Admissibility of Fixpoint Induction over Partial Types

Partial types allow the reasoning about partial functions in type theory. The partial functions of main interest are recursively computed functions, which are commonly assigned types using xpoint induction. However, xpoint induction is valid only on admissible types. Previous work has shown many types to be admissible, but has not shown any dependent products to be admissible. Disallowing recursion on dependent product types substantially reduces the expressiveness of the logic; for example, it prevents much reasoning about modules, objects and algebras. In this paper I present two new tools, predicate-admissibility and monotonicity, for showing types to be admissible. These tools show a wide class of types to be admissible; in particular, they show many dependent products to be admissible. This alleviates di culties in applying partial types to theorem proving in practice. I also present a general least upper bound theorem for xed points with regard to a computational approximation relation, and show an elegant application of the theorem to compactness. This research was conducted while the author was at Cornell University. This material is based on work supported in part by ARPA/AF grant F30602-95-1-0047, and AASERT grant N00014-95-1-0985. Any opinions, ndings, and conclusions or recommendations expressed in this publication are those of the author and do not re ect the views of these agencies.