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A guest post by mathematician Thomas Hales, published by Terence Tao on Oct. 9, 2026, examines Lean’s role in checking formal mathematical proofs and the rapid growth of AI-assisted formalization. Hales describes major projects and Lean’s expanding mathlib library, but the supplied post excerpt does not establish that AI-generated proofs are error-free or settle broader questions about reliability.
Mathematician Thomas Hales has published a survey of the Lean theorem prover, its use in checking formal mathematical proofs and a wave of AI-assisted formalization projects. The guest post, published by Terence Tao on Oct. 9, 2026, describes a rapidly expanding body of Lean mathematics while raising the reliability questions that accompany the conversion of research proofs into machine-checked code.
Hales describes a formal proof as one checked against the foundations of mathematics and the rules of logic by specialized software. Lean is one of several proof assistants, and the post focuses on it because Hales says it is the most popular among mathematicians. Developed by Leo de Moura and introduced in 2013, Lean is open-source.
The post gives figures for Lean’s shared mathematical library, mathlib: nearly 300,000 theorems, more than 100,000 definitions, about 2.5 million lines of code and over 700 contributors. Hales explains that formalized results can be reused: a later proof can call on an existing result such as the Cauchy-Schwarz inequality rather than formalizing it again.
Hales says AI-assisted formalization became a practical reality in 2026. He lists projects announced between September 2025 and September 2026, including textbook formalization, the 24-dimensional sphere-packing problem, and formalizations associated with Fermat’s Last Theorem and a Navier-Stokes blowup result with forcing. The post reports that the Fermat project announced by Anthropic generated 13 million lines of Lean in 11 days; that is a reported code volume, not a measure of correctness or mathematical significance.
AI Changes the Scale of Formal Proof
Formal proof offers mathematicians a way to check whether a proof, as encoded, follows from specified definitions and rules. That can make errors in lengthy derivations easier to detect and can let other researchers build on verified library results. It does not, by itself, show that a formal statement captures the intended theorem or that the original argument was translated faithfully.
The reported acceleration matters because traditional formalization has required substantial human effort. Hales cites the Kepler conjecture formalization as taking about 20 human work-years and producing roughly 500,000 lines of proof scripts. AI systems that generate formal code could reduce some of that labor and broaden the range of mathematics checked in proof assistants. At the same time, growing code output makes careful review of statements, assumptions and verification processes more important.
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From Lean to Large-Scale Formalization
Lean emerged in 2013, and work on mathlib began in 2017, when Mario Carneiro and Johannes Hölzl started a separate mathematical library using existing Lean material. The library’s collaborative model lets contributors add definitions and results that later proofs can reuse. Hales places Lean alongside other proof assistants, including HOL Light, Isabelle, Rocq (formerly Coq), Metamath and Mizar.
The post traces AI-assisted work through a series of announcements. A September 2025 project on the prime number theorem still required human guidance when the system stalled, Hales says. A January 2026 preprint by J. Urban described formalizing substantial parts of a topology textbook in a set-theory-based proof assistant. Later examples in Hales’s account include the Meta/Facebook Research ATLAS project and announcements from Math Inc., Anthropic and OpenAI. The projects used different systems and methods, so their reported outputs are not direct comparisons of Lean performance.
“Autoformalization is the formalization of mathematics by AI.”
— Thomas Hales, guest post on Terence Tao’s blog
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What the Project Claims Do Not Resolve
The supplied post excerpt introduces the question “Is Lean reliable?” and begins an explanation of its type-theory foundations, but it ends before Hales’s full discussion of reliability. It therefore does not provide enough material to report his complete analysis of Lean’s kernel, implementation risks or limits.
Several distinctions also remain important. The project descriptions and output figures are reported by Hales from announcements and other sources; the excerpt does not independently evaluate each result or provide a common test of accuracy. It is not clear from the material provided how much human intervention each project required, how all formal statements were checked against their intended textbook or research claims, or what independent review each received. Large amounts of generated code alone do not answer those questions.
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Scrutiny of AI-Written Proofs
The next step for mathematicians is to examine the formal statements, dependencies and checking records behind individual projects, rather than treating code volume or an announcement as proof of overall reliability. Researchers will also need to assess how much human guidance is involved and whether a formalized statement faithfully represents the result being claimed.
Hales’s post frames autoformalization as a fast-developing field, with projects appearing across proof assistants and AI systems. The pace and breadth of that work will become clearer as more formalizations are made available for inspection and as researchers report on their verification and maintenance. The post excerpt does not identify a specific next technical milestone or timetable.
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Key Questions
What is Lean?
Lean is an open-source proof assistant used to encode mathematical statements and proofs in a form that specialized software can check against formal rules.
What is autoformalization?
In Hales’s usage, autoformalization is the use of AI to turn mathematical material into formal proof code for Lean or another proof assistant. The amount of human guidance can differ between projects.
Does a large AI-generated Lean proof automatically establish that a result is correct?
No. Code volume is not a measure of correctness. Readers need to know what statement was formalized, what assumptions it uses, whether the code passes the relevant proof checker and how faithfully it represents the intended mathematical result.
How large is mathlib, according to Hales?
Hales reports that mathlib contains nearly 300,000 theorems, over 100,000 definitions and about 2.5 million lines of code, with more than 700 contributors.
Source: hn
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