On the Strength of Proof-Irrelevant Type Theories

We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is particularly useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. Finally we show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.