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A Certificate-Based Approach to Formally Verified Approximations

Authors Florent Bréhard, Assia Mahboubi, Damien Pous

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Author Details

Florent Bréhard
  • Plume and AriC teams, LIP, ENS de Lyon, Université de Lyon, Lyon, France
  • MAC team, LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France
Assia Mahboubi
  • Gallinette team, LS2N, INRIA, Université de Nantes, Nantes, France
Damien Pous
  • Plume team, LIP, CNRS, ENS de Lyon, Université de Lyon, Lyon, France

Funding

This work has been funded by the European Research Council (ERC) under the European Union’s Horizon 2020 programme (CoVeCe, grant agreement No 678157), and was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). This work was supported in part by the project FastRelax ANR-14-CE25-0018-01.

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Florent Bréhard, Assia Mahboubi, and Damien Pous. A Certificate-Based Approach to Formally Verified Approximations. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ITP.2019.8

Abstract

We present a library to verify rigorous approximations of univariate functions on real numbers, with the Coq proof assistant. Based on interval arithmetic, this library also implements a technique of validation a posteriori based on the Banach fixed-point theorem. We illustrate this technique on the case of operations of division and square root. This library features a collection of abstract structures that organise the specfication of rigorous approximations, and modularise the related proofs. Finally, we provide an implementation of verified Chebyshev approximations, and we discuss a few examples of computations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • approximation theory
  • Chebyshev polynomials
  • Banach fixed-point theorem
  • interval arithmetic
  • Coq

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