# * 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.