A reasonably gradual type theory

Gradualizing the Calculus of Inductive Constructions (CIC) involves dealing with subtle tensions between normalization, graduality, and conservativity respect to CIC. Recently, GCIC has been proposed as a parametrized gradual type theory that admits three variants, each sacrificing one these properties. For devising proof assistant based on CIC, normalization CIC are key, but tension graduality needs be addressed. Additionally, several challenges remain: (1) The presence two wildcard terms at any type---the error unknown terms---enables trivial proofs theorem, jeopardizing use in assistant; (2) Supporting general indexed inductive families, most prominently equality, is an open problem; (3) Theoretical accounts typing so far do not support handling mismatches detected during reduction; (4) Precision external notions amenable reasoning within theory. All issues manifest primally CastCIC, cast calculus used define GCIC. In this work, we present extension CastCIC called GRIP. GRIP reasonably addresses above, featuring internal precision exception handling. By adopting novel interpretation term carefully for universe levels, satisfies large well-defined class terms, addition being normalizing conservative Internal supports about itself, instance characterize exception-handling subset types. We develop metatheory using model formalized Coq, provide prototype implementation Agda.