This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly infinitary operators and equations. We prove that QWI types can be derived from quotient types and inductive types in the type theory of toposes with natural number object and universes, provided those universes satisfy the Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types as colimits of a family of approximations to them defined by well-founded recursion over a suitable notion of size, whose definition involves the WISC axiom. We developed the proof and checked it using the Agda theorem prover.

中文翻译：

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商，归纳类型和商归纳类型

本文在构造类型理论的框架内介绍了一种称为QWI类型的索引商归纳类型的表达类。它们是方程式理论索引族的初始代数，可能带有无限式算子和方程式。我们证明了QWI类型可以从自然数对象和宇宙的姿势类型理论中的商类型和归纳类型派生，只要这些宇宙满足弱初始覆盖集（WISC）公理。我们通过将QWI类型构造为它们的一个近似族的共极限来实现，它们由对适当大小概念的充分基础的递归定义，该大小概念涉及WISC公理。我们开发了证明，并使用Agda定理证明者对其进行了检验。