  Let P be any pure type system, we are going to show how we can extend P into
a PTS P' which will be used as a proof system whose formulas express properties
about sets of terms of P. We will show that P' is strongly normalizable if and
only if P is. Given a term t in P and a formula F in P', P' is expressive
enough to construct a formula "t ||- F" that is interpreted as "t is a realizer
of F". We then prove the following adequacy theorem: if F is provable then by
projecting its proof back to a term t in P we obtain a proof that "t ||- F".
