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One free scan finds every outdated or missing driver and matches the right update for your exact hardware.Free scan · exact hardware matchAI can now generate formal proofs that a proof assistant such as Lean checks, but that achievement is narrower than “AI can prove mathematics.” The hard limits are finding a proof, translating an informal theorem into a faithful formal statement, and knowing whether benchmark scores transfer to unfamiliar research problems. A checker certifies a proof of the statement it receives in the stated formal environment; it does not certify that the statement captures what a mathematician meant.
What AI theorem proving actually does
An AI theorem-proving system usually operates inside a pipeline rather than taking a sentence and producing a universally trustworthy mathematical result. The pipeline has at least four stages:
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- Interpretation: the system reads an informal problem and identifies its objects, assumptions and desired conclusion.
- Formalization: it expresses that problem in a formal language, often Lean, with precise types, definitions and hypotheses.
- Proof search: a model proposes tactics or a complete proof term.
- Verification: the proof assistant checks the term against its kernel, libraries and the exact formal proposition.
Failure at any stage can invalidate the end result. A fluent explanation, an algebraic derivation that looks persuasive, or a proof sketch is not equivalent to a formal proof accepted by Lean. Large language models can hallucinate a lemma, omit a condition or make an invalid inference; formal checking is designed to reject those errors rather than overlook them.
Automated theorem proving is therefore best understood as constrained generation followed by independent checking. The proof assistant can reject an invalid proof term, but it cannot repair a mistranslated theorem. If an informal statement says that a sequence has a property under one set of assumptions and the formal version accidentally adds a stronger assumption, proves a different quantifier order or changes a domain, a valid proof may solve the wrong problem.
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The biggest bottleneck is often the statement, not the proof
Formal proof generation and informal-to-formal translation are separate tasks. A system may be very good at proving propositions that are already written correctly in Lean while still being unreliable at deciding what an informal theorem means. The complete pipeline can consequently perform much worse than scores reported for either component in isolation.
The NeurIPS 2025 study miniF2F-Lean Revisited: Reviewing Limitations and Charting a Path Forward illustrates the gap. On its original miniF2F evaluation, the best end-to-end accuracy was about 36%. The same paper cites separate literature results of roughly 97% for autoformalization and 69% for theorem proving. Those figures are not contradictory: they measure different tasks and do not imply that a system successfully translates and proves 97% of informal problems.
The authors found discrepancies between formal and informal statements in more than half of the original problems. After correcting the problems and publishing miniF2F-v2, the reported best full-pipeline accuracy rose to 70%. That improvement is evidence that dataset quality and statement fidelity materially affect measured capability; it is not evidence that the underlying model suddenly acquired a general ability to prove arbitrary mathematics.
What a checker can and cannot certify
- It can establish that a submitted proof term follows from the exact formal hypotheses and definitions available in the specified environment.
- It can catch many syntactic, type-theoretic and logical errors that an informal reader might miss.
- It cannot determine whether the formal theorem faithfully represents the informal question.
- It cannot show that the selected definitions, axioms or imported libraries match the mathematical context a reader intended.
- It cannot turn a proof of a stronger, weaker or simply different proposition into a proof of the original theorem.
This distinction is the central limit of AI-assisted formal mathematics: verification is extremely strong relative to a formal claim, while the choice of that claim remains a human- and model-dependent activity.
Why benchmark percentages are easy to misread
Reported scores are conditional measurements. Before comparing two numbers, identify the benchmark version, split, formal system, model, sampling budget, correction method and success metric. A percentage without those details does not describe a model’s general mathematical ability.
| System or study | Reported result | How to interpret it |
|---|---|---|
| miniF2F-Lean Revisited (NeurIPS 2025) | About 36% on the original miniF2F full pipeline | End-to-end understanding, formalization and proving on the study’s original evaluation |
| miniF2F-Lean Revisited (NeurIPS 2025) | 70% best accuracy on corrected miniF2F-v2 | Full-pipeline result on the revised statements; not a universal mathematics rate |
| Goedel-Prover-V2 (ICLR 2026) | 88.1% on MiniF2F at pass@32 in standard mode | At least one successful candidate among up to 32 generated attempts, under the paper’s protocol |
| Goedel-Prover-V2 (ICLR 2026) | 90.4% on MiniF2F at pass@32 with self-correction | Same benchmark-style metric with the reported self-correction procedure |
| Goedel-Prover-V2-8B (ICLR 2026) | 84.6% on MiniF2F at pass@32 | Pass@32 result for the 8-billion-parameter variant under that evaluation |
| Goedel-Prover-SFT (arXiv preprint, 2025) | 57.6% on miniF2F at Pass@32 | Preprint result using the stated model and sampling setup |
These values should not be arranged as a simple improvement curve. The studies can differ in benchmark version, problem split, model, inference compute, proof-search strategy and whether self-correction is enabled. Even the capitalization of “MiniF2F” versus “miniF2F” can conceal different reporting conventions; the protocol matters more than the label.
Pass@32 is not one-shot accuracy
In pass@32 evaluation, the system generates multiple candidates for each problem. The problem counts as solved if at least one of those candidates passes within the stated 32 attempts. A score of 88.1% therefore does not mean the model succeeds 88.1% of the time on its first answer, nor that it can produce a reliable proof on demand without the associated sampling and verification budget.
Self-correction changes the procedure again: the model can use proof-assistant feedback or its own critique to revise candidates. That can be useful in practice, but it makes the compute budget and interaction loop part of the result. A comparison between one-shot and self-correcting systems is not an apples-to-apples comparison unless those resources are reported.
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miniF2F is useful, but it is not all of mathematics
The miniF2F repository describes a benchmark of formalized mathematics drawn from olympiad problems, including AMC, AIME and IMO material, together with high-school and undergraduate mathematics. It is intended to compare automated theorem-proving systems in the formal systems targeted by the benchmark.
That scope makes miniF2F valuable for reproducible testing: the propositions are defined, the proof assistant can check submissions and different systems can be evaluated on a common collection. It also sets a boundary on what a high score means. Olympiad-style and undergraduate exercises are not a representative sample of arbitrary research mathematics. They typically have a known statement, a manageable formal vocabulary and a benchmark-compatible proof target.
A system can perform strongly on miniF2F while lacking the abilities needed for open research: selecting a worthwhile conjecture, discovering useful definitions, locating relevant literature, deciding which abstraction will scale, inventing a new proof strategy or coordinating a proof that depends on large, evolving libraries. The cited benchmark work does not establish broad performance across arbitrary research domains.
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Generalization remains unproven
Current evidence supports a carefully bounded claim: AI systems have become capable assistants and solvers on defined formal tasks, and the result is independently checked when the proof assistant accepts the proof. It does not support the stronger claim that AI has general mathematical competence or can prove any theorem a mathematician can state.
The reviewed studies do not provide a universal threshold beyond which AI cannot prove a theorem, nor a complete list of theorem classes that remain out of reach. Difficulty depends on the formal language, available libraries, theorem length, abstraction level, amount of search, interaction with the checker and whether the statement was already formalized. A benchmark result can show what happened under one protocol; it cannot establish a boundary for every future system or mathematical field.
Formal problem-solving research also continues to report persistent hallucination or erroneous logic in language-model mathematical reasoning outside the narrow setting where every final proof term is checked. This is why a natural-language answer should be treated as a proposal until its formal artifact passes verification.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How to evaluate an AI-produced theorem responsibly
If you are using an AI system for a real proof, separate the claim you want to trust from the convenience of the generated explanation.
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- Write the intended theorem independently. State domains, quantifiers, side conditions and definitions in ordinary mathematical language before asking for formalization.
- Inspect the formal statement. Check every hypothesis, type, coercion, imported definition and quantifier order. Ask whether it is equivalent to the intended theorem, not merely whether it compiles.
- Record the environment. Note the Lean version, library revision, imported modules and any axioms or admitted declarations. A proof is relative to that environment.
- Require kernel-checked acceptance. Treat a proof assistant’s accepted proof term as the verification event. Do not count a model’s prose, a tactic trace that was not run or an unverified proof sketch as success.
- Test on held-out problems. Use problems outside the prompts and examples used to guide the system. Report whether evaluation is one-shot, pass@k or self-correcting, and state the sampling budget.
- Have a mathematician review meaning. Human review is still needed for statement fidelity, relevance of definitions and whether the formal result answers the original question.
This process does not make AI infallible. It makes the remaining uncertainty visible: the checker addresses proof validity relative to the formal claim, while review addresses whether that claim is the one you meant.
What AI is realistically good for today
Within a defined formal setting, AI can propose tactics, fill routine proof steps, search for lemmas, translate some textbook-style statements and use compiler feedback to repair failed attempts. Scaffolded data synthesis, proof-assistant feedback and test-time exploration are active design strategies in recent systems. These capabilities can reduce the mechanical cost of formalization and help a human explore alternatives.
The strongest practical role is therefore an assistant whose output is checked, inspected and iterated—not an oracle whose explanation is accepted because it sounds mathematical. For routine or benchmark-shaped propositions, that distinction may save substantial time. For a new research theorem, the central risks move upstream to choosing the right statement, finding a genuinely new idea and judging whether the formalization captures the mathematics.
Bottom line on the limits
AI theorem proving has crossed an important threshold: systems can generate proofs that a proof assistant independently verifies. The limits are equally important. Proof search is not the same as mathematical understanding; formalization can silently change the problem; pass@32 measures multi-attempt search rather than one-shot reliability; and miniF2F results describe a bounded collection of olympiad, school and undergraduate problems, not arbitrary research mathematics. Use AI as a checked formal assistant, and treat both the theorem statement and the benchmark protocol as objects that require human scrutiny.
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