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lean-lang-org · lean-fro · leonardo-de-moura · mathlib · lean-for-programmers · lean-certificate · autoformalization · flt-formalization-anthropic · formal-verification
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lean-for-programmers.md

Lean (theorem prover)

An interactive proof assistant and programming language for writing and machine- checking formal mathematical proofs. Introduced to this wiki by lean-for-programmers.

What it is

  • A proof assistant, not an automatic theorem prover: its kernel checks that expressions are well-formed and proof steps are legal; the IDE helps the user build proofs and suggests steps.
  • Built on the Curry-Howard correspondence: propositions are types, proofs are terms of those types. Uses dependent types (types that depend on values) and universe levels to avoid paradoxes.
  • Proofs are written with tactics (intro, cases, rw, simp, ring, omega, exact, …) in by mode. Ships Mathlib, which the project now describes as over a million lines of formalized mathematics.
  • Prominent in AI mathematics: alphaproof (DeepMind, IMO silver-medalist), LeanDojo, miniF2F, ProofNet.

New role, 2026-08-01 (ten-proofs). Lean stopped being only the place AI mathematics is done and became the place it is audited. OpenAI released a Lean 4.32.0 + mathlib formalization for each of ten claimed advances, so a reader checks the kernel instead of the press release. See lean-certificate.

And the place a 350-year-old proof got machine-checked, 2026-09-04. flt-formalization-anthropic: Claude wrote 13 million lines of Lean over 11 days to produce the first end-to-end verified proof of Fermat's Last Theorem, using Lean's three standard axioms and 5× more code than Mathlib itself. Lean was the substrate for autoformalization at a scale the community had budgeted years for.

Honest bridge to the knowledge-management cluster

The article's history of mechanizing reasoning — Leibniz's dream of a universal formal language plus a calculating machine, through Hilbert, Gödel, Church-Turing — overlaps directly with as-we-may-think: Bush (1945) also invoked Leibniz's calculator and foresaw "a machine which will manipulate premises in accordance with formal logic." Lean is, in a real sense, the realization of the mechanized formal reasoning that Bush gestured at, just as the memex anticipated the llm-wiki. The AI-proof angle (AlphaProof) also rhymes with the LLM-as-agent theme behind gbrain. These are genuine threads, not forced links — but Lean's core domain (formal methods) remains distinct (cluster E).

lean-for-programmers · as-we-may-think · ten-proofs · lean-certificate · alphaproof · flt-formalization-anthropic · autoformalization

Re-seen 2026-09-06 — the project's own site, and the tagline has changed

Arrived via Telegram as https://lean-lang.org/, headed "Lean Programming Language." Read with /fro/about and the GitHub API; summarized at lean-lang-org. Three things this page did not have.

The self-description leads with the language. "Lean is an open-source programming language and proof assistant." This page was written from a tutorial and called Lean a "proof assistant and programming language" in that order. The project puts them the other way round, and the featured projects back it: five of the six are software and hardware verification, not mathematics — AWS's Cedar authorization language, Microsoft Research's verification of SymCrypt Rust cryptography via the Aeneas toolchain, the Ethereum Foundation's ArkLib, Veil for distributed protocols, and SampCert differential-privacy primitives running in AWS Clean Rooms. AWS says it runs the Lean formalization as an executable oracle against production Rust in CI, which is a use of Lean this wiki had no instance of: not proving a theorem, but continuously differential-testing shipped code against a specification that happens to be a Lean program.

grind. The automation tactic shipped in Lean 4.22.0 (August 2025) and now carries most of the front page's sample proofs, including the infinitude of primes. It closes goals by pattern matching and case analysis, and solves systems of linear integer inequalities outright. This page's original tactic list (intro, cases, rw, simp, ring, omega) predates it. The distinction the page draws between an assistant that checks and an automatic prover that finds still holds, but grind moves the line: more of a routine proof is now found rather than written.

Governance, funding and licence. Development runs through the Lean FRO, a non-profit formed July 2023 under Convergent Research, led by Leonardo de Moura (who created Lean at Microsoft Research in 2013, and created Z3 before it) with Sebastian Ullrich. Funders include Alex Gerko of XTX Markets ($10M in July 2025), Amazon Automated Reasoning, the Sloan and Simons foundations, and Harmonic. The site never states a licence; the GitHub API gives Apache-2.0 for leanprover/lean4 and for Mathlib.

Scale, read the same day: leanprover/lean4 at 9,007 stars / 964 forks, created 2018-04-15, pushed 2026-09-06. Lean 4.0 was released September 2023. Awards in 2025: the ACM SIGPLAN Programming Languages Software Award, and the Skolem Award for the 2015 CADE paper describing the system.

One number worth holding at arm's length. 9,007 stars is small. The corpus paged a coding agent the same day at 29,778 stars on an eight-month-old repository. Stars measure attention inside a population, and the population that installs a proof assistant is not the population that installs a terminal tool — the comparison says nothing about importance, and lean-lang-org works through why none of Lean's published figures can be put on an adoption curve at all.

lean-lang-org · lean-fro · leonardo-de-moura · mathlib · formal-verification · adoption-curve-measurement

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