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AI can produce formal proofs that a proof assistant such as Lean accepts, but that does not mean it can reliably prove arbitrary new theorems. The main limits are finding a proof, translating the intended mathematical claim accurately into a formal language, and determining whether results on a particular benchmark generalize to other mathematics. A proof assistant checks the formal claim it is given; it cannot tell whether that claim captures what the mathematician meant.
What does it mean for AI to prove a theorem?
The phrase “AI proves a theorem” can refer to several different jobs. A system might generate a proof for a statement already written in a formal language; it might translate a natural-language problem into that language; or it might attempt the entire chain, from interpreting the question to producing a checked proof. These are related but distinct tasks, and a result for one should not be mistaken for success at all of them.
In a formal workflow, the theorem is expressed as a proposition in a system such as Lean, and the AI proposes a proof. The proof assistant checks whether the proof satisfies the rules of the formal environment and establishes that proposition. If it does, the result is stronger than a plausible explanation generated in ordinary prose: the formal proof has passed a mechanical check rather than relying on a reader to spot a hidden gap.
That assurance has a defined boundary. The checker establishes a relationship between a proof and a formal statement. It does not establish that the formal statement is the right translation of the original question, nor does acceptance by itself tell a reader whether the theorem is important, useful, or the theorem they intended to ask about.
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Where does the proof process break down?
A complete system has to handle a chain of tasks. It must interpret the problem, express it precisely in a formal language, find a proof, and submit that proof for checking. A failure at any stage can prevent a useful answer, even when another stage works well.
- Interpret the informal problem. Mathematical prose can rely on context, definitions, conventions, and implicit assumptions. The system has to identify what is being claimed and what objects and conditions are involved.
- Formalize the statement. The intended claim must be rendered as a precise proposition in a formal system such as Lean. An inaccurate translation can change the problem before proof search begins.
- Search for a proof. The system must construct a proof that follows from the formal statement and the available mathematical environment. A natural-sounding derivation is not enough if the checker rejects it.
- Verify the candidate. The proof assistant checks the formal proof against the proposition. It can reject an invalid proof term; it does not repair a mistaken proposition or decide whether that proposition matches the original problem.
This creates an important asymmetry: formal verification can be exact about the formal task while the end-to-end result is still wrong for the reader’s task. If an informal problem is mistranslated, an accepted proof can rigorously prove the wrong claim. The weak link may be the bridge between mathematical language and formal language, not the proof checker.
Why component scores can overstate end-to-end ability
The distinction is visible in the 2025 NeurIPS study miniF2F-Lean Revisited: Reviewing Limitations and Charting a Path Forward. On its original miniF2F evaluation, the authors report about 36% best accuracy for the complete problem-understanding, formalization, and proving pipeline. They also cite separate literature results of 97% for autoformalization and 69% for theorem proving. Those component results do not combine into a comparable end-to-end success rate: the full system has to get the statement right and then prove that statement.
The authors attribute much of the drop to discrepancies between formal and informal statements in more than half of the original problems. After correcting problems and introducing miniF2F-v2, they report 70% best full-pipeline accuracy. These results show why benchmark quality and statement fidelity matter, as well as why an individual component score cannot be read as a measure of the whole process. They are study-specific measurements, not estimates of how often AI can prove mathematics in general.
For a practical evaluation, ask what the reported system was actually given. If the benchmark supplies a formal proposition, the result measures proof generation under that condition. If the system must begin from an informal question, evaluation also includes understanding and formalization. A number from the first setup does not answer how well the system performs in the second.
What benchmark scores do—and do not—show
miniF2F is a benchmark of formal mathematics problems translated across formal systems. The repository describes its problems as drawing from olympiad sources such as AMC, AIME, and IMO, as well as high-school and undergraduate mathematics. That makes it useful for comparing automated theorem-proving systems on the problems and formal systems included in the benchmark. It is not a sample of every area of mathematics, and performance on it alone cannot establish an ability to prove arbitrary research theorems.
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Published percentages also depend on how a system is evaluated. Pass@32 means the system tries multiple candidate proofs for a problem and receives credit if at least one succeeds within 32 attempts. It is not a claim that the system succeeds on every problem in a single attempt. Self-correction and test-time exploration can likewise affect how many candidates are generated and how a result is obtained.
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1Clear out junk files and repair common Windows errors2Scan for outdated or missing drivers - takes under a minute3Repair Windows errors before they cause bigger problemsThe following reported results provide context, but should not be treated as a clean time series or ranked as if they came from one identical test. The benchmark version or split, model, method, sampling budget, and success metric all matter.
| Study and system | Reported result | What the figure describes |
|---|---|---|
| Goedel-Prover-V2, ICLR 2026 | 88.1% on MiniF2F at pass@32 in standard mode; 90.4% in self-correction mode | Benchmark result with multiple candidate attempts; the latter uses self-correction. |
| Goedel-Prover-V2-8B, ICLR 2026 | 84.6% on MiniF2F at pass@32 | Reported result for the 8B system under the stated multi-attempt metric. |
| Goedel-Prover-SFT, arXiv preprint, 2025 | 57.6% on miniF2F at Pass@32 | Reported result under a 32-attempt pass metric. |
| miniF2F-Lean Revisited, NeurIPS 2025 | About 36% on the original miniF2F evaluation; 70% on corrected miniF2F-v2 | Best reported full-pipeline accuracy in that study, including understanding, formalization, and proving. |
The entries do not provide a controlled comparison across a single shared protocol. In particular, pass@32 and full-pipeline accuracy describe different evaluation setups. Read each percentage together with its benchmark version and task scope, rather than as a universal capability score.
Does a proof assistant guarantee that the theorem is right?
A proof assistant offers a rigorous check of the formal proof relative to the formal proposition and environment. This makes it useful for rejecting invalid proof attempts and for distinguishing accepted proofs from persuasive-looking but unchecked text. It does not independently verify the correspondence between the proposition and an informal theorem stated elsewhere.
There are therefore two separate questions to ask about an accepted result:
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- Is the formal proof valid for the formal proposition? The proof assistant addresses this by checking the candidate in its formal environment.
- Does the formal proposition express the intended mathematical claim? That requires attention to the definitions, assumptions, and translation from the original problem.
A “yes” to the first question does not automatically answer the second. In practical terms, a proof checked by Lean is evidence about the statement encoded in Lean. A reader who cares about an informal theorem must also establish that the encoding faithfully represents it.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How to judge a claim that AI has proved a theorem
When evaluating a headline, paper, or demonstration, separate the achievement into a few concrete checks. This avoids treating all uses of “theorem proving” as equivalent.
- Identify the input. Was the system given a formal statement, an informal problem, or both? A proof-generation result that starts with a formal proposition does not measure the full translation pipeline.
- Check the benchmark and version. Original miniF2F and corrected miniF2F-v2 are not interchangeable. Ask which dataset, split, and formal system produced the reported result.
- Inspect statement fidelity. Determine whether the evaluation checks that a formal proposition matches the intended informal problem, rather than only whether the formal proposition is provable.
- Read the success metric and compute budget. One attempt, pass@32, self-correction, and test-time exploration represent different conditions. A multi-attempt success rate should not be described as single-shot reliability.
- Confirm what verification means. Ask whether the final formal proof was accepted by a specified proof assistant, and which formal statement it proved.
- Match the conclusion to the domain. Olympiad, school, and undergraduate problems are valuable test cases, but they do not by themselves establish broad performance across research mathematics.
What remains unproven about AI mathematicians
The studies establish meaningful capability on defined formal tasks: AI systems can produce checked proofs for a growing subset of benchmark problems, and some approaches use proof-assistant feedback, self-correction, or test-time exploration. They also show that benchmark construction and mismatches between informal and formal statements can materially affect measured full-pipeline performance.
They do not establish a universal threshold beyond which AI cannot prove a theorem, a complete inventory of theorem types that remain out of reach, or general competence at proving arbitrary new research mathematics. The available benchmark evidence is bounded by the problem sets and protocols used. Nor does a high result on one dataset alone answer how reliably a system will handle a theorem from another area, an unfamiliar formulation, or a problem that requires formalization as well as proof search.
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The defensible conclusion is narrower and more useful: AI is a capable assistant and solver for some specified formal theorem-proving tasks, and a proof accepted by a proof assistant has been checked against its formal claim. Whether that amounts to proving the theorem a person intended depends on statement fidelity; whether it generalizes depends on evidence beyond the particular benchmark and evaluation setup.
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