Exercising Nuprl’s Open-Endedness

Abstract

Abstract. 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 illustrates the process of extending Nuprl with versions of the axiom of choice.

Bar induction: The good, the bad, and the ugly

“…, is an infinite sequence of terms, which are used in a similar fashion as above to prove that some bar recursion operator realizes the negative translation of the axiom of choice. Similarly, as mentioned in [57], using our choice sequences, we have proved the validity of versions of the axiom of choice. In [11] the authors write: "The infinite terms are not for computational purposes, they only play a role in the termination proof".…”