A nominal exploration of intuitionism

Abstract

This papers extends the Nuprl proof assistant (a system representative of the class of extensional type theoriesà la Martin-Löf) 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 metatheoretical 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.

Exercising Nuprl’s Open-Endedness

“…For example, the non-truncated version of Brouwer's continuity principle, i.e., where the existential quantifier is interpreted constructively, is false in type theories such as Agda or Nuprl [29,39,18,34,36], while its truncated version is true in Nuprl [34]. Similar results hold about AC as discussed in Sec.…”

“…They are also similar to Howe's set-theoretical functions in [24,25,23], which he used to provide a set-theoretical semantics of both Nuprl and HOL, allowing the shallow embedding of HOL in Nuprl. (4) We have proved the validity of truncated versions of AC [34,Sec.5.3]. Open-endedness.…”

“…Allen's PER semantics, and (2) proving that False is not inhabited. Using these two facts, we derive that there cannot be a proof derivation of False, i.e., Nuprl is consistent (see [6,5,33, Appx.A] for more details). We are using our Coq formalization to prove the validity of all the inference rules of Nuprl, and have already verified a large number of them.…”

“…We have recently added several operators to Nuprl: (1) named exceptions [34]; (2) a try/catch operator [34]; (3) a fresh operator to generate fresh names [34]; (4) choice sequences; (5) an eager application operator; as well as (6) various values denoting types such as our PER types [4]. In the case of exceptions, which are some sorts of values in the sense that they do not compute further, we had to modify one inference rule [33, Appx.C].…”

“…w.r.t. Nuprl's PER semantics [34]. For that, following Longley's method [31], we used named exceptions as a probing mechanism to compute the modulus of continuity of a function.…”