We advocate the use of de Bruijn's universal abstraction $\lambda^\infty$ for the quantification of schematic variables in the predicative setting and we present a typed $\lambda$-calculus featuring the quantifier $\lambda^\infty$ accompanied by other practically useful constructions like explicit substitutions and expected type annotations. The calculus stands just on two notions, i.e., bound rt-reduction and parametric validity, and has the expressive power of $\lambda\rightarrow$. Thus, while not aiming at being a logical framework by itself, it does enjoy many desired invariants of logical frameworks including confluence of reduction, strong normalization, preservation of type by reduction, decidability, correctness of types and uniqueness of types up to conversion. This calculus belongs to the $\lambda\delta$ family of formal systems, which borrow some features from the pure type systems and some from the languages of the Automath tradition, but stand outside both families. In particular, the calculus includes and evolves two earlier systems of this family. Moreover, a machine-checked specification of its theory is available.

中文翻译：

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图解变量通用量化的正式系统

我们提倡使用 de Bruijn 的通用抽象 $\lambda^\infty$ 在谓词设置中量化图解变量，我们提出了一个类型化的 $\lambda$-演算，其特征是量词 $\lambda^\infty$ 伴随着其他实际有用的结构，如显式替换和预期类型注释。该微积分仅基于两个概念，即边界 rt-reduction 和参数有效性，并且具有 $\lambda\rightarrow$ 的表达能力。因此，虽然它本身并不旨在成为一个逻辑框架，但它确实享有许多所需的逻辑框架不变量，包括归约汇合、强规范化、归约保留类型、可判定性、类型正确性和类型在转换之前的唯一性。这个微积分属于形式系统的 $\lambda\delta$ 族，它从纯类型系统中借用了一些功能，从 Automath 传统语言中借用了一些功能，但不属于这两个家族。特别是，微积分包括并发展了这个家族的两个早期系统。此外，还提供了其理论的机器检查规范。