We revisit occurrence typing, a technique to refine the type of variables occurring in type-cases and, thus, capture some programming patterns used in untyped languages. Although occurrence typing was tied from its inception to set-theoretic types---union types, in particular---it never fully exploited the capabilities of these types. Here we show how, by using set-theoretic types, it is possible to develop a general typing framework that encompasses and generalizes several aspects of current occurrence typing proposals and that can be applied to tackle other problems such as the reconstruction of intersection types for unannotated or partially annotated functions and the optimization of the compilation of gradually typed languages.

中文翻译：

---

重温事件打字

我们重新审视了出现类型，这是一种改进类型案例中出现的变量类型的技术，从而捕获一些在无类型语言中使用的编程模式。尽管出现类型从一开始就被绑定到集合论类型——特别是联合类型——它从未充分利用这些类型的功能。在这里，我们展示了如何通过使用集合论类型来开发一个通用类型框架，该框架包含和概括当前出现类型建议的几个方面，并且可以应用于解决其他问题，例如未注释的交叉类型的重建或部分注释的函数和渐进式语言编译的优化。