Univariate and bivariate integral roots certificates based on Hensel's lifting

If it is quite easy to check a given integer is a root of a given polynomial with integer coefficients, verifying we know all the integral roots of a polynomial requires a different approach. In both univariate and bivariate cases, we introduce a type of integral roots certificates and the corresponding checker specification, based on Hensel's lifting. We provide a formalization of this iterative algorithm from which we deduce a formal proof of the correctness of the checkers, with the help of the COQ proof assistant along with the SSReflect extension. The ultimate goal of this work is to provide a component that will be involved in a complete certification chain for solving the Table Maker's Dilemma in an exact way

Some issues related to double roundings

International audienceDouble rounding is a phenomenon that may occur when different floating- point precisions are available on the same system. Although double rounding is, in general, innocuous, it may change the behavior of some useful small floating-point algorithms. We analyze the potential influence of double rounding on the Fast2Sum and 2Sum algorithms, on some summation algorithms, and Veltkamp's splitting

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

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

International audienceThe verification of floating-point mathematical libraries requires computing numerical bounds on approximation errors. Due to the tightness of these bounds and the peculiar structure of approximation errors, such a verification is out of the reach of generic tools such as computer algebra systems. In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them. In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions. It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions. All the computations are performed inside Coq's logic, in a reflexive setting. This paper also compares our tactic with various existing tools on a large set of examples

Coq Community Survey 2022: Summary of Results

Affiliated with ITP 2022, part of FLoC 2022International audienceThe Coq Community Survey 2022 was an online public survey conducted during February 2022. Its main goal was to obtain an updated picture of the Coq user community and inform future decisions taken by the Coq team. In particular, the survey aimed to enable the Coq team to make effective decisions about the development of the Coq software, and also about matters that pertain to the ecosystem maintained by Coq users in academia and industry. In this presentation abstract, we outline how the survey was designed, its content, and some initial data analysis and directions. Not least due to free-text answers to some questions requiring a more lengthy summary, the full presentation includes additional data and conclusions

Formalization of Hensel's lemma in Coq

Suppose we want to find the integral simple roots of a univariate polynomial with integer coefficients. We can use Hensel's lifting, which can be viewed as the $p$-adic variant of the Newton--Raphson's iteration. It enables us to compute the roots modulo increasing powers of a prime $p$. As soon as the considered power of $p$ becomes greater than a known bound on the roots, we easily obtain the sought integral roots. The bivariate version of this root-finding algorithm is extensively used by the Stehlé--Lefèvre--Zimmermann algorithm, designed to solve the so-called Table Maker's Dilemma in an exact way. Consequently, the formal verification of Hensel's lemma (i.e., the correctness lemma of Hensel's lifting method) will contribute to the validation of this kind of algorithm. In this talk, we describe the various steps we met during the formalization of Hensel's lemma with the Coq proof assistant along with the SSReflect extension

Formalization of Hensel's lemma in Coq

Suppose we want to find the integral simple roots of a univariate polynomial with integer coefficients. We can use Hensel's lifting, which can be viewed as the $p$-adic variant of the Newton--Raphson's iteration. It enables us to compute the roots modulo increasing powers of a prime $p$. As soon as the considered power of $p$ becomes greater than a known bound on the roots, we easily obtain the sought integral roots. The bivariate version of this root-finding algorithm is extensively used by the Stehlé--Lefèvre--Zimmermann algorithm, designed to solve the so-called Table Maker's Dilemma in an exact way. Consequently, the formal verification of Hensel's lemma (i.e., the correctness lemma of Hensel's lifting method) will contribute to the validation of this kind of algorithm. In this talk, we describe the various steps we met during the formalization of Hensel's lemma with the Coq proof assistant along with the SSReflect extension

Formal proofs and certified computation in Coq (The French Symposium on Games - Theory and Applications, Université Paris Diderot, 26/05/15-30/05/15)

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Cumulative Effects in Learning

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