research-article

A nominal exploration of intuitionism

Authors:

Mark BickfordAuthors Info & Claims

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

Pages 130 - 141

https://doi.org/10.1145/2854065.2854077

Published: 18 January 2016 Publication History

Abstract

This papers extends the Nuprl proof assistant (a system representative of the class of extensional type theories a la Martin-Lof) with named exceptions and handlers, as well as a nominal fresh operator. Using these new features, we prove a version of Brouwer's Continuity Principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's meta-theory using our formalization of Nuprl in Coq and show how we can reflect these meta-theoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key meta-theoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem and bar induction on monotone bars.

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Cited By

STERLING J(2021)Higher order functions and Brouwer’s thesisJournal of Functional Programming10.1017/S095679682100009531Online publication date: 19-May-2021
https://doi.org/10.1017/S0956796821000095
Cohen L(2021)Formally Computing with the Non-computableConnecting with Computability10.1007/978-3-030-80049-9_12(135-145)Online publication date: 2-Jul-2021
https://doi.org/10.1007/978-3-030-80049-9_12
Rahli VBickford MCohen LConstable R(2019)Bar Induction is Compatible with Constructive Type TheoryJournal of the ACM10.1145/330526166:2(1-35)Online publication date: 24-Apr-2019
https://dl.acm.org/doi/10.1145/3305261
Show More Cited By

Index Terms

A nominal exploration of intuitionism
1. Software and its engineering
  1. Software notations and tools
    1. Formal language definitions
2. Theory of computation

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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.

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Published: 18 January 2016

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January 18 - 19, 2016

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Overall Acceptance Rate 18 of 26 submissions, 69%

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Cited By

STERLING J(2021)Higher order functions and Brouwer’s thesisJournal of Functional Programming10.1017/S095679682100009531Online publication date: 19-May-2021
https://doi.org/10.1017/S0956796821000095
Cohen L(2021)Formally Computing with the Non-computableConnecting with Computability10.1007/978-3-030-80049-9_12(135-145)Online publication date: 2-Jul-2021
https://doi.org/10.1007/978-3-030-80049-9_12
Rahli VBickford MCohen LConstable R(2019)Bar Induction is Compatible with Constructive Type TheoryJournal of the ACM10.1145/330526166:2(1-35)Online publication date: 24-Apr-2019
https://dl.acm.org/doi/10.1145/3305261
Bickford MCohen LConstable RRahli V(2018)Computability Beyond Church-Turing via Choice SequencesProceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3209108.3209200(245-254)Online publication date: 9-Jul-2018
https://dl.acm.org/doi/10.1145/3209108.3209200
Bickford M(2018)Connectedness of the continuum in intuitionistic mathematicsMathematical Logic Quarterly10.1002/malq.20170005764:4-5(387-394)Online publication date: 18-Oct-2018
https://doi.org/10.1002/malq.201700057
Rahli VBickford MConstable RAceto LIngólfsdóttir A(2017)Bar inductionProceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/3329995.3330009(1-12)Online publication date: 20-Jun-2017
https://dl.acm.org/doi/10.5555/3329995.3330009
Angiuli CHarper RWilson T(2017)Computational higher-dimensional type theoryACM SIGPLAN Notices10.1145/3093333.300986152:1(680-693)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.1145/3093333.3009861
Bohrer RRahli VVukotic IVölp MPlatzer ABertot YVafeiadis V(2017)Formally verified differential dynamic logicProceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs10.1145/3018610.3018616(208-221)Online publication date: 16-Jan-2017
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Angiuli CHarper RWilson TCastagna GGordon A(2017)Computational higher-dimensional type theoryProceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages10.1145/3009837.3009861(680-693)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.1145/3009837.3009861
Rahli VBickford MConstable R(2017)Bar induction: The good, the bad, and the ugly2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2017.8005074(1-12)Online publication date: Jun-2017
https://doi.org/10.1109/LICS.2017.8005074
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