A logical framework with explicit conversions

The type theory ¿P corresponds to the logical framework LF. In this paper we present ¿H, a variant of ¿P where convertibility is not implemented by means of the customary conversion rule, but instead type conversions are made explicit in the terms. This means that the time to type check a ¿H term is proportional to the size of the term itself. We define an erasure map from ¿H to ¿P, and show that through this map the type theory ¿H corresponds exactly to ¿P: any ¿H judgment will be erased to a ¿P judgment, and conversely each ¿P judgment can be lifted to a ¿H judgment. We also show a version of subject reduction: if two ¿H terms are provably convertible then their types are also provably convertible.