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AI can help you find likely gaps, test edge cases, explain difficult steps, and rewrite a proof more clearly. It cannot certify an informal proof simply because its answer sounds convincing. For higher assurance, translate the theorem into Lean or another proof assistant and run the formal checker—then verify that the encoded statement really matches the mathematics you intended.
The safest workflow separates three jobs: use a language model for targeted criticism and exposition, check every mathematical response yourself, and use formal verification when the claim justifies the extra setup.
What an AI proof check actually tells you
A conversational model reads the statement and proof as text. It may identify a missing hypothesis, expose an invalid inference, suggest a counterexample, or propose a different route. It may also produce a polished explanation of an argument that is false. Fluency is not a proof certificate.
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The Leiden Declaration on Artificial Intelligence and Mathematics captures the risk precisely: “Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs.” Treat an AI response as a review comment or brainstorming aid, not as an independent referee report.
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There are two fundamentally different checks:
| Approach | What is checked | What the tool returns | What you must still do | Setup and fit |
|---|---|---|---|---|
| Language-model critique | Plausibility, explanations, local inferences and possible counterexamples in ordinary language | Critique, alternative argument, proof-step suggestion or rewrite | Recompute the mathematics and confirm every hypothesis, quantifier, domain and boundary case | Works directly on an informal proof; little setup |
| Formal proof assistant such as Lean | A derivation encoded in the assistant’s formal language | A machine-checked proof term or a precise failure | Check the formal statement, definitions, imports and assumptions against the intended theorem | Requires formalization and a suitable library; effort varies by domain |
A second model response can provide another perspective, but model-to-model agreement is not the same as an independent formal check or expert review.
A seven-step workflow for checking and improving a proof
1. State the target precisely
Give the model the exact theorem, every assumption, definitions of nonstandard terms, the domain of each variable, and the complete proof. Ask it to restate the claim before criticizing the argument. This catches a surprisingly common failure: the model silently changes “for every real number” to “for every positive real number,” or drops a condition that the proof uses.
Use a prompt such as:
Restate the theorem below, including all domains and hypotheses. List the conclusion in your own notation. Do not judge the proof yet. Point out any ambiguity in the statement.Compare the restatement line by line with your original statement. Correct the problem specification before asking for a proof review.
2. Request local checks, one at a time
Do not begin with “Is this proof correct?” That invites a broad, confident verdict with little audit trail. Select one inference or lemma and ask for an explicit derivation.
- “Justify the implication from line 4 to line 5, naming the theorem used.”
- “Where is the assumption that x is nonzero used?”
- “Check the quantifiers in this lemma and give a counterexample if any are too strong.”
- “Recompute this algebra independently and show every transformation.”
- “Test the statement at boundary values, including zero, equality cases and limiting cases.”
- “Find a counterexample to the intermediate claim, even if you think it is probably true.”
Ask for one task per response. Narrow questions make it easier to inspect what the model actually checked.
3. Verify every critique yourself
An AI-generated objection can be wrong just as an AI-generated proof can be wrong. Recalculate the disputed step, test the proposed counterexample, and check that cited lemmas apply under the stated assumptions. If the model says a division is invalid, identify whether the denominator can really be zero. If it says a function is continuous, check the domain and the definition being used.
For algebra, write the transformation on paper or in a computer algebra system you understand. For analysis, inspect the order of limits, quantifiers and uniformity conditions. For probability, verify the sample space and independence assumptions. For geometry, check orientation, degeneracy and excluded configurations. The subject-specific audit remains yours.
4. Separate validity from clarity
After the logical audit, ask the model to improve exposition as a separate job. Request an outline, clearer notation, a motivation paragraph or a version aimed at a specified audience. Then compare the rewrite line by line with the checked proof.
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A stylistic improvement does not repair a logical gap. Require the model to preserve the theorem, assumptions, definitions and order of dependencies. If it introduces a “without loss of generality” statement, ask it to explain why the reduction is valid and what symmetries or cases are being used.
5. Formalize when a stronger check is justified
For an important result, encode the theorem and proof in Lean or another proof assistant. OpenAI describes Lean as a programming language in which mathematical proofs can be checked by a computer, and theorem-proving research uses language models for tactic suggestions, proof search and premise selection.
You can ask an AI tool to translate definitions, locate library lemmas or suggest tactics, but inspect every generated declaration. The proof assistant checks the formal derivation, not the informal document sitting beside it.
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Before treating a successful check as evidence about your original proof, compare the formal and informal versions:
- Are the variables quantified over the same sets?
- Are all hypotheses present, including finiteness, positivity, nondegeneracy or measurability conditions?
- Do the formal definitions mean what the informal terms mean in context?
- Did an imported theorem add an axiom or typeclass assumption that was not intended?
- Is the conclusion exactly the one you wanted, rather than a weaker or different proposition?
- Did automation simplify a case away by changing the domain or a definition?
A successful Lean check should therefore be reported narrowly: the encoded statement and encoded proof were checked. It should not be presented as automatic endorsement of an unrelated natural-language argument.
7. Seek independent review for consequential work
For research-level mathematics, keep a record of assumptions, tool assistance, formal files and unresolved questions, then ask a qualified mathematician to review the argument. Expert review matters because correctness can depend on interpretation, definitions and omitted context that neither a language model nor a formal checker can infer safely.
Prompt patterns that produce useful feedback
Statement and dependency audit
Here is a theorem and proof. First list every definition, hypothesis and previously established result the proof uses. For each line, identify the exact dependency. Do not rewrite the proof or give a final verdict until the dependency list is complete.This exposes hidden uses of assumptions and circular reasoning.
Quantifier and domain audit
Check all quantifiers and domains in this statement and proof. Look for illegal substitutions, division by zero, empty domains, non-invertible maps, and changes from “for all” to “there exists.” Give a concrete counterexample for each suspected error.Ask the model to distinguish a genuine counterexample from one that violates an assumption.
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Lemma stress test
Consider only Lemma 2. Try to disprove it by testing the smallest, largest, zero, negative, equality and degenerate cases allowed by its hypotheses. If it survives, explain why each test is covered; do not infer that testing proves the lemma.Finite tests can reveal a bug but cannot establish a universal theorem.
Alternative-route request
Give two independent approaches to the stated theorem. For each, list the required hypotheses and identify the step most likely to fail. Do not claim either approach is correct without checking its inferences.Different routes can reveal that the original proof relies on an unnecessarily strong assumption or misses a simpler invariant.
Exposition rewrite with a preservation contract
Rewrite this proof for a graduate-level reader. Preserve the exact theorem, assumptions, definitions, case split and dependency order. Mark any sentence that is explanatory rather than deductive. After the rewrite, provide a line-by-line correspondence to the original proof.The correspondence makes it harder for a fluent rewrite to introduce an unnoticed mathematical change.
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A manual checklist for accepting or rejecting a proof step
- Statement: Does the line assert exactly what follows from the preceding line, or has a stronger claim appeared?
- Domains: Are variables still in the sets or spaces required by the theorem being applied?
- Quantifiers: Have existential choices been made legally, and has a universal claim been weakened or strengthened?
- Hypotheses: Can you point to where each assumption is used? Is any assumption missing at the step where it is needed?
- Definitions: Are terms such as convergence, independence, rank or compactness being used with their stated definitions?
- Algebra: Were signs, denominators, inequalities and exponents handled correctly?
- Cases: Are zero, negative, boundary, degenerate and equality cases included or explicitly excluded?
- Dependencies: Does a cited lemma precede the result and avoid assuming the conclusion indirectly?
- Conclusion: Does the final sentence match the original target, not merely a related proposition?
Ask the AI to fill this checklist, but do not outsource the final decision. Its answers are leads for your inspection.
Using AI to improve exposition without hiding uncertainty
Once validity has been checked, AI is useful for communication. It can suggest a motivating example, reorder definitions, replace vague references such as “clearly,” and flag notation that changes meaning. Request explicit labels for intuition, formal deduction and unproved conjecture.
Do not let it erase qualifications. “Assume the denominator is nonzero” is not interchangeable with silently cancelling it. “Choose an arbitrary element” is not interchangeable with choosing an element that exists. Keep the original proof beside the rewrite and restore any condition the editor removed.
What Lean-style formal verification establishes
In a proof assistant, you define objects and propositions in a formal language and construct a proof term. The kernel checks that term against the encoded proposition and its permitted foundations. This is stronger than judging whether prose appears persuasive because the checker verifies a derivation with explicit types and dependencies.
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A small illustrative Lean shape
The exact syntax and available lemmas depend on the version and imported library. Conceptually, a theorem has an explicit type, assumptions and conclusion:
theorem example (x y : ℝ) (h : x ≤ y) : x + 1 ≤ y + 1 := by
exact add_le_add_right h 1The checker can validate the encoded implication. It cannot determine whether your informal problem was really about real numbers, whether “add one” was the intended operation, or whether a hidden condition was omitted from the formal statement.
Where AI helps in a formal workflow
- Suggesting a theorem from the library that may match a proof step.
- Proposing tactics or a sequence of smaller lemmas.
- Finding premises needed by an existing result.
- Translating an informal definition into candidate formal declarations.
- Explaining an error message in ordinary language.
Run the actual checker yourself and read its output. A generated script that looks plausible is not evidence until the assistant accepts it under the intended imports and assumptions.
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| Use chat-first review when… | Invest in formalization when… |
|---|---|
| You are exploring an idea, debugging a local step, looking for examples or improving exposition. | The result is central, likely to be reused, difficult to audit informally or valuable as a machine-checkable artifact. |
| The proof is short or still changing rapidly. | Definitions and dependencies are stable enough to encode. |
| You need explanations in natural language for collaborators or students. | You need the assistant’s kernel to reject invalid derivations under explicit foundations. |
Formalization has a real domain-fit and setup cost: the theorem, proof and supporting definitions must be expressed in the assistant’s language and available library. The source material does not establish a universal time estimate, so treat the effort as project-specific.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Common failure modes and recovery steps
The model gives a confident “correct” verdict
Ask it to replace the verdict with a line-by-line derivation, list every assumption, and produce attempted counterexamples. Then verify those responses independently.
The critique invents a gap
Request the exact line and a concrete counterexample. If the alleged counterexample violates a hypothesis, explain that violation and continue checking the original step yourself.
The model silently changes the theorem
Have it restate the statement first, then compare symbols, domains and quantifiers. Include the full theorem in every focused prompt rather than relying on conversation memory.
Automation proves an unintended statement
Inspect the formal declaration, imports, typeclass assumptions and definitions. Compare the formal conclusion with the natural-language target before citing the machine check.
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A generated Lean script fails
Read the first error, isolate the smallest failing lemma, and ask for an explanation rather than a replacement script. Check namespaces, types and missing premises. A failure may indicate a coding issue, an incorrect suggested theorem or a mismatch between your intended statement and the encoded one.
The proof is too long for one prompt
Split it into named lemmas. Give the model the statement and dependencies of one lemma at a time, then perform a final pass over how the lemmas compose. Local approval does not remove the need to check the assembled proof.
What current evidence says about AI and difficult mathematics
OpenAI’s 2026 First Proof account says it ran an internal model on all 10 problems. After expert feedback, OpenAI assessed at least five of the ten attempts as having a high chance of correctness. It 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. These are task-specific assessments from that effort, not a general accuracy rate for everyday proof checking.
In a separate mathematics release dated October 6, 2026, OpenAI reported roughly three hours of ChatGPT Pro thinking-equivalent compute per average result. That figure describes the average result in that release and should not be generalized to other tasks or models.
OpenAI summarized the difficulty of such work this way: “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.” The statement is an institutional description of its First Proof submissions, not a claim that every AI-assisted proof needs the same amount of review.
A practical record for collaborative or research work
Keep a compact audit trail:
- The exact theorem and version of each definition.
- All assumptions, including those added during debugging.
- Prompts that materially influenced a proof step or formal declaration.
- AI suggestions that were rejected and why.
- Manual checks, counterexamples tested and unresolved uncertainties.
- The formal source files, imports and checker result, if formalized.
- Names or roles of independent reviewers and the scope of their review.
This record helps collaborators distinguish an original deduction from a model suggestion and makes later corrections possible when an assumption changes.
Bottom line
Use AI as a tireless critic and writing assistant: make it restate the theorem, audit one inference at a time, search for counterexamples and explain alternatives. Verify every response yourself. When the stakes justify it, formalize the claim in Lean and run the checker, while separately confirming that the formal statement faithfully represents the mathematics you meant. Neither a fluent explanation nor a successful formal check excuses that translation and review.
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