Revisiting Soundness for Occurrence Typing, Semantically

2026-09-14

AI Summary

本文针对类型化Racket中的代入类型问题，使用分步索引逻辑关系修正了其核心演算并证明了语义类型健全性。

Abstract

Over the past two decades, numerous systems have brought some of the benefits of dependent typing to a wide variety of new programming languages, often by restricting which terms can appear inside types. Such techniques are known as refinement types, occurrence typing, liquid types, and path dependent types, among others. However, the restrictions adopted by these systems often break the substitution property, because they explicitly disallow the ability to substitute arbitrary terms for variables inside types. This leads to significant complexity in the design and metatheory of these systems, increasing the possibility of significant errors. We consider a specific line of work on occurrence typing, namely, the calculus underlying Typed Racket due to Tobin-Hochstadt and Felleisen 2010. We show that the fundamental challenge of substitution into types resulted in multiple flaws in the formalism and the syntactic type soundness theorem of this work. These flaws are replicated in several other papers building on this work, and also surface as a soundness bug in Typed Racket itself. We identify and repair these problems, revising the core calculus of Typed Racket and giving a \emph{semantic type soundness} proof using step-indexed logical relations, formalized in Lean. We argue that this approach is simpler than it may seem, and easily scales to handle the complexity of the occurrence typing in Typed Racket.

Problem

• occurrence typing
• substitution property
• type soundness
• typed racket

Innovation

• semantic type soundness
• step-indexed logical relations
• occurrence typing