In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type$_0$ : Type$_1$ : $\cdots$ . Such type systems are called cumulative if for any type $A$ we have that $A$ : Type$_{i}$ implies $A$ : Type$_{i+1}$. The predicative calculus of inductive constructions (pCIC) which forms the basis of the Coq proof assistant, is one such system. In this paper we present and establish the soundness of the predicative calculus of cumulative inductive constructions (pCuIC) which extends the cumulativity relation to inductive types.

中文翻译：

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累积归纳结构的谓词演算的一致性 (pCuIC)

为了避免与自我参照定义相关的众所周知的悖论，高阶依赖类型理论使用可数无限的宇宙层次结构（也称为排序）对理论进行分层，Type$_0$ : Type$_1$ : $\ cdots$ 。如果对于任何类型 $A$ 我们有 $A$ : Type$_{i}$ 意味着 $A$ : Type$_{i+1}$ ，则这种类型系统被称为累积型系统。构成 Coq 证明助手基础的归纳结构的谓词演算 (pCIC) 就是这样一个系统。在本文中，我们提出并建立了累积归纳结构的谓词演算 (pCuIC) 的合理性，它将累积性关系扩展到归纳类型。