A modular construction of type theories

The lambda-Pi-calculus modulo theory is a logical framework in which manytype systems can be expressed as theories. We present such a theory, the theoryU, where proofs of several logical systems can be expressed. Moreover, weidentify a sub-theory of U corresponding to each of these systems, and provethat, when a proof in U uses only symbols of a sub-theory, then it is a proofin that sub-theory