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.
@InProceedings{brehard_et_al:LIPIcs.ITP.2019.8, author = {Br\'{e}hard, Florent and Mahboubi, Assia and Pous, Damien}, title = {{A Certificate-Based Approach to Formally Verified Approximations}}, booktitle = {10th International Conference on Interactive Theorem Proving (ITP 2019)}, pages = {8:1--8:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-122-1}, ISSN = {1868-8969}, year = {2019}, volume = {141}, editor = {Harrison, John and O'Leary, John and Tolmach, Andrew}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.8}, URN = {urn:nbn:de:0030-drops-110638}, doi = {10.4230/LIPIcs.ITP.2019.8}, annote = {Keywords: approximation theory, Chebyshev polynomials, Banach fixed-point theorem, interval arithmetic, Coq} }
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