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Validating Brouwer's continuity principle for numbers using named exceptions

Published online by Cambridge University Press:  02 November 2017

VINCENT RAHLI  and
MARK BICKFORD
VINCENT RAHLI
Affiliation:
SnT, University of Luxembourg, Esch-sur-Alzette, Luxembourg Email: [email protected]
MARK BICKFORD
Affiliation:
Cornell University, Computer Science Department, Ithaca, NY, USA Email: [email protected]
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Abstract

This paper extends the Nuprl proof assistant (a system representative of the class of extensional type theories with dependent types) 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 metatheory using our formalization of Nuprl in Coq and reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key metatheoretical 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, bar induction on monotone bars and the negation of the law of excluded middle.

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Mathematical Structures in Computer Science , Volume 28 , Issue 6 , June 2018 , pp. 942 - 990
Copyright
Copyright © Cambridge University Press 2017 

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