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Yes. AI systems can produce mathematical proofs that a proof assistant such as Lean checks mechanically—but they do not yet prove arbitrary mathematics on demand. The strongest results come from systems that search for proof steps after a theorem has been translated into a precise formal statement. AlphaProof, for example, solved three of the five non-geometry problems at the 2024 International Mathematical Olympiad, including its hardest problem, according to a 2025 Nature paper. That is meaningful evidence of competition-level ability, not evidence that AI can independently read, understand and solve any mathematical question.
What does it mean for AI to prove a theorem?
In ordinary mathematical writing, a proof is an argument that a reader can inspect and judge. In AI theorem proving, “proof” often has a narrower, mechanically verifiable meaning: the system produces a formal proof term that a trusted proof assistant accepts for a precise theorem statement.
Lean is one such system. Its project documentation describes it as “a functional programming language and theorem prover built for formalizing math and for formal verification.” A person or AI supplies definitions, a theorem statement and candidate proof steps. Lean elaborates those steps into a formal object and its kernel checks that the object follows from the system’s logic and the imported libraries.
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Lean’s dependent type theory is expressive enough to formalize almost any conventional mathematical theorem, as explained in Theorem Proving in Lean 4 by Jeremy Avigad, Leonardo de Moura, Soonho Kong and Sebastian Ullrich. That is a statement about what the proof assistant can represent; it does not mean that an automated system can find a proof for every theorem. The hard part is often discovering the argument, expressing the problem correctly and supplying the definitions and supporting lemmas needed to make the argument check.
How does AI theorem proving work?
Most current AI theorem provers combine a language model with a proof assistant. The model proposes a tactic or proof fragment; the assistant checks it and returns feedback about whether it works. The system can then use that feedback to continue, revise or abandon a candidate path.
- Formalize the problem. Translate the informal claim into a precise statement with definitions, quantifiers and assumptions. For example, a statement about “every” object differs from one about “some” object, so translation errors can change the theorem.
- Search for a proof. A model proposes tactics, lemmas or proof terms. Search may explore multiple branches, using successful and unsuccessful assistant feedback to guide the next attempts.
- Check the candidate. Lean verifies whether the proof term establishes the formal theorem from the available logic and libraries. A failed check is not a proof, even if the proposed reasoning sounds plausible.
- Review what was actually proved. A valid proof establishes the formal statement as written. A person still needs to check that the formal statement captures the intended informal problem and that the imported assumptions and definitions are appropriate.
AI methods can make the search stage much faster or broader than a person’s manual trial and error, especially when a system has been trained on formal proofs. But search does not eliminate the need for a correct statement, suitable mathematical knowledge encoded in libraries or enough compute to explore promising possibilities.
Which current tools and results matter most?
For readers who want to experiment, Lean 4 and Mathlib form the clearest practical ecosystem in the evidence available here. AlphaProof is a significant research result, while benchmark systems show how results can look strong under one evaluation and more modest when the full translation-and-proof task is measured.
Lean 4 and Mathlib
Lean 4 is an interactive theorem prover and programming language. Mathlib is its extensive mathematical library, containing definitions and results that formal proofs can reuse. The official Lean learning portal points readers to Theorem Proving in Lean, Mathematics in Lean, Mathlib references, natural-language theorem search, LeanDojo, Pantograph and other machine interfaces.
The practical advantage of an interactive environment is feedback: a user can write or request a proof step and see whether Lean accepts it. A large library also reduces the need to formalize every basic result from scratch. The limitation is that a library is not all of mathematics. If the relevant definition or lemma is missing, the proof may require additional formalization before an AI can use it.
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AlphaProof and the 2024 IMO
AlphaProof paired a 3-billion-parameter encoder–decoder proof network with an AlphaZero-inspired search procedure over Lean tactic states. A 2025 Nature paper from Google DeepMind reported that it solved three of the five non-geometry problems at the 2024 IMO, including the most difficult problem. These were formal proofs checked in Lean, rather than answers judged only by the plausibility of a natural-language explanation.
The result is important because olympiad problems demand substantial reasoning, not merely routine symbolic manipulation. Its scope is still specific: it was a specialized research system using extensive training and inference compute, and the reported IMO result excluded geometry. It is evidence that AI can reach a high level on a demanding, bounded competition set—not that the same system can reliably handle any theorem a mathematician might ask.
Benchmarks and reported scores
Benchmarks provide controlled collections of formal mathematics problems, but their scores are not interchangeable measures of general mathematical ability. The original miniF2F release contains 244 statements. A later miniF2F-v2 repository describes 488 statements and says the collection was frozen in October 2022. Systems may differ in the version they use, the formal statements they receive, the amount of search they perform and how many attempts they are allowed.
| System or evaluation | Reported result | How to interpret it |
|---|---|---|
| AlphaProof, 2024 IMO | Three of five non-geometry problems solved; the 2025 Nature paper identifies one as the hardest problem. | A competition-level demonstration on the reported non-geometry problems, not a score across all mathematics. |
| BFS-Prover-V2 | 95.08% on miniF2F and 41.4% on ProofNet test sets, as reported on the ByteDance Seed project page. | Benchmark results for the named test sets; they do not establish the same success rate on informal questions or new research theorems. |
| Full informal-to-formal pipeline | About 36% on original miniF2F and 70% on corrected miniF2F-v2 in a 2025 NeurIPS study. | Includes the challenge of translating an informal problem into a formal statement before proving it. |
The difference between the last row and results obtained when formalization and proving are assessed separately is especially instructive. A prover given a clean Lean statement has already bypassed a difficult task: deciding exactly what the human-language problem means. The NeurIPS 2025 re-evaluation found that discrepancies between informal and formal statements caused many failures. Thus, a high theorem-proving score on formal inputs does not automatically show that a system can solve the same problems from natural language.
Benchmarks also need careful reading because reported figures can depend on sampling and search budgets, training data, and benchmark version. The available descriptions do not establish a single shared evaluation setup for all the scores above. Compare systems only when the benchmark version, input format and evaluation conditions match.
What can AI theorem provers do well now?
Current systems are most useful where the problem can be stated precisely and the proof can draw on established formal definitions and lemmas. In those conditions, the assistant’s checker provides a strong guarantee about the validity of the proof term for the statement it was given.
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- Find proof steps in established libraries. Guided tactic generation and search can solve many formalized algebra, number-theory and combinatorics exercises when the relevant definitions and results are available.
- Reject invalid proof terms. Lean’s kernel checks the final formal object. This prevents a persuasive but logically invalid natural-language explanation from being mistaken for an accepted Lean proof.
- Explore alternatives. Neural models, reinforcement learning and tree search can try many candidate paths. This can help locate a proof route that a person might not choose immediately, although it can require substantial computation.
- Support learning and formalization. Lean’s interactive feedback, tutorials and Mathlib can help a learner turn a mathematical argument into checked code one piece at a time.
These strengths make AI useful as a proof-search assistant and a formalization aid. They do not remove the need to understand what the result says, particularly when the formal statement is produced automatically.
What can’t AI theorem provers reliably do yet?
The central limits are not simply that models sometimes make mistakes. They are that theorem proving is a chain of dependent tasks: understand the intended claim, express it faithfully, find a proof using available formal knowledge, and verify that the formal result answers the original question.
Translate every informal problem correctly
Formalization requires choices about definitions, quantifiers and assumptions. A small mismatch can make the formalized problem easier, harder or simply different from the original. The 2025 NeurIPS study’s gap between separate-stage results and full-pipeline results shows that this translation is a major source of failure, not a clerical detail that can be ignored.
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Most public evaluation described here uses curated benchmarks or competition problems. A benchmark result does not establish that a system can autonomously develop definitions, find a new conceptual approach and prove a previously unknown research theorem. The ACL benchmark authors note that models often struggle with novel and complex theorems where human insight is essential.
Work without suitable libraries
A true theorem can remain out of reach if the formal environment lacks the needed definitions, domain-specific results or intermediate lemmas. In that case, someone may need to build the formal foundation or prove supporting facts before the main result is approachable. Library coverage affects practical capability, even when the proof assistant itself is expressive enough to represent the mathematics.
Cover geometry and every other domain equally
AlphaProof’s reported IMO result covered the five non-geometry problems, not geometry. The available data attributes the exclusion to formalization and library limitations. That example should not be generalized into a claim that all AI systems fail at geometry, but it does show that success in algebraic or number-theoretic tasks does not prove broad coverage across mathematical domains.
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Make benchmark scores directly comparable
Pass-at-k sampling, search budget, training data and benchmark versions can all affect a reported score. A system allowed many candidate attempts may have a different measured result from one given a single attempt. Unless those conditions are aligned, a percentage is not a reliable ranking of general theorem-proving ability.
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How can you try AI-assisted proving with Lean?
A sensible first experiment is to prove a small, familiar statement in Lean and treat the assistant as a collaborator whose work must pass the checker. The official Lean learning portal provides books, references and machine interfaces; the exact setup depends on the environment or interface you choose.
- Learn Lean’s basic notation and proof style. Start with the official Theorem Proving in Lean or Mathematics in Lean materials, then use the Mathlib references to understand available definitions and results.
- Choose a small theorem with a precise statement. A familiar exercise makes it easier to spot when the formal claim does not match the mathematics you intended.
- Check the statement before asking for a proof. Confirm the variables, assumptions, definitions and quantifiers. An accepted proof is only as relevant as the statement it proves.
- Ask for candidate tactics or proof steps, then submit them to Lean. Use Lean’s feedback to identify syntax, type or logical failures rather than assuming generated text is correct.
- Inspect dependencies and the final theorem. Make sure the proof relies on appropriate library results and that the checked conclusion is the one you wanted to establish.
This workflow keeps the strongest feature of formal proving—the independent check—while using AI for the parts it can help with. It also reveals the real cost of the task: a user may need to learn enough Lean to diagnose an incorrect statement or a failed proof attempt.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How should you judge a claim that an AI has proved a theorem?
Ask what input the system received and what output was actually checked. “Solved a theorem” can mean proving a ready-made formal statement, translating an informal problem and then proving it, or merely generating a natural-language solution; these are materially different achievements.
- Was there a formal proof? Look for a proof assistant and confirmation that its checker accepted the result, not only an explanation.
- Was the statement faithful? Determine whether a human supplied the formal statement or the AI had to translate the original problem.
- Which benchmark and version? A result on one miniF2F release should not be silently compared with another version or with a different test set.
- What search resources were used? Check whether the figure reflects one attempt or extensive sampling and search, when that information is reported.
- What does the result not show? Success on curated formal problems does not by itself demonstrate general autonomy, new mathematical discovery or coverage of untested domains.
These questions distinguish a genuine machine-checked result from an overstated claim without diminishing what formal verification does establish. If the checker accepts a proof term, the formal theorem follows within the stated system and dependencies. The separate question is whether that formal theorem is the one the original problem meant.
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AI can search for proof paths and may help people explore ideas, but the evidence here does not establish autonomous discovery of new mathematics in the broad research sense. The strongest cited results concern competition problems and curated benchmarks, while evaluation research still identifies informal interpretation and human insight as important obstacles.
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A system could contribute to discovery if it found a useful conjecture, definition or proof that mathematicians had not recognized. But such a contribution still needs verification and mathematical interpretation: researchers must establish that the formal result is meaningful, novel and relevant. A checked proof can certify a claim; it cannot, by itself, establish that the claim is an important new insight.
Frequently Asked Questions
Can ChatGPT prove math in Lean?
A language model can suggest Lean code or proof steps, but that output becomes a verified proof only when Lean accepts it for the intended theorem. The available evidence here does not establish a general success rate for any particular chat product or model.
Is Lean a proof checker or an automated prover?
It is both a programming language and a theorem-proving environment, but it is useful to separate its roles: Lean checks proof terms, while tactics, libraries and external AI systems can help find them. The checker’s acceptance is not the same thing as Lean independently discovering the proof.
What makes a formal proof useful to mathematicians?
A formal proof can provide a machine-checkable certificate and make the exact assumptions and conclusion explicit. Its practical value depends on whether the formalized statement captures the intended mathematics and whether the proof can be understood or reused by other researchers.
Can a Lean proof be wrong if Lean accepts it?
Acceptance means the proof term follows from Lean’s logic and its imported dependencies for the formal statement as written. It does not guarantee that the statement was translated correctly from an informal problem or that its assumptions match what a reader intended.
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