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Use AI as a proof-development partner, not as a proof referee. A language model can restate a theorem, explain a difficult implication, expose a missing hypothesis, suggest a counterexample, or propose a cleaner route. Its polished prose is not evidence that the argument is valid. For stronger assurance, formalize the theorem and derivation in a proof assistant such as Lean, run the checker, and then verify that the formal statement really matches the mathematics you intended.
Can AI actually check whether a proof is correct?
That depends on what “check” means. A general-purpose AI system reads natural-language mathematics and predicts a plausible response. It can find obvious algebraic mistakes, point out undefined terms, test simple edge cases, and suggest missing steps. It can also confidently accept an invalid argument or invent a justification that sounds rigorous. The Leiden Declaration on Artificial Intelligence and Mathematics puts the danger plainly: “Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs.”
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A proof assistant performs a different job. Lean and similar systems check a formal proof object against a formally stated theorem, using a trusted kernel and the definitions and axioms available in the environment. A successful check is stronger evidence about that encoded derivation than a convincing paragraph from a language model. It is not, however, an automatic endorsement of an informal proof. If a definition, hypothesis, quantifier, or conclusion was translated incorrectly, the checker may certify a different theorem from the one you meant.
| Approach | What the tool checks | What you must still check |
|---|---|---|
| AI critique of an informal proof | Plausibility, explanations, possible gaps, alternative arguments and examples | Every mathematical inference, condition, domain, quantifier and calculation |
| AI-assisted formalization | Suggested definitions, tactics, lemmas or proof-search steps | The generated statement, imports, assumptions and whether the formal model matches the intended problem |
| Proof assistant run | The formal derivation accepted by the checker for the encoded statement | That the encoding, axioms and definitions faithfully represent the informal theorem |
| Independent expert review | The argument’s mathematical meaning, novelty and hidden assumptions | Usually still needed for research-level or consequential claims |
Prepare the proof before asking for help
AI feedback is only as precise as the problem description. Put the complete claim and its context in one place instead of pasting isolated lines.
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State the theorem exactly
Write the conclusion with all quantifiers, domains and dependencies. Distinguish “for every” from “there exists,” and say whether variables are real numbers, integers, vectors, functions or elements of a particular structure. Include definitions that are not standard or that have a specialized meaning in your field.
List every assumption
Include positivity, nonzero denominators, continuity, differentiability, boundedness, independence, finiteness, ordering and boundary conditions. An AI may silently supply a familiar assumption that your proof never states. Making the list explicit lets you ask where each condition is used.
Show the whole argument and mark uncertainty
Number the proof’s steps or separate its lemmas. If one implication feels weak, label it as a question rather than disguising it as a fact. Tell the tool which steps are established results and which are your own claims. This gives you a map for targeted review.
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1. Ask for a restatement before asking for a verdict
Start with a prompt such as: Restate the theorem, all assumptions, the domain of each variable, and the conclusion in your own words. Do not judge the proof yet. Compare the response with your statement. If the model drops a quantifier, changes an inequality, treats a partial function as total, or turns a conditional claim into an unconditional one, correct the prompt before proceeding. This catches a specification mismatch at the cheapest point in the process.
2. Request one local check at a time
Do not begin with “Is this proof correct?” Ask about a single transition:
- “Derive line 4 from lines 2 and 3, naming the theorem or algebraic identity used.”
- “Where exactly is the assumption that x is nonzero used?”
- “Does this lemma remain true when the variable is zero, negative, infinite, or at an endpoint?”
- “List the quantifiers in this step and check that their order has not changed.”
- “Find a concrete counterexample to the intermediate claim if it is false.”
Require an explicit derivation. A bare label such as “by standard results” is a prompt to inspect the step yourself, not a certificate.
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3. Make the model attack the proof
Ask for adversarial checks rather than reassurance: Try to falsify the lemma. Search small integer cases, boundary cases and degenerate structures. If you cannot find a counterexample, state what you actually checked. A failure to find an example does not prove the lemma, but a valid counterexample identifies a real defect quickly. Ask separately whether the counterexample satisfies every original assumption; AI-generated objections can also be invalid.
4. Verify every criticism independently
Recompute algebra on paper or in a trusted calculator, expand definitions, and test the disputed case yourself. Check that a cited theorem has the required hypotheses and that its conclusion has the right direction and domain. When the model says a step is invalid, determine whether it found a genuine gap, misunderstood notation, or made a different mistake.
5. Separate validity from exposition
After the logic is stable, ask for a clearer outline, notation audit or reader-friendly explanation. Compare any rewrite line by line with the original proof. A shorter proof may omit a condition; a polished paragraph may merge two non-equivalent cases. Treat style assistance as a second task, never as evidence that the mathematics has been repaired.
6. Escalate to formalization when the risk justifies it
For a theorem that matters, encode its definitions and statement in Lean or another proof assistant. You may ask AI for a first translation, tactic suggestions, proof search or likely library lemmas. Inspect the generated code before running it. Then run the actual checker; do not report a proof as verified merely because the model predicted that the code should compile.
7. Close the translation loop and obtain independent review
Read the formal statement beside the informal one. Confirm that every intended assumption appears, no unintended axiom was imported, and no definition was simplified into a different concept. For research-level work, publication claims, safety-critical mathematics or results used in software, ask a qualified mathematician to review the argument and the formalization. Keep a record of the assumptions, AI assistance, checker output and unresolved questions.
Prompt patterns that produce useful mathematical feedback
Focused prompts make the model’s role auditable. Supply notation and context once, then ask for one operation.
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Restatement and scope
Restate this theorem with all quantifiers, domains and assumptions. Identify any ambiguity in the notation. Do not add assumptions without labeling them.Line-by-line implication
For the transition from line 5 to line 6, list each equality or implication separately. Name the rule used and state the hypotheses needed for that rule. If a hypothesis is missing, stop rather than repair it silently.Hypothesis-use audit
Make a table with one row for each assumption and the exact proof step where it is used. If an assumption is unused, say so. If a step needs an assumption not listed, identify it.Counterexample search
Attempt to disprove the lemma by checking boundary, zero, negative, repeated, infinite and minimal cases where they are meaningful. Give a fully verified counterexample or explain why the proposed cases do not satisfy the assumptions.Alternative route
Suggest two independent proof strategies. For each, state the key lemma and any additional assumptions. Do not claim either strategy is complete until every step is justified.Exposition review
Rewrite this proof for a reader who knows the definitions but not the trick. Preserve every hypothesis and conclusion. Mark any sentence that changes the logical strength of the original.These prompts are not a tested protocol or a guarantee of correctness. They are ways to force the tool to expose assumptions and intermediate reasoning instead of producing a yes-or-no impression.
Manual checks that catch the most common gaps
Algebra and cancellation
Whenever a proof divides, cancels, takes a square root, applies a logarithm or takes an inverse, check the domain and exceptional values. From ab = ac, cancellation of a requires a ≠ 0 in the usual number systems. Squaring an equation can introduce solutions; taking a square root can require non-negativity. Ask AI to list the operation’s preconditions, then verify them yourself.
Quantifiers and dependencies
“For every x there exists a y” is generally weaker than “there exists a y for every x.” Check whether a quantity chosen in one step is allowed to depend on a variable introduced later. Have the model rewrite the proof with explicit quantifiers, then inspect the order manually.
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Induction
Verify the base cases, the induction hypothesis and the exact range in which the step is applied. A step proved for n ≥ 1 cannot establish a claim whose induction starts at zero unless the gap is handled. Ask for a check of the first few values, but do not confuse examples with the induction step.
Limits, continuity and probability
Check that limit arguments use the correct mode of convergence and that an interchange of limit, sum, integral or expectation has a stated justification. In probability, distinguish almost-sure, in-probability and expectation claims. AI often recognizes familiar patterns while overlooking the condition that permits an interchange.
Definitions and hidden cases
Expand specialized definitions at the point of use. Terms such as “connected,” “measurable,” “invertible,” “optimal” or “generic” have technical meanings that can differ by field. Ask the model to produce a case where each definition is true and a nearby case where it fails; then check that the proof handles both.
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Using Lean for a stronger machine check
What Lean verifies
Lean is a programming language and proof assistant that lets mathematical proofs be checked by a computer. You express a theorem in Lean’s language, construct a proof term using definitions, lemmas and tactics, and run the checker. The result concerns the formal statement and derivation that Lean accepted, not an unencoded page of natural-language prose.
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An AI copilot can suggest theorem names, tactics, premises and proof-search steps. Research such as Towards Large Language Models as Copilots for Theorem Proving in Lean describes these assistive functions, including tactic suggestion, proof search and premise selection. Such systems support a human theorem-proving workflow; they do not guarantee that the intended theorem has been formalized or proved.
A practical formalization sequence
- Choose the exact statement. Decide the types of all variables, the structures they belong to and the assumptions that are genuinely needed.
- Encode definitions before the theorem. Avoid replacing a domain-specific definition with an easier informal synonym unless you can prove the equivalence.
- Write the theorem with explicit hypotheses. Make ranges, inequalities, nonzero conditions and finiteness visible in the formal statement.
- Ask AI for small translations. Request one definition, lemma or tactic at a time and inspect the generated code and imports.
- Run Lean’s checker. Treat a successful run as confirmation of the encoded derivation. A failed run is diagnostic information, not evidence that the informal theorem is false.
- Review dependencies. Check imported axioms, coercions, typeclass assumptions and simplifications that might alter the intended claim.
- Compare both versions. Read the formal theorem and the informal theorem side by side, including edge cases and the conclusion’s exact strength.
Why a successful check is not the end
The Leiden Declaration warns about the translation between computer-encoded and human presentations. LeanTutor, a 2025 proof-of-concept system, likewise documents limitations in autoformalization: an incorrect translation can produce misleading feedback or no response. Therefore, the honest report is “Lean checked this formal statement and proof under these imports and assumptions.” It is not “Lean proved my original paragraph” unless you have independently established that the paragraph and formal statement are equivalent.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Choosing between chat critique, formalization and expert review
| Question | Chat-based critique | Lean or another proof assistant | Expert review |
|---|---|---|---|
| Best use | Find local gaps, explain steps, generate examples and improve exposition | Check a formal derivation against a formal statement | Assess meaning, assumptions, novelty and research significance |
| Output | Natural-language critique or suggested argument | Machine-checked proof term or an error | Mathematical judgment and contextual feedback |
| Human responsibility | Verify every inference and every AI objection | Verify the derivation and the translation | Evaluate the whole claim and its place in the field |
| Setup | Paste an informal proof and its context | Express theorem and proof in the assistant’s language and supported library | Provide a complete, readable argument and relevant background |
| Independence | A second model response is another suggestion, not an independent proof | The checker is independent of the model’s prose once code is run | Independent review is especially valuable for consequential claims |
| Time or cost figures | Not established generally | Not established generally | Not established generally |
The sources do not quantify the time or effort required for formalization. In practice, the choice is qualitative: use chat for rapid local diagnosis, formalize the parts where a machine-checked derivation is worth the setup, and seek human review when interpretation or stakes exceed what either tool can establish.
What recent AI mathematics results do—and do not—show
OpenAI’s October 6, 2026 account of mathematical progress describes AI-generated work accompanied by Lean formalizations and reports roughly three hours of ChatGPT Pro thinking-equivalent compute per average result in that release. That figure applies to the specific release and its average result; it is not a general estimate for checking an everyday proof.
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In its 2026 First Proof account, OpenAI says it ran its internal model on all 10 problems. After expert feedback, it assessed at least five of the ten attempts as having a high chance of correctness. OpenAI also says its initial view of problem 2 changed: an attempt first believed likely correct was later judged incorrect after official commentary and community analysis. The account emphasizes, “Unlike short-answer or competition-style math, these problems require building end-to-end arguments in specialized domains, and correctness is hard to establish without expert review.” These are task-specific institutional assessments, not a benchmark for ordinary prompts or a general accuracy rate.
Failure modes and recovery steps
The model says “looks correct”
Replace the verdict request with a local derivation, hypothesis-use table and counterexample search. A confidence statement contains no additional proof.
The model invents a theorem or citation
Ask for the theorem’s exact statement and assumptions. Verify it in a trusted textbook, paper or formal library. If the source cannot be checked, treat the name as a hint, not authority.
The critique identifies a gap but cannot explain it
Ask for the smallest concrete example that breaks the disputed step, then test whether that example satisfies the original assumptions. If it does not, discard the objection and continue reviewing.
Two AI systems disagree
Do not decide by majority vote. Expand the disputed definitions, recompute the step and, where practical, formalize it. Model-to-model back-and-forth can reveal questions, but it is not the same as independent formal checking or expert review.
Lean rejects an apparently obvious proof
Inspect types, coercions, missing imports, implicit assumptions and the exact theorem statement. A rejection may expose a genuine gap, or simply a mismatch between your informal notation and the formal definitions. Fix the statement only after deciding which version represents the intended mathematics.
Lean accepts a suspiciously easy proof
Review imported axioms, definitions, simplification lemmas and typeclass instances. Confirm that the theorem is not vacuous, that the variables have the intended types and that no assumption was accidentally strengthened until the result became trivial.
A repeatable checklist
- Have I stated the exact conclusion, domains and quantifiers?
- Are all assumptions explicit, including nonzero, positivity, finiteness and regularity conditions?
- Did I ask the AI to restate the claim before critiquing it?
- Did I request local derivations instead of a global confidence judgment?
- Did I test boundary and degenerate cases and verify any counterexample myself?
- Did I check every cited theorem and its hypotheses?
- Did I keep exposition editing separate from validity checking?
- If I used Lean, did I inspect the formal statement, imports, axioms and generated definitions?
- Can I explain precisely what the checker verified and what remains a semantic judgment?
- For a consequential or research-level result, has an independent mathematician reviewed it?
The practical bottom line
Ask AI to make a proof easier to inspect, not to make uncertainty disappear. Use focused prompts to expose assumptions, derive individual steps and search for counterexamples; independently verify every suggestion. When the claim warrants stronger assurance, formalize it in Lean, run the checker and audit the translation from informal mathematics. Only that combination—targeted AI assistance, human mathematical judgment and, where appropriate, a machine-checked formal derivation—supports a responsible claim about correctness.
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