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A guest post by mathematician Thomas Hales, published on Terence Tao’s blog on October 9, 2026, surveys Lean’s mathematical library and recent AI-assisted formalization projects. The post frames reliability as a central question, but the supplied text ends just as Hales begins discussing Lean’s foundations, so it does not provide his assessment of the prover’s reliability.
Mathematician Thomas Hales published a guest post on Terence Tao’s blog on October 9, 2026, surveying the growth of the Lean theorem prover and recent AI efforts to translate mathematics into formal proofs. The post asks what mathematicians should know about reliability and AI, as large formalization projects have drawn attention to computer-checked proofs; the supplied text, however, stops before Hales gives his assessment of Lean’s reliability.
Hales describes a formal proof as one checked against mathematical foundations and rules of logic, usually by software because of the number of steps involved. Such systems are known as proof assistants or theorem provers. The post lists formalizations including the four-color theorem, the Feit–Thompson theorem, the Kepler conjecture, Fermat’s Last Theorem, and sphere-packing results. It says three of the listed projects were completed in 2026, though the excerpt does not identify which three.
The post focuses on Lean, an open-source proof assistant introduced by Leo de Moura in 2013 while he was at Microsoft. Hales says the associated mathematical library, mathlib, has grown to nearly 300,000 theorems, more than 100,000 definitions and 2.5 million lines of code, with over 700 contributors. Researchers can use existing results in the library, such as the Cauchy–Schwarz inequality, in further proofs.
Hales calls autoformalization the use of AI to read mathematical material and produce formal proofs in a system such as Lean. He describes a sequence of projects, including an announced Fermat’s Last Theorem formalization by Anthropic on September 4, 2026, which the post says produced 13 million lines of Lean in 11 days. It also cites an OpenAI announcement on September 8 concerning a formalization of a Navier–Stokes blowup result with forcing. These are accounts of project announcements in Hales’s post; the excerpt does not independently document their verification or technical scope.
AI Projects Put Proof Checking in Focus
The projects described could change how mathematicians approach the labor involved in formal proof. Hales contrasts recent AI-assisted efforts with the Kepler conjecture formalization, which he says required about 20 human work-years and produced roughly 500,000 lines of proof scripts. If AI tools can help translate existing mathematical arguments into proof-assistant code, more results may become available for checking and reuse.
That possibility matters because formal proofs are checked by software against specified foundations and logic. A large library such as mathlib can also let formalized results serve as building blocks for later work. Yet growth in the volume of generated code is not, by itself, evidence that a particular mathematical claim is correct or that an entire project has been independently reviewed. The article’s framing puts the reliability of the checking process alongside the speed and scale of AI-assisted formalization.
For working mathematicians, the immediate development is a changing research landscape: AI-assisted formalization is being reported across several projects and proof systems, while Lean is the focus of Hales’s discussion. The excerpt offers milestones and scale figures, but it does not establish how broadly the methods work, how much expert guidance they require, or how the resulting formal proofs should be evaluated in each case.
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From Lean’s Launch to AI Milestones
Lean began in 2013, according to Hales, and its open-source release helped make it available to a wider research community. In 2017, Mario Carneiro and Johannes Hölzl began a separate mathematical library, mathlib, using existing parts of Lean’s core library. Hales presents its current size as evidence of a substantial shared base that users can draw on when formalizing further mathematics.
The post places recent AI work on a timeline. It describes a September 2025 prime number theorem effort by Math Inc. as “quasi-autoformalization,” because humans had to guide the system when it got stuck. In January 2026, J. Urban posted an arXiv preprint about formalizing large parts of a topology textbook in a proof assistant based on set theory. Hales also cites a May 2026 Meta/Facebook Research project, ATLAS, involving material from 26 mathematics textbooks.
For Lean specifically, Hales points to a March 2026 announcement concerning the 24-dimensional sphere-packing problem and to September announcements involving Fermat’s Last Theorem and a Navier–Stokes result. The examples span different projects and, in some cases, different proof assistants. The post therefore presents a field-wide trend while concentrating its discussion on Lean and its mathematical library.
“Autoformalization is the formalization of mathematics by AI.”
— Thomas Hales, guest post on Terence Tao’s blog
Reliability Assessment Is Not Included
The supplied post text ends after Hales introduces Lean’s basis in type theory, specifically the calculus of inductive constructions, and says he will be brief. It does not include the ensuing discussion or a conclusion about Lean’s reliability. The title raises that issue, but the excerpt alone cannot establish what qualifications or risks Hales identifies.
Several project descriptions are reported as announcements or claims in the post. The excerpt does not specify the precise checking status, scope, independent review, or human involvement for each project. It also does not explain which three formalization projects were completed in 2026, despite saying that three of the examples were completed that year. Those details remain unconfirmed in the material provided.
The Reliability Discussion Remains
The next step for readers following Hales’s argument is to consult the complete October 9 post, particularly the portion on type theory and Lean’s reliability, which is absent from the supplied text. That section would be needed to report his answer to the question in the title.
Further reporting on the cited AI projects would need to distinguish announcements from completed, checked formalizations and establish what each project’s code covers, which proof assistant it uses, and what human direction or review it involved. The post’s milestones point to a fast-moving area, but its excerpt does not settle those questions.
Key Questions
What is the development covered in the post?
Mathematician Thomas Hales published a guest post on Terence Tao’s blog on October 9, 2026. It surveys Lean, its mathlib library, and recent AI-assisted efforts to formalize mathematics.
What does autoformalization mean here?
Hales uses the term for AI translating mathematical material, such as a paper, into a formal proof in Lean or another proof assistant. The post’s examples include projects whose process involved human guidance or whose details are reported as announcements.
Does the supplied text say whether Lean is reliable?
No. The excerpt begins a section on Lean’s type-theory foundations, then ends before giving Hales’s reliability discussion or conclusion.
How large is Lean’s mathlib library, according to Hales?
Hales says mathlib contains nearly 300,000 theorems, more than 100,000 definitions and 2.5 million lines of code, with over 700 contributors. These are figures stated in the post.
Source: hn
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