Relativizing statements in Higher-Order Logic (HOL) from types to sets is useful for improving productivity when working with HOL-based interactive theorem provers such as HOL4, HOL Light and Isabelle/HOL. This paper provides the first comprehensive definition and study of types-to-sets relativization in HOL, done in the more general form of types-to-PERs (partial equivalence relations). We prove that, for a large practical fragment of HOL which includes container types such as datatypes and codatatypes, types-to-PERs relativization is admissible, in that the provability of the original, type-based statement implies the provability of its relativized, PER-based counterpart. Our results also imply the admissibility of a previously proposed axiomatic extension of HOL with local type definitions. We have implemented types-to-PERs relativization as an Isabelle tool that performs relativization of HOL theorems on demand.

@article{paperbot2072,
  title = {Admissible Types-to-PERs Relativization in Higher-Order Logic},
  author = {Andrei Popescu and Dmitriy Traytel},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {7},
  number = {POPL},
  year = {2023},
  doi = {10.1145/3571235}
}