Skip to main content
PRL Project

Validating Brouwer’s Continuity Principle for Numbers Using Named Exceptions

by Vincent Rahli, Mark Bickford
2018


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.