Impredicative Observational Equality

Abstract

In dependent type theory, impredicativity is a powerful logical principle that allows the definition of propositions that quantify over arbitrarily large types, potentially resulting in self-referential propositions. Impredicativity can provide a system with increased logical strength and flexibility, but in counterpart it comes with multiple incompatibility results. In particular, Abel and Coquand showed that adding definitional uniqueness of identity proofs (UIP) to the main proof assistants that support impredicative propositions (Coq and Lean) breaks the normalization procedure, and thus the type-checking algorithm. However, it was not known whether this stems from a fundamental incompatibility between UIP and impredicativity or if a more suitable algorithm could decide type-checking for a type theory that supports both. In this paper, we design a theory that handles both UIP and impredicativity by extending the recently introduced observational type theory TTobs with an impredicative universe of definitionally proof-irrelevant types, as initially proposed in the seminal work on observational equality of Altenkirch et al. We prove decidability of conversion for the resulting system, that we call CCobs, by harnessing proof-irrelevance to avoid computing with impredicative proof terms. Additionally, we prove normalization for CCobs in plain Martin-Löf type theory, thereby showing that adding proof-irrelevant impredicativity does not increase the computational content of the theory.