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 $\mathcal{U}$, where proofs of several logical systems can be expressed. Moreover, we identify a sub-theory of $\mathcal{U}$ corresponding to each of these systems, and prove that, when a proof in $\mathcal{U}$ uses only symbols of a sub-theory, then it is a proof in that sub-theory.

翻译：$\ lambda\ Pi- calculus modulo 理论是一个逻辑框架, 许多类型系统都可以在其中表现为理论。 我们提出了这样一个理论, 即 $\ mathcal{ U}$ 理论, 其中可以表达多个逻辑系统的证据。 此外, 我们确定了一个与上述每个系统相对应的 $\ mathcal{ U} $的子理论, 并证明当 $\ mathcal{ U} $ 的证据只使用子理论的符号时, 那么它就是该子理论的证明 。