Quantitative continuity and computable analysis in Coq

Florian Steinberg, Laurent Théry, Holger Thies

研究成果: Chapter in Book/Report/Conference proceedingConference contribution

抄録

We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the sense of computable analysis. The development is done in the proof assistant COQ and heavily relies on the INCONE library for information theoretic continuity. This library is developed by one of the authors and the results of this paper extend the library. While full executability in a formal development of mathematical statements about real numbers and the like is not a feature that is unique to the INCONE library, its original contribution is to adhere to the conventions of computable analysis to provide a general purpose interface for algorithmic reasoning on continuous structures. The paper includes a brief description of the most important concepts of INCONE and its sub libraries MF and METRIC. The results that provide complete computational content include that the algebraic operations and the efficient limit operator on the reals are computable, that the countably infinite product of a space with itself is isomorphic to a space of functions, compatibility of the enumeration representation of subsets of natural numbers with the abstract definition of the space of open subsets of the natural numbers, and that continuous realizability implies sequential continuity. We also describe many non-computational results that support the correctness of definitions from the library. These include that the information theoretic notion of continuity used in the library is equivalent to the metric notion of continuity on Baire space, a complete comparison of the different concepts of continuity that arise from metric and represented space structures and the discontinuity of the unrestricted limit operator on the real numbers and the task of selecting an element of a closed subset of the natural numbers.

本文言語英語
ホスト出版物のタイトル10th International Conference on Interactive Theorem Proving, ITP 2019
編集者John Harrison, John O'Leary, Andrew Tolmach
出版社Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN(電子版)9783959771221
DOI
出版ステータス出版済み - 9 2019
イベント10th International Conference on Interactive Theorem Proving, ITP 2019 - Portland, 米国
継続期間: 9 9 20199 12 2019

出版物シリーズ

名前Leibniz International Proceedings in Informatics, LIPIcs
141
ISSN(印刷版)1868-8969

会議

会議10th International Conference on Interactive Theorem Proving, ITP 2019
Country米国
CityPortland
Period9/9/199/12/19

All Science Journal Classification (ASJC) codes

  • Software

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