Validating Brouwer's continuity principle for numbers using named exceptions

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.

A unifying framework for continuity and complexity in higher types

“…Moreover, explicit moduli of continuity can be defined within System T itself. In recent years there has been a renewed interest in syntactic approaches to continuity which provide explicit moduli of continuity, and several new proofs of the continuity of System T definable functionals have been given, including [7,11,13] and most recently [16,28,39]. We provide another proof of this fact via our abstract framework, which applies not just to System T but to arbitrary languages which satisfy the relevant conditions.…”

“…The approach presented here is more closely connected to recent work, where continuity and the extraction of corresponding moduli are established by appealing to monads and effects (e.g. [7,11,13,16,28,39], though this list is not exhaustive). A related monadic translation in all finite types due to the present author, but using the state monad and working in an equational setting, is also the basis for [27].…”

$\text{TT}^{\Box}_{\mathcal C}$: a Family of Extensional Type Theories with Effectful Realizers of Continuity

“…In [RB16,RB17], the authors proved that Brouwer's continuity principle is consistent with Nuprl [CAB + 86, ABC + 06] by realizing the modulus of continuity of functions on the Baire space also using Longley's method [Lon99], but using exceptions instead of references. The realizer there is more complicated than the one presented in this paper as it involves an effectful computation that repeatedly checks whether a given number is at least as high as the modulus of continuity, and increasing that number until the modulus of continuity is reached.…”

Russian Constructivism in a Prefascist Theory

“…Let us also mention that our prompt mechanism is reminiscent of a technique recently used by Rahli and Bickford [42] to implement Brouwer's continuity principle based on the generation of fresh exceptions. Their work suggests new uses for our exceptional presheaf model.…”