Garlene: Guarded Recursion in Lean

2026-09-21

AI Summary

本文解决了在Lean中支持受保护递归的问题，通过实现一种嵌入式语言，结合元编程技术，使用户能够编写和证明关于受保护递归定义的性质。

Abstract

Extending the recursion principles of a formal system is an enticing but dangerous endeavour with a well-documented history of leading to consistency bugs. Nakano's guarded recursion is an elegant, type-based approach to soundly extend type theory with a powerful recursion principle. This makes guarded recursion useful for many applications, from programming with infinite structures such as streams to reasoning about advanced programming language features using synthetic guarded domain theory. Sadly, guarded recursion is not directly supported by any major interactive theorem prover, which leaves users of guarded recursion with unmechanised pen-and-paper proofs or mechanisations that depend on unmaintained theorem provers. In this paper, we present an implementation of guarded recursion as an embedded language in Lean consisting of a simply-typed lambda calculus for definitions and a higher-order logic for reasoning. Using Lean's excellent support for metaprogramming, our language allows users to write guarded recursive definitions in an intuitive syntax and to prove properties about them using a dedicated proof mode. We give our language a presheaf model, which we use to prove the soundness of our language and to allow users to export guarded recursive definitions and their theorems into standard Lean developments. To demonstrate the usefulness of our language, we present several case studies for programming and reasoning with guarded recursion.

Problem

• guarded recursion
• interactive theorem prover
• formal system

Innovation

• guarded recursion
• embedded language
• Lean
• metaprogramming
• presheaf model