nombreuses définitions et lemmes permettant de faire de l'analyse, mais ceux-ci se limitent à l'étude de fonctions totales à une seule variable. Même si l'étude des dérivées partielles de fonctions à deux variables peut le plus souvent réutiliser les lemmes existants, l'étude de la formule de d'Alembert comme solution de l'équation des ondes en dimension 1 montre les limites de cette approche. La formalisation en Coq de cette étude a ainsi nécessité de développer une théorie sur la dérivation des intégrales à paramètre et des fonctions à deux variables. De plus, démontrer en Coq la dérivabilité de la formule de d'Alembert s'est révélé être excessivement long. Nous avons donc construit une tactique par réflexion permettant de résumer l'expression "par application des théorèmes généraux de dérivation" souvent utilisée dans les démonstrations sur papier

Self-Doping of Gold Chains on Silicon: A New Structural Model for Si(111)5x2-Au

A new structural model for the Si(111)5x2-Au reconstruction is proposed and analyzed using first-principles calculations. The basic model consists of a "double honeycomb chain" decorated by Si adatoms. The 5x1 periodicity of the honeycomb chains is doubled by the presence of a half-occupied row of Si atoms that partially rebonds the chains. Additional adatoms supply electrons that dope the parent band structure and stabilize the period doubling; the optimal doping corresponds to one adatom per four 5x2 cells, in agreement with experiment. All the main features observed in scanning tunneling microscopy and photoemission are well reproduced.Comment: 4 pages, 4 figures, to appear in Phys. Rev. Lett. (preprint with high quality figures available at http://cst-www.nrl.navy.mil/~erwin/papers/ausi111

Formalization of Real Analysis: A Survey of Proof Assistants and Libraries

International audienceIn the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis

PHARAO Laser Source Flight Model: Design and Performances

In this paper, we describe the design and the main performances of the PHARAO laser source flight model. PHARAO is a laser cooled cesium clock specially designed for operation in space and the laser source is one of the main sub-systems. The flight model presented in this work is the first remote-controlled laser system designed for spaceborne cold atom manipulation. The main challenges arise from mechanical compatibility with space constraints, which impose a high level of compactness, a low electric power consumption, a wide range of operating temperature and a vacuum environment. We describe the main functions of the laser source and give an overview of the main technologies developed for this instrument. We present some results of the qualification process. The characteristics of the laser source flight model, and their impact on the clock performances, have been verified in operational conditions.Comment: Accepted for publication in Review of Scientific Instrument

Modulated contact frequencies at gene-rich loci support a statistical helix model for mammalian chromatin organization

International audienceABSTRACT: BACKGROUND: Despite its critical role for mammalian gene regulation, the basic structural landscape of chromatin in living cells remains largely unknown within chromosomal territories below the megabase scale. RESULTS: Here, using the 3C-qPCR method, we investigate contact frequencies at high resolution within the interphase chromatin at several mouse loci. We find that, at several gene-rich loci, contact frequencies undergo a periodical modulation (every 90-100 kb) that affects chromatin dynamics over large genomic distances (few hundred kb). Interestingly, this modulation appears to be conserved in human cells and bioinformatic analyses of locus-specific, long-range cis-interactions suggest that it may underlie the dynamics of a significant number of gene-rich domains in mammals, thus contributing to genome evolution. Finally, using an original model derived from polymer physics, we show that this modulation can be understood as a fundamental helix shape that chromatin tends to adopt in gene-rich domains when no significant locus-specific interaction takes place. CONCLUSIONS: Altogether, our work unveils a fundamental aspect of chromatin dynamics in mammals and contributes to a better understanding of genome organization within chromosomal territories

Long-range chromatin interactions at the mouse Igf2/H19 locus reveal a novel paternally expressed long non-coding RNA

Parental genomic imprinting at the Igf2/H19 locus is controlled by a methylation-sensitive CTCF insulator that prevents the access of downstream enhancers to the Igf2 gene on the maternal chromosome. However, on the paternal chromosome, it remains unclear whether long-range interactions with the enhancers are restricted to the Igf2 promoters or whether they encompass the entire gene body. Here, using the quantitative chromosome conformation capture assay, we show that, in the mouse liver, the endodermal enhancers have low contact frequencies with the Igf2 promoters but display, on the paternal chromosome, strong interactions with the intragenic differentially methylated regions 1 and 2. Interestingly, we found that enhancers also interact with a so-far poorly characterized intergenic region of the locus that produces a novel imprinted long non-coding transcript that we named the paternally expressed Igf2/H19 intergenic transcript (PIHit) RNA. PIHit is expressed exclusively from the paternal chromosome, contains a novel discrete differentially methylated region in a highly conserved sequence and, surprisingly, does not require an intact ICR/H19 gene region for its imprinting. Altogether, our data reveal a novel imprinted domain in the Igf2/H19 locus and lead us to propose a model for chromatin folding of this locus on the paternal chromosome

H19 Antisense RNA Can Up-Regulate Igf2 Transcription by Activation of a Novel Promoter in Mouse Myoblasts

It was recently shown that a long non-coding RNA (lncRNA), that we named the 91H RNA (i.e. antisense H19 transcript), is overexpressed in human breast tumours and contributes in trans to the expression of the Insulin-like Growth Factor 2 (IGF2) gene on the paternal chromosome. Our preliminary experiments suggested that an H19 antisense transcript having a similar function may also be conserved in the mouse. In the present work, we further characterise the mouse 91H RNA and, using a genetic complementation approach in H19 KO myoblast cells, we show that ectopic expression of the mouse 91H RNA can up-regulate Igf2 expression in trans despite almost complete unmethylation of the Imprinting-Control Region (ICR). We then demonstrate that this activation occurs at the transcriptional level by activation of a previously unknown Igf2 promoter which displays, in mouse tissues, a preferential mesodermic expression (Pm promoter). Finally, our experiments indicate that a large excess of the H19 transcript can counteract 91H-mediated Igf2 activation. Our work contributes, in conjunction with other recent findings, to open new horizons to our understanding of Igf2 gene regulation and functions of the 91H/H19 RNAs in normal and pathological conditions