We present a new model of guarded dependent type theory (GDTT), a type theory with guarded recursion and multiple clocks in which one can program with and reason about coinductive types. Productivity of recursively defined coinductive programs and proofs is encoded in types using guarded recursion and can therefore be checked modularly, unlike the syntactic checks implemented in modern proof assistants. The model is based on a category of covariant presheaves over a category of time objects, and quantification over clocks is modelled using a presheaf of clocks. To model the clock irrelevance axiom, crucial for programming with coinductive types, types must be interpreted as presheaves internally right orthogonal to the object of clocks. In the case of dependent types, this translates to a lifting condition similar to the one found in homotopy theoretic models of type theory, but here with an additional requirement of uniqueness of lifts. Since the universes defined by the standard Hofmann–Streicher construction in this model do not satisfy this property, the universes in GDTT must be indexed by contexts of clock variables. We show how to model these universes in such a way that inclusions of clock contexts give rise to inclusions of universes commuting with type operations on the nose.

中文翻译：

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保护依赖类型理论的指称语义

我们提出了一种新的受保护依赖类型理论 (GDTT) 模型，这是一种具有保护递归和多个时钟的类型理论，人们可以在其中编程和推理协约类型。递归定义的共归纳程序和证明的生产力使用受保护的递归编码在类型中，因此可以模块化检查，这与现代证明助手中实施的句法检查不同。该模型基于一类时间对象上的一类协变预层，并且使用时钟的预层对时钟上的量化进行建模。为模型建模时钟无关公理，对于使用互归纳类型进行编程至关重要，类型必须被解释为内部与时钟对象正交的预滑轮。在依赖类型的情况下，这转化为类似于在类型论的同伦理论模型中发现的提升条件，但这里对提升的唯一性有额外的要求。由于此模型中由标准 Hofmann-Streicher 构造定义的宇宙不满足此属性，因此 GDTT 中的宇宙必须由时钟变量的上下文索引。我们展示了如何以这样一种方式对这些宇宙进行建模，即时钟上下文的包含引起宇宙的包含与鼻子上的类型操作进行交换。