Sharing a Library between Proof Assistants: Reaching out to the HOL Family

We observe today a large diversity of proof systems. This diversity has the negative consequence that a lot of theorems are proved many times. Unlike programming languages, it is difficult for these systems to co-operate because they do not implement the same logic. Logical frameworks are a class of theorem provers that overcome this issue by their capacity of implementing various logics. In this work, we study the STTforall logic, an extension of Simple Type Theory that has been encoded in the logical framework Dedukti. We present a translation from this logic to OpenTheory, a proof system and interoperability tool between provers of the HOL family. We have used this translation to export an arithmetic library containing Fermat's little theorem to OpenTheory and to two other proof systems that are Coq and Matita.Comment: In Proceedings LFMTP 2018, arXiv:1807.0135

Matemática,1º ano, 5ª edição, 1934.

O livro possui dimensões 220 mm X 160 mm, 394 páginas. O exemplar pertence ao acervo do GHEMAT na cidade de Osasco- SP. Doado pela professora Circe Dynnikov.O livro destina-se ao uso de professores e traz considerações teóricas e práticas de abordagens de conteúdos de matemática para o ensino primário. São temas tratados: numeração, adição subtração, multiplicação, divisão, potencia de número, múltiplo e divisor, números primos, frações, álgebra etc

Exporter une librairie d'arithmétique depuis Dedukti vers HOL

Today, we observe a large diversity of proof systems. This diversity has the negative consequence that a lot of theorems are proved many times. Unlike programming languages, it is difficult for these systems to cooperate because they do not implement the same logic. Logical frameworks are a class of theorems provers that overcome this issue by their capacity of implementing various logics. In this work, we study the STT∀ βδ logic, an extension of the Simple Type Theory that has been encoded in the logical framework Dedukti. We show that this new logic is a good candidate to export proofs to other provers. As an example, we show how this logic has been encoded into Dedukti and how we used it to export proofs to the HOL family provers via OpenTheory

Photodynamiques moléculaires sondées par imagerie de vecteurs vitesses et génération d'harmoniques d'ordre élevé

Cette thèse traite de l'étude de phénomènes ultra-rapides, et plus particulièrement de dynamiques de relaxation initiées par une impulsion femtoseconde (10-15 s). Dans la première partie, l'imagerie de vecteurs vitesses a permis d'étudier deux systèmes : l'iodure de méthyle (CH3I), excité dans son premier état de Rydberg et le tetrathiafulvalène (TTF -C6H4S4). Dans le TTF, le but est de caractériser les deux bandes majoritaires d'absorption vers 300 nm. Durant l'étude de la prédissociation de CH3I, la distribution vibrationnelle du fragment CH3 a été caractérisée, alors que pour le TTF une force de liaison du dimère a été extraite. L'interprétation des spectres de photoélectrons de CH3I permet une bonne compréhension des dynamiques depuis et/ou en résonance avec des états proches du potentiel d'ionisation. Dans la seconde partie, la structure électronique d'atomes ou de molécules est sondée par la génération d'harmoniques d'ordre élevé (GHOE). Le minimum de la section efficace totale de photoionisation de l'argon (minimum de Cooper) a été observé et étudié dans le spectre harmonique. Cette étude a permis, entre autre, la mise au point d'un modèle théorique fiable et complet. Dans une autre étude, la GHOE a montré la sensibilité du rayonnement harmonique à la chiralité lors de interaction des 2 énantiomères de la fenchone (C10H16O), avec un rayonnement polarisé elliptiquement. Pour finir, l'utilisation de la GHOE pour sonder une dynamique moléculaire, amorcée par un réseau transitoire d’excitation à 400 nm, a été réalisée sur le dioxyde d'azote (NO2). Ceci à l'échelle picoseconde : photodissociation, et femtoseconde : transfert de population par le biais d’une intersection conique.This thesis deals with the study of ultrafast phenomena, particularly the relaxation dynamics initiated by a femtosecond pulse (10-15 s). In the first part, velocity map imaging was used to study two systems: methyl iodide (CH3I), in its first excited Rydberg state and tetrathiafulvalene (TTF -C6H4S4). In the TTF, the aim is to characterize the two major absorption bands at 300 nm. During the study of the predissociation of CH3I, the vibrational distribution of the CH3 fragment was characterized,whereas as far as the TTF is concerned bond strength of the dimer was extracted. The interpretation of photoelectron spectra of CH3I provides a good understanding of the dynamics from and/or in resonance with states near the ionization potential. In the second part, the electronic structure of atom or molecule is probed by high order harmonics generation (HHG). The minimum of the total photoionization cross section of argon (Cooper minimum) was observed and studied in the harmonic spectrum. This study allowed, among other things, for the design of a reliable and comprehensive theoretical model. Then the HHG in a chiral molecule: the fenchone (C10H16O), clearly showed the sensitivity of the harmonic process to the chirality, while interacting with an elliptically polarized field. Finally, HHG was used to probe molecular dynamics initiated by a transient excitation grating at 400 nm in nitrogen dioxide (NO2). This at picosecond time scale : photodissociation, and femtosecond time scale : population transfer via a conical intersectio

Internship report MPRI 2 Reverse engineering on arithmetic proofs

International audiencededukti is a logical framework that implements the λΠ− modulo theory, an extension of the simply typed lambda calculus with dependent types and rewriting rules. It aims to be a back-end for other proof checkers by compiling proofs from these proof checkers to dedukti. This may also increase re-usability of proofs between proof checkers. However if a logic is more powerful than an other, a theorem in the first logic may not be a theorem in the second. During this internship, we consider arithmetic theorems since many proof checker are able to check arithmetic proofs. One problem that we study in this master thesis is to translate arithmetic proofs coming from a powerful proof checker, -- in our casematita -- to a less powerful proof checker -- HOL-- . This translation needs to modify the logic used in proofs and that is why dedukti is handy here. But a lot of arithmetic theorems are proved also by automatic provers. Indeed, today a lot of easy arithmetic theorems are proved by this kind of tool. But most of them do not give a proof if it claims to prove a theorem. Since for these kind of tool, constructing a full proof may be tiresome, they prefer to give a certificate , a sketch of a proof. However, any automatic prover can implement its own certificate format. To answer this problem, Zakaria Chihani & Dale Miller proposed a certificate framework: Foundational Proof Certificate (FPC) [CMR13]. This framework aims to provide a certificate format shared by many automatic provers so that from the latter, a full proof might be reconstructed.However, for now, no certificate format is given for arithmetic proofs. A second problem addressed in this internship is to answer what kind of certificate is needed for arithmetic proofs (arithmetic without multiplication)

Some axioms for type theories

The $\lambda\Pi$-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory $\mathcal{U}$, where proofs of several logical systems can be expressed. Moreover, we identify a sub-theory of $\mathcal{U}$ corresponding to each of these systems, and prove that, when a proof in $\mathcal{U}$ uses only symbols of a sub-theory, then it is a proof in that sub-theory

L'intéropérabilité entre des preuves d'arithmétique en utilisan Dedukti

International audienc