research-article

Formal proofs of transcendence for e and pi as an application of multivariate and symmetric polynomials

Authors:

Sophie Bernard,

Laurence Rideau,

Pierre-Yves StrubAuthors Info & Claims

CPP 2016: Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs

Pages 76 - 87

https://doi.org/10.1145/2854065.2854072

Published: 18 January 2016 Publication History

Abstract

We describe the formalisation in Coq of a proof that the numbers `e` and `pi` are transcendental. This proof lies at the interface of two domains of mathematics that are often considered separately: calculus (real and elementary complex analysis) and algebra. For the work on calculus, we rely on the Coquelicot library and for the work on algebra, we rely on the Mathematical Components library. Moreover, some of the elements of our formalized proof originate in the more ancient library for real numbers included in the Coq distribution. The case of `pi` relies extensively on properties of multivariate polynomials and this experiment was also an occasion to put to test a newly developed library for these multivariate polynomials.

References

[1]

Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4(1):71–84, 2011.

[2]

Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Coquelicot: A user-friendly library of real analysis for Coq. Mathematics in Computer Science, 9(1):41–62, 2015.

[3]

Cyril Cohen. Formalized algebraic numbers: construction and firstorder theory. PhD thesis, École polytechnique, Nov 2012.

[4]

Cyril Cohen, Maxime Dénès, and Anders Mörtberg. Refinements for free! In Georges Gonthier and Michael Norrish, editors, Certified Programs and Proofs, CPP 2013, Melbourne, Australia, Springer, 2013.

Digital Library

[5]

Julian Lowell Coolidge. The number e. The American Mathematical Monthly, 55(9):591–602, Nov 1950.

[6]

Coq development team. The coq proof assistant, 2008. http: //coq.inria.fr.

[7]

Maxime Dénès, Anders Mörtberg, and Vincent Siles. A refinementbased approach to computational algebra in coq. In Lennart Beringer and Amy Felty, editors, Interactive Theorem Proving, ITP2012, volume 7406 of LNCS, pages 83–98, Princeton, USA, Springer, 2012.

[8]

Georges Gonthier. Formal proof—the Four Color Theorem. Notices of the AMS, 55(11):1382–1393, 2008.

[9]

Georges Gonthier, et al. A Machine-Checked Proof of the Odd Order Theorem. In Sandrine Blazy, Christine Paulin, and David Pichardie, editors, Interactive Theorem Proving, ITP2013, volume 7998 of LNCS, pages 163–179, Rennes, France, Springer, 2013.

Digital Library

[10]

Florian Haftmann, Andreas Lochbihler, and Wolfgang Schreiner. Towards abstract and executable multivariate polynomials in isabelle. Isabelle Workshop 2014, http://www.infsec.ethz.ch/people/ andreloc/publications/haftmann14iw.pdf, 2014.

[11]

John Harrison. HOL Light: A Tutorial Introduction. In FMCAD, pages 265–269, 1996.

Digital Library

[12]

Charles Hermite. Sur la fonction exponentielle. Gauthier-Villars, Paris, 1874.

[13]

Paul Jackson. Exploring abstract algebra in constructive type theory. In Proceedings of the 12th International Conference on Automated Deduction, CADE-12, pages 590–604, London, UK, UK, 1994. Springer-Verlag.

Digital Library

[14]

Ferdinand von Lindemann. Über die Zahl π. Mathematische Annalen 20 (1882): pp. 213-225. Springer-Verlag.

[15]

Assia Mahboubi. Proving formally the implementation of an efficient gcd algorithm for polynomials. In Ulrich Furbach and Natarajan Shankar, editors, Automated Reasoning, IJCAR 2006, volume 4130 of LNCS, pages 438–452, Seattle, WA, USA, Springer. 2006.

Digital Library

[16]

César Muñoz and Anthony Narkawicz. Formalization of a representation of Bernstein polynomials and applications to global optimization. Journal of Automated Reasoning, 51(2):151–196, August 2013.

Digital Library

[17]

Tobias Nipkow, Lawrence C. Paulson, and Markus Wenzel. Isabelle/HOL — A Proof Assistant for Higher-Order Logic, volume 2283 of LNCS. Springer, 2002.

Digital Library

[18]

Ivan Niven. The transcendence of π. American Mathematical Monthly, 46(8):469–471, October 1939.

[19]

Ivan Niven. A simple proof that π is irrational. Bulletin of the AMS, 53(6):509, 1947.

[20]

Laurent Théry. A machine-checked implementation of buchberger’s algorithm. Journal of Automated Reasoning, 26(2):107–137, 2001.

Digital Library

Cited By

Slagel JWhite LDutle AHriţcu CPopescu A(2021)Formal verification of semi-algebraic sets and real analytic functionsProceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3437992.3439933(278-290)Online publication date: 17-Jan-2021
https://dl.acm.org/doi/10.1145/3437992.3439933
Lewis RMahboubi AMyreen M(2019)A formal proof of Hensel's lemma over the p-adic integersProceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3293880.3294089(15-26)Online publication date: 14-Jan-2019
https://dl.acm.org/doi/10.1145/3293880.3294089
Bentkamp ABlanchette JKlakow D(2019)A Formal Proof of the Expressiveness of Deep LearningJournal of Automated Reasoning10.1007/s10817-018-9481-563:2(347-368)Online publication date: 1-Aug-2019
https://dl.acm.org/doi/10.1007/s10817-018-9481-5
Show More Cited By

Index Terms

Formal proofs of transcendence for e and pi as an application of multivariate and symmetric polynomials
1. Theory of computation
  1. Logic
  2. Models of computation
    1. Computability

Recommendations

A Combination of a Dynamic Geometry Software With a Proof Assistant for Interactive Formal Proofs

This paper presents an interface for geometry proving. It is a combination of a dynamic geometry software, Geogebra Geogebra development team, Introduction to GeoGebra. http://www.geogebra.org/book/intro-en/ with a proof assistant, Coq Coq development ...
Interactive proofs in higher-order concurrent separation logic
POPL '17

When using a proof assistant to reason in an embedded logic -- like separation logic -- one cannot benefit from the proof contexts and basic tactics of the proof assistant. This results in proofs that are at a too low level of abstraction because they ...
A Graphical User Interface for Formal Proofs in Geometry

We present in this paper the design of a graphical user interface to deal with proofs in geometry. The software developed combines three tools: a dynamic geometry software to explore, measure, and invent conjectures; an automatic theorem prover to check ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

CPP 2016: Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs

January 2016

196 pages

ISBN:9781450341271

DOI:10.1145/2854065

Program Chairs:
Jeremy Avigad
Carnegie Mellon University, USA
,
Adam Chlipala
MIT, USA

Copyright © 2016 ACM.

Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.

Sponsors

SIGPLAN: ACM Special Interest Group on Programming Languages

In-Cooperation

SIGLOG: ACM Special Interest Group on Logic and Computation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 18 January 2016

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Cofund Action AMAROUT II
French National Agency for Research (ANR)

Conference

CPP 2016

Sponsor:

SIGPLAN

CPP 2016: Certified Proofs and Programs

January 18 - 19, 2016

FL, St. Petersburg, USA

Acceptance Rates

Overall Acceptance Rate 18 of 26 submissions, 69%

Upcoming Conference

POPL '25

Sponsor:
sigplan

The 52nd Annual ACM SIGPLAN Symposium on Principles of Programming Languages

January 19 - 25, 2025

Denver , CO , USA

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
136
Total Downloads

Downloads (Last 12 months)6
Downloads (Last 6 weeks)0

Reflects downloads up to 13 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

Slagel JWhite LDutle AHriţcu CPopescu A(2021)Formal verification of semi-algebraic sets and real analytic functionsProceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3437992.3439933(278-290)Online publication date: 17-Jan-2021
https://dl.acm.org/doi/10.1145/3437992.3439933
Lewis RMahboubi AMyreen M(2019)A formal proof of Hensel's lemma over the p-adic integersProceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3293880.3294089(15-26)Online publication date: 14-Jan-2019
https://dl.acm.org/doi/10.1145/3293880.3294089
Bentkamp ABlanchette JKlakow D(2019)A Formal Proof of the Expressiveness of Deep LearningJournal of Automated Reasoning10.1007/s10817-018-9481-563:2(347-368)Online publication date: 1-Aug-2019
https://dl.acm.org/doi/10.1007/s10817-018-9481-5
Korniłowicz ANaumowicz AGrabowski A(2017)All Liouville Numbers are TranscendentalFormalized Mathematics10.1515/forma-2017-000425:1(49-54)Online publication date: 11-May-2017
https://doi.org/10.1515/forma-2017-0004
Grabowski AKorniłowicz A(2017)Introduction to Liouville NumbersFormalized Mathematics10.1515/forma-2017-000325:1(39-48)Online publication date: 11-May-2017
https://doi.org/10.1515/forma-2017-0003
Martin-Dorel ÉRoux PBertot YVafeiadis V(2017)A reflexive tactic for polynomial positivity using numerical solvers and floating-point computationsProceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs10.1145/3018610.3018622(90-99)Online publication date: 16-Jan-2017
https://dl.acm.org/doi/10.1145/3018610.3018622
Bernard S(2017)Formalization of the Lindemann-Weierstrass TheoremInteractive Theorem Proving10.1007/978-3-319-66107-0_5(65-80)Online publication date: 2017
https://doi.org/10.1007/978-3-319-66107-0_5
Bentkamp ABlanchette JKlakow D(2017)A Formal Proof of the Expressiveness of Deep LearningInteractive Theorem Proving10.1007/978-3-319-66107-0_4(46-64)Online publication date: 2017
https://doi.org/10.1007/978-3-319-66107-0_4

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents