Terence Tao-hosted post highlights rapid advances in AI-driven autoformalization of mathematics using Lean

A guest post hosted by Fields Medalist Terence Tao surveys recent progress in autoformalization — the use of AI to convert mathematical papers into formally verified proofs within proof assistants such as Lean. The author emphasizes that reliability in mathematics hinges on consistency and formal verification, and frames autoformalization as a practical breakthrough in 2025–2026.

Lean, a proof assistant introduced in 2013 by Leo de Moura at Microsoft and later open-sourced, has become the most widely used system among mathematicians. Its mathematical library, mathlib, now contains nearly 300,000 theorems, over 100,000 definitions, and 2.5 million lines of code contributed by more than 700 people. Any definition or theorem in mathlib can be reused to prove further results, enabling cumulative formalization at scale.

The post documents several 2025–2026 milestones in autoformalization. In September 2025, Math Inc. reported a quasi-autoformalization of the prime number theorem, requiring human guidance when the AI stalled. In January 2026, J. Urban described autoformalizing large parts of Munkres’s topology textbook in two weeks. In March 2026, Math Inc. announced an autoformalization of the sphere-packing problem in 24 dimensions following the proof by Viazovska, generating roughly 500,000 source lines of code that were later reduced to about 200,000. In May 2026, a Meta/Facebook Research team autoformalized a large part of 26 mathematical textbooks in a project called ATLAS.

Notable later milestones include Anthropic’s September 4, 2026 announcement of an autoformalization of Fermat’s Last Theorem, which produced 13 million lines of Lean in 11 days, and OpenAI’s September 8, 2026 announcement of an autoformalization of Navier–Stokes blowup with forcing, also in Lean. In January 2026, Urban predicted that autoformalization would become "quite easy and ubiquitous" in 2026 across proof assistants. On September 8, 2026, Jared Lichtman launched the Mathematics Autoformalization Project (MAP), aiming to translate "all known math into formal code."

The reliability of Lean is tied to its foundations in type theory, specifically the Calculus of Inductive Constructions (CIC). The post notes that type theory avoids paradoxes like Russell’s by structuring mathematical objects into disjoint types, and cites a 1997 result by B. Werner showing that ZFC set theory can be encoded into CIC and back, offering reassurance to mathematicians accustomed to set-theoretic foundations.