Jordan Curve Theorem reformalized: Mizar and HOL Light to Lean

Simon Guilloud, Sankalp Gambhir and Samuel Chassot submitted a paper titled "Reformalization of the Jordan Curve Theorem" to arXiv on 2 July 2026 (arXiv:2607.01734). The paper presents a case study in reformalization that reports three concrete transfers of the Jordan Curve Theorem between proof assistants and evaluates the design choices behind those pipelines.

The paper defines reformalization as "a variant of autoformalization in which the input proof is not natural language but a formal development in a different proof assistant." The authors document three reformalizations: from Mizar to Lean, from HOL Light to Lean, and from HOL Light to Agda. The submission appears in the cs.AI category and the arXiv entry lists a PDF and TeX source; the version submitted on 2 July 2026 is labeled arXiv:2607.01734 and the submission record notes a 207 KB upload.

What did the paper do?

The paper carries out three concrete reformalizations of the Jordan Curve Theorem and analyses the outcomes and pipelines used. Specifically, the study moves formal proofs between Mizar, HOL Light and two target assistants: Lean and Agda, producing Mizar→Lean, HOL Light→Lean, and HOL Light→Agda reformalizations. The authors then analyse the results and identify pipeline design choices that matter for practical reformalization tasks.

Beyond listing the three ports, the arXiv entry shows the authors provide a PDF and TeX source for the work and notes an arXiv-issued DOI via DataCite (pending registration). The paper is filed under Artificial Intelligence (cs.AI) on arXiv.

How does reformalization differ from autoformalization?

Reformalization starts from a formal development in one proof assistant and targets a different assistant, whereas autoformalization normally begins with natural language proofs. The paper states this distinction succinctly: reformalization is "a variant of autoformalization in which the input proof is not natural language but a formal development in a different proof assistant." This framing shifts the technical challenge from natural-language understanding to translating between formal semantics, libraries and proof styles.

The submission documents three cases that instantiate that idea: two separate source systems (Mizar and HOL Light) were taken as inputs and translated into two different targets (Lean and Agda), producing three reformalizations that let the authors compare how pipelines perform across source/target pairs.

Why it matters

Reformalization addresses a practical bottleneck in formal mathematics: reuse. By transferring formal proofs from one assistant to another, developers can leverage existing developments rather than redoing work from scratch. The paper’s comparative approach—three distinct source/target ports—lets practitioners see where translation pipelines succeed and where design choices create friction. That empirical focus helps clarify which pipeline components are likely to affect practical adoption of cross-assistant workflows.

The authors’ decision to publish the case study and to include source materials on arXiv (PDF and TeX) and to note the pending DataCite DOI increases the work’s accessibility for researchers exploring inter-assistant tooling.

What to watch

Look for the arXiv DOI registration to appear via DataCite and for the paper’s TeX source and any linked code or data on the arXiv page. Also watch whether other teams adopt the specific pipeline design choices the authors identify when attempting further reformalizations between systems beyond Mizar, HOL Light, Lean and Agda.