This article presents a bidirectional type system for the Calculus of Inductive Constructions (CIC). It introduces a novel judgement intermediate between the usual inference and checking, dubbed constrained inference, to handle the presence of computation in types. The key property is the completeness of the system with respect to the usual undirected one, which has been formally proven in Coq as a part of the MetaCoq project. Although it plays a central role in an ongoing completeness proof for a realistic typing algorithm, the interest of bidirectionality is much wider, as it clarifies previous works in the area and gives strong insights and structure when trying to prove properties on CIC or design variations and extensions.

中文翻译：

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归纳构造演算的完整双向键入

本文介绍了用于感应构造微积分（CIC）的双向类型系统。它引入了一种新颖的判断方法，该方法介于通常的推理和检查之间（称为约束推理），以处理类型中计算的存在。关键特性是系统相对于通常的非定向系统的完整性，这是Coq作为MetaCoq项目的一部分已正式证明。尽管它在实现逼真的类型算法的持续完整性证明中发挥着核心作用，但是双向性的兴趣却更为广泛，因为它澄清了该领域的先前工作，并在试图证明CIC或设计变体的特性时提供了深刻的见解和结构。扩展名。