Reformalization Study of Jordan Curve Theorem in Proof Assistants

This paper delves into the concept of reformalization, a specialized form of autoformalization where the input is not natural language but an existing formal proof developed in a different proof assistant. The study focuses on the Jordan Curve Theorem, a foundational result in topology, as a complex case study for this process. The researchers detail three specific reformalizations: translating a proof from Mizar into Lean, and two translations from HOL Light into Lean and Agda, respectively. This multi-faceted approach allows for a comparative analysis of the challenges and successes involved in porting highly intricate formal mathematical developments across different formal verification systems. The analysis identifies and discusses various pipeline design choices that are critical for the practical success of reformalization tasks. This includes aspects like parsing, semantic mapping, and handling differences in logical foundations and library structures between the source and target proof assistants. The findings contribute to the broader goal of making formal mathematics more interoperable and accessible across different verification tools.