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Yes—but only in a precise sense. AI systems can generate formal proofs that a proof assistant such as Lean checks, including difficult competition problems. They cannot yet be treated as universally reliable mathematicians: the theorem must first be translated correctly into a formal statement, the search may require substantial computation, and a checked proof guarantees the formal statement—not necessarily that it captured the original question’s intended meaning.
What does it mean for AI to prove a theorem?
A mathematical proof has two separate stages that are easy to conflate. First, a system must find an argument. Second, someone or something must verify every inference. A language model can produce fluent mathematical prose that looks convincing while silently assuming a missing condition, misusing a theorem, or making an invalid algebraic step. Fluency is not verification.
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Automated theorem proving addresses the verification problem with a formal proof assistant. In Lean, a proposition is written in a precise formal language, and the system produces a proof object. Lean’s kernel checks that object against its rules and imported definitions. If the checker accepts it, the formal proposition follows within that trusted framework. The check does not certify that an informal problem was translated correctly, that the definitions match a textbook’s conventions, or that an explanatory paragraph accurately describes the machine-generated proof.
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Repair common Windows errors and clear accumulated junk for a smoother, more stable PC - no reinstall needed.Free scan · no reinstallA 2024 Association for Computational Linguistics workshop paper by Vanessa Lama, Catherine Ma and Tirthankar Ghosal summarized the central constraint: “Theorem proving presents a significant challenge for large language models (LLMs) due to the requirement for formal proofs to be rigorously checked by proof assistants, such as Lean, eliminating any margin for error or hallucination.” The statement concerns the proof object after formal checking; it should not be read as saying that formalization or the entire system is error-free.
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How AI theorem proving works
1. Formalize the question
An informal sentence such as “there are infinitely many primes” must become a precisely typed proposition. Variables, domains, quantifiers, hypotheses and definitions all need explicit treatment. This translation is itself mathematical work. In the 2024 International Mathematical Olympiad (IMO) evaluation, experts formalized the non-geometry problems immediately after release. The experiment therefore measured solving of formally stated problems, not an end-to-end system’s ability to read the natural-language contest paper under competition conditions.
2. Search for a proof
The AI proposes tactics, lemmas or complete proof terms. Some systems use language-model guidance, reinforcement learning, symbolic search, or combinations of these methods. Search can branch dramatically: a promising-looking tactic may fail deep inside a proof, while a short proof may depend on a lemma that is difficult to discover automatically.
3. Let the proof assistant check every step
Lean or another verifier rejects an object that does not type-check. This makes the accepted formal proof much stronger evidence than an unchecked natural-language answer. It still depends on the axioms, definitions and library imported by the formal development.
4. Explain the result to people
A readable explanation is a separate output. A system may have a valid proof term but produce an incomplete or misleading prose summary. Conversely, a polished explanation can conceal a gap if it was never connected to a checked proof.
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What happened at the 2024 IMO?
The strongest public demonstration in the cited work is the combined AlphaProof and AlphaGeometry 2 system at the 2024 IMO. AlphaProof formally solved algebra and number-theory problems P1, P2 and P6; P6 was the contest’s most difficult problem. AlphaGeometry 2 solved the geometry problem, P4. Together they earned 28 of 42 points, a score within the official human silver-medal range and one point below the gold threshold.
This was described as the first AI system to reach medal-level performance at the IMO, but “the system” matters: the result combines two different approaches, and the score is a comparison with human medal thresholds rather than an official contestant medal awarded to AlphaProof.
- Human formalization: experts translated the five non-geometry problems into Lean after the statements were released.
- Computation: solving used multiple days, unlike the tightly bounded human contest schedule.
- Division of labor: AlphaProof handled domains represented in Lean’s mathematical library; AlphaGeometry 2 supplied a specialized geometry capability.
- Library coverage: gaps in the relevant Lean geometry library made it impractical to express some Olympiad-style geometry problems for AlphaProof alone.
AlphaProof’s main reinforcement-learning phase used approximately 80,000 TPU-days. That training figure is separate from the multi-day computation used to solve the IMO problems and should not be presented as a contestant’s time or hardware budget.
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| System | Primary domain | Verification or method | Human and technical limits |
|---|---|---|---|
| AlphaProof | Formal algebra, number theory and other propositions supported by Lean’s library | Generates proof objects checked by Lean’s kernel | Needed expert formalization for the IMO evaluation; library coverage affects which problems can be stated conveniently |
| AlphaGeometry 2 | Olympiad geometry | Specialized geometry reasoning, used for IMO problem P4 in the combined result | Designed for a narrower problem family rather than arbitrary formal mathematics |
The practical comparison is not simply “which AI is smarter?” A meaningful evaluation asks which mathematical domains a system supports, whether every result is formally checked, how much human formalization is required, what library it can use, and how much time or search budget it receives.
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What do theorem-proving benchmarks actually measure?
Benchmark percentages are meaningful only with the dataset, split, attempt budget and verifier attached. The AlphaProof work reports results on corrected miniF2F, an expert-formalized collection of historical non-geometry IMO problems, and Putnam problems. Its formal-imo benchmark contains 258 problems: 107 algebra, 77 combinatorics and 74 number theory. The study also describes a held-out Putnam test split and measures intended to keep training and evaluation data separate.
| Reported result | What it measures | Qualification |
|---|---|---|
| 28/42 points | Combined AlphaProof and AlphaGeometry 2 performance at IMO 2024 | Silver-medal score range; expert formalization and multi-day computation; not an official contestant medal |
| 258 problems | Size of the formal-imo historical non-geometry IMO set | 107 algebra, 77 combinatorics and 74 number theory |
| 65.0% on miniF2F-test | Self-play Theorem Prover (STP) whole-proof generation | Reported in a 2025 ICML paper using Lean and Isabelle verifiers |
| 23.9% on ProofNet-test | STP performance on the ProofNet test split | Same study and evaluation setup |
| 8 of 644 on PutnamBench | STP problems solved at pass@3200 | Up to 3,200 attempts per problem; not comparable to a single-attempt rate |
The 2025 STP paper also identifies two persistent obstacles: high-quality correct proofs are scarce, and theorem proving has sparse rewards because most attempted proof paths fail before completion. Its results should be read as measurements of that paper’s setup, not as a permanent ranking of every theorem-proving system.
Can ChatGPT prove a mathematical theorem?
A general-purpose chatbot can suggest a proof, translate parts of an argument into Lean, or help debug a failed tactic. Without a formal checker, however, its answer is a conjectured argument, not a certified proof. Even when it writes Lean code, acceptance depends on the exact imports, Lean version, definitions and theorem statement.
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A safer workflow is:
- State the proposition precisely. Specify types, assumptions and the intended conclusion before asking for tactics.
- Choose a proof assistant and version. Keep the project reproducible; a tactic that works with one library revision may fail with another.
- Ask the model for a small step. Request a lemma or tactic sequence rather than trusting an entire long proof.
- Run the code locally. Treat the assistant’s output as untrusted until Lean accepts it without warnings or unsolved goals.
- Review the formalization. Check that the proposition says what the original mathematics meant, including edge cases and hidden assumptions.
- Explain and document the proof. Separate the machine-checked term from any informal narrative added for readers.
Are AI-generated proofs reliable?
For the exact formal statement accepted by a soundly configured checker, verification provides a strong guarantee: an invalid proof term should be rejected. Reliability is weaker at the surrounding layers.
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- Translation risk: an incorrectly formalized problem can be proved perfectly while answering the wrong question.
- Definition risk: overloaded words such as “continuous,” “finite,” or “prime” need explicit formal definitions.
- Library risk: a proof may rely on imported axioms or results whose scope the reader has not inspected.
- Search risk: failure to find a proof is not evidence that no proof exists; success may require a large attempt budget.
- Communication risk: generated prose can omit cases or misdescribe a valid proof.
- Reproducibility risk: results can change with model version, random seed, library revision, hardware and time limit.
The ACL study notes that models struggle with novel, complex theorems where human insight is essential. The cited evidence therefore supports AI assistance and verified formal results, not autonomous discovery and validation of arbitrary new research mathematics.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What is an automated theorem prover?
An automated theorem prover is software that searches for formal derivations and submits them to a logical verifier. Some systems are general-purpose and operate inside assistants such as Lean; others specialize in domains such as Olympiad geometry. Interactive assistants let humans guide the search with definitions, lemmas and tactics while the kernel checks the final object. This combination is often more useful in practice than asking a model to emit an unchecked essay-length proof.
To compare two provers fairly, record the formal language and library, benchmark name and split, number of attempts, time and hardware limits, whether humans supplied intermediate lemmas, and whether every reported solution was independently checked. A headline percentage without those details can give a false impression of capability.
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Lean’s official documentation includes Theorem Proving in Lean 4, an online, book-length introduction to dependent type theory, tactics and interactive proof development. The documentation version identified in the cited material assumes Lean 4.33.0. Readers should verify the version installed in their own project because syntax and library APIs can change.
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Begin with elementary propositions, inspect the goals after each tactic, and deliberately test false statements to see the checker reject them. Then formalize a familiar textbook proof and compare the formal statement with the original wording. This practice exposes the most important lesson of automated mathematics: checking a proof is powerful, but deciding exactly what should be proved remains a human responsibility.
What AI theorem proving can—and cannot—claim today
AI theorem proving is demonstrably real. Systems have generated formally checked proofs of difficult contest problems, and specialized and general-purpose approaches can complement one another. The 2024 IMO result is substantial evidence of progress, not evidence that a chatbot can solve every theorem on demand.
Current capabilities remain bounded by the mathematical domain represented in a formal library, the quality of the translation, available computation, search-attempt limits and the evaluation protocol. No cited result establishes generally reliable autonomous research-level mathematics across arbitrary informal questions. The most defensible use today is a human–AI workflow in which people formulate and inspect the mathematics while a proof assistant supplies an exact, mechanically checked verdict on the formal statement.
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