Pure Type Systems with an Internalized Parametricity Theorem
Parametricity results have recently been proved for dependently-typed calculi such as the Calculus of Constructions. However these results are meta theorems, and although they can be stated as internal propositions, they cannot be proved internally. In this thesis we define for any sufficiently strong Pure Type System O (such as the Calculus of Constructions) an extension P in which each instance of the parametricity theorem, including those corresponding to open terms, can be proved internally. As a consequence we can prove inside the system that each term of type forall A. A -> A is an identity. Furthermore, our system P is proved to be strongly normalizing by a reduction-preserving interpretation into O. We also prove Church-Rosser and Subject Reduction properties; consistency follows.