Proofs for Free in the $λΠ$-Calculus Modulo Theory

Parametricity allows the transfer of proofs between different implementations of the same data structure. The lambdaPi-calculus modulo theory is an extension of the lambda-calculus with dependent types and user-defined rewrite rules. It is a logical framework, used to exchange proofs between different proof systems. We define an interpretation of theories of the lambdaPi-calculus modulo theory, inspired by parametricity. Such an interpretation allows to transfer proofs for free between theories that feature the notions of proposition and proof, when the source theory can be embedded into the target theory.