Towards a computational interpretation of parametricity

Preprint, 2011

Reynolds' abstraction theorem has recently been extended to lambda-calculi with dependent types. In this paper, we show how this theorem can be internalized. More precisely, we describe an extension of the Calculus of Constructions with a special parametricity rule (with computational content), and prove fundamental properties such as Church-Rosser's and strong normalization. The instances of the abstraction theorem can be both expressed and proved in the calculus itself.