We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism, which were developed in blame calculi and to state the "gradual guarantee" theorem of gradual typing. Combined with the ordinary extensionality ($\eta$) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs, which is a semantic analogue of Findler and Felleisen's definitions of contracts, and use it to build some concrete domain-theoretic models of gradual typing.

中文翻译：

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逐名渐进类型理论

我们提出了渐进类型理论，这是一种按名称调用渐进类型的逻辑和类型理论。我们以一种新的方式定义了渐进类型的中心结构（动态类型、类型转换和类型错误），通过与渐进类型和术语动态的新判断相关的普遍属性，这些新判断是在推算中发展起来的，并陈述“渐进”保证”渐进式定理。结合类型理论提供的普通外延性 ($\eta$) 原则，我们表明强制转换的大多数标准操作行为都是由渐进保证唯一确定的。这为强制转换的定义提供了语义上的理由，并表明强制转换的非标准定义必须违反这些原则。我们的类型理论是某种称为设备的先序类别的内部语言。