Internalizing Parametricity

Parametricity results have recently been proved for dependently-typed calculi such as the Calculus of Constructions. However these results are meta theorems, and although the theorems can be stated as internal propositions, they cannot be proved internally. In this thesis, we develop a dependent type-theory in which each instance of the parametricity theorem, including those for open terms, can be proved internally. For instance we can prove inside the system that each term of type (X : *) -> X -> X is an identity. We show three successive proposals for a solution to this problem, each an improvement of the previous one. In the first one we introduce a dependent type theory with special syntax for hypercubes. In the second proposal we outline a colored type theory, which provides a simplification of the type theory with hypercubes. In the third and final proposal, we give a more definite presentation of the colored type theory. We also prove its consistency by a modification of the presheaf model of dependent type theory. We believe that our final colored type theory is simple enough to provide a basis for a proof assistant where proofs relying on parametricity can be performed internally