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The lambda-Pi-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory U, where proofs of several logical systems can be expressed. Moreover, we identify a sub-theory of U corresponding to each of these systems, and prove that, when a proof in U uses only symbols of a sub-theory, then it is a proof in that sub-theory.
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Blanqui, F., Dowek, G., Grienenberger, E., Hondet, G., & Thiré, F. (2023). A modular construction of type theories. Logical Methods in Computer Science e.V.. https://library.shubhaarti.in/content/676a8232-e980-463a-99f3-4dfa076910bf
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