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Use AI to explore a proof, not to certify one. A language model can suggest an argument, lemmas, or formal proof code, but its confident explanation is not evidence that the mathematics is correct. For stronger assurance, formalize the intended theorem in a proof assistant such as Lean, Isabelle/HOL, or Coq, then inspect the statement and dependencies the system checked.
What counts as checking a proof?
There are two different tasks: finding a plausible argument and verifying that an argument follows from stated assumptions. AI can help with the first by proposing proof strategies or filling in steps. A proof assistant checks a formal proof against a formal theorem statement. Scholarly coverage describes systems including Lean, Coq, and Isabelle; OpenAI describes Lean as a language for computer-checkable proofs. Communications of the ACM’s 2026 survey discusses this relationship, while OpenAI’s 2026 account describes Lean formalizations.
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A successful check means the proof assistant accepted a derivation of the encoded proposition under its rules and accepted dependencies. It does not, by itself, show that the proposition faithfully captures the original question in ordinary mathematical language.
A practical workflow for checking an AI-assisted proof
- State the exact claim. Write down the domain, quantifiers, hypotheses, and definitions. Ask an AI to flag ambiguities if useful, but resolve them against the original problem or source.
- Use AI to explore. Request a proof outline, candidate lemmas, alternative approaches, and a step-by-step derivation. Ask it to name the dependencies behind nontrivial steps. Treat every suggestion as unverified material.
- Try to break the argument. Check boundary and degenerate cases, look for hidden assumptions, and test small finite instances when that is useful. A counterexample can disprove a universal claim; passing examples cannot prove one. A second reviewer or tool can help challenge the reasoning, but it is not a substitute for verification.
- Formalize when the stakes or complexity justify it. Encode the theorem and proof in Lean, Isabelle/HOL, Coq, or another suitable proof assistant. Read the formal theorem statement before interpreting the checker’s result. Generated proof code or tactics must also pass the assistant’s checker.
- Inspect what the check depends on. Note the assistant and version, imported libraries, axioms, admitted placeholders, and external automation where relevant. For high-assurance work, preserve a reproducible build and consider independent checking appropriate to the system.
- Describe the evidence accurately. Distinguish an AI suggestion from a human-reviewed argument, example testing, or a formal check. Say which system and formal statement were checked only if that check actually occurred.
What a proof assistant can—and cannot—establish
Formal checking can provide strong assurance that a derivation follows from the assistant’s formal rules and accepted dependencies. The guarantee is deliberately narrower than “the AI proved the original claim”: it applies to the encoded proposition. A missing hypothesis or mistaken translation can leave a perfectly accepted proof of a different, weaker, or otherwise unintended theorem.
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The trusted base also matters. Proof assistants are software, and NIST’s 2021 Ockham criteria note that theorem provers have had coding errors. That is a reason to consider the checker, kernel, dependencies, and reproducibility—not a reason to treat all formal checks as worthless. NIST’s SATE VI Ockham Sound Analysis Criteria discusses this trust concern.
Common failure modes and how to respond
- Confident but false steps: Language models can produce hallucinations, including confidently incorrect claims. Ask for the justification of each nontrivial inference and verify it independently. OpenAI’s 2025 discussion of language-model hallucinations explains the general risk.
- Omitted conditions or edge cases: Make domains and hypotheses explicit, then inspect boundary values and degenerate cases.
- Formalization drift: Compare the encoded theorem line by line with the intended claim. Check that no assumption was silently added and none was accidentally dropped.
- Misleading proof-script success: A tactic or generated script may fail, rely on an unexpected lemma, or obscure dependencies. Inspect what the assistant accepted rather than relying on the AI’s description. A 2024 study of LLM-assisted Isabelle/HOL work describes integrating generated steps with verification and discusses failure modes. Read the EMNLP/ACL Anthology paper.
- Overgeneralizing a benchmark: A result on one benchmark, model release, or proof library does not establish performance on a different theorem or current tool version. No comparable, sufficiently contextualized accuracy figure is established here.
Choosing a proof assistant
Lean, Isabelle, and Coq are established options, but there is no basis here for ranking one as universally easiest or safest. Choose based on the logic and libraries your subject needs, how naturally the theorem can be formalized, available automation and AI integrations, proof readability and maintenance, and the system’s trusted components and reproducibility requirements. The 2026 Communications of the ACM survey covers these systems in the context of formal reasoning and language models.
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How to report what you checked
Use wording that identifies the evidence and its limits. For example: “AI suggested this proof outline; I checked the algebra and boundary cases,” or “Lean accepted a formal proof of the stated theorem using the listed dependencies.” Avoid saying “verified” when all that happened was that a chatbot said the proof looked correct. A formal check is meaningful only when readers can see what statement and trust boundary it refers to.
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