When we investigate a type system, it is helpful if we can establish the well-foundedness of types or terms with respect to a certain hierarchy, and the Extended Calculus of Constructions (called ECC, defined and studied comprehensively in [Luo, 1994]) is no exception. However, under a very natural hierarchy relation, the well-foundedness of terms does not hold generally. In this article, the well-foundedness are established for two natural families of terms (namely, types in a valid context and terms having normal forms). Also, we give an independent proof of the existence of principal types in ECC since it is used in the proof of well-foundedness of types in a valid context although it is often proved by utilizing the well-foundedness of terms, which would make our argument circular if adopted.

中文翻译：

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ECC术语的主要类型和有根据的

当我们研究一个类型系统时，如果我们能够建立关于某个层次结构的类型或术语的良好基础，以及扩展的构造演算（称为 ECC，在 [Luo, 1994] 中进行了全面的定义和研究）是有帮助的。也不例外。然而，在非常自然的等级关系下，术语的有根据的并不普遍。在本文中，为两个自然的术语族（即有效上下文中的类型和具有范式的术语）建立了良好的基础。此外，我们给出了 ECC 中主要类型存在的独立证明，因为它被用于证明有效上下文中类型的有根据的证明，尽管它经常通过利用术语的有根据的证明来证明，这将使我们的论据循环，如果被采纳。