Exercising Nuprl's open-endedness

Nuprl is an interactive theorem prover that implements an extensional constructive type theory, where types are interpreted as partial equivalence relations on closed terms. Nuprl is both computationally and type-theoretically open-ended in the sense that both its computation system and its type theory can be extended as needed by checking a handful of conditions. For example, Doug Howe characterized the computations that can be added to Nuprl in order to preserve the congruence of its computational equivalence relation. We have implemented Nuprl’s computation and type systems in Coq, and we have showed among other things that it is consistent. Using our Coq framework we can now easily and rigorously add new computations and types to Nuprl by mechanically verifying that all the necessary conditions still hold. We have recently exercised Nuprl’s open-endedness by adding nominal features to Nuprl in order to prove a version of Brouwer’s continuity principle, as well as choice sequences in order to prove truncated versions of the axiom of choice and of Brouwer’s bar induction principle. This paper illustrate the process of extending Nuprl with versions of the axiom of choice.