Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory

We develop a uniform type theory that integrates intensionality, extensionality and proof irrelevance as judgmental concepts. Any object may be treated intensionally (subject only to α-conversion), extensionally (subject also to βη-conversion), or as irrelevant (equal to any other object at the same type), depending on where it occurs. Modal restrictions developed by R. Harper et al. (2000) for single types are generalized and employed to guarantee consistency between these views of objects. Potential applications are in logical frameworks, functional programming and the foundations of first-order modal logics. Our type theory contrasts with previous approaches that, a priori, distinguished propositions (whose proofs are all identified - only their existence is important) from specifications (whose implementations are subject to some definitional equalities)