To verify an AI-generated mathematical proof, first check the argument and its exact claim, then formalize that claim in Lean, compile it, and inspect the theorem’s dependencies and axioms. A successful Lean check shows that the encoded proposition follows from the declarations Lean accepts; it does not, by itself, show that the encoding matches the original mathematical question.
1. Write down the exact claim
Before reviewing the generated proof, record the result it is meant to establish. Preserve the definitions, assumptions, quantifiers, domains and conclusion. Those details determine what a proof must show: a claim about positive integers, for example, is not interchangeable with one about all integers.
Keep the original wording beside the formal statement as you work. A proof can be valid for a proposition that has been weakened, misstated or translated incorrectly.
2. Review the informal reasoning
Break the argument into its meaningful mathematical inferences. For each step, ask whether it follows from the preceding claims and the stated assumptions. Pay particular attention to:
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- Assumptions that appear without justification or are used without being stated.
- Changes of variable or domain that alter what is being proved.
- Division by an expression that could be zero.
- A generalization that has not been established for all required cases.
- A final conclusion that is weaker than the original claim.
This review can reveal gaps that a polished explanation conceals. It is a human audit, not a guarantee that every flaw will be found.
3. Formalize the claim and proof in Lean
Lean is a programming language and theorem prover used to formalize mathematics and verify proofs. Encode the proposition you wrote down, then compare the Lean theorem declaration with the original claim before treating a proof as evidence.
This comparison matters because type-checking establishes the proposition represented in Lean, not whether a person translated the intended mathematical meaning correctly. Lean’s Validating a Lean Proof guide explicitly separates a theorem’s valid proof from the meaning of its statement.
4. Compile the proof and confirm kernel acceptance
In Lean’s editor workflow, the blue double check marks indicate that the theorem statement was elaborated and the kernel accepted a proof following from declarations in the file and its imports. The Lean reference also documents running lake build on the module; a build that completes without errors or warnings provides the same baseline assurance.
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1Fix the driver behind crashes, sound loss and screen glitches2Repair Windows errors before they cause bigger problems3Scan for outdated or missing drivers - takes under a minuteThese checks are meaningful: they establish that the encoded theorem follows from the definitions, theorems and axioms Lean is using. They do not establish that the AI’s prose is faithful to the theorem declaration, nor do they remove assumptions inherited from imported declarations.
5. Inspect axioms and dependencies
Ask Lean to print the axioms on which the theorem depends, and review important imported lemmas as well. A dependency on sorryAx indicates an incomplete proof somewhere in the dependency chain. A custom axiom means the theorem is established only relative to that axiom’s soundness.
Do not assume that a clean-looking editor check means every dependency is complete: blue checks can appear even when an imported dependency contains sorry. The axiom inspection described in Lean’s validation guide helps expose this limitation.
6. Use a stronger replay check for high-stakes or adversarial cases
When a proof could be misleading or deliberately adversarial, Lean’s reference recommends building the project and then running lean4checker --fresh on the relevant module. Check that it finishes without errors. This replays stored declarations and proofs through the kernel, adding assurance beyond the ordinary editor workflow.
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It is an additional check, not an escape from the trust boundary: the result still depends on the stored files and on the declarations and assumptions used by the project.
7. Check intermediate steps when the final theorem is not enough
A final theorem can verify an endpoint without making the route through a natural-language argument transparent. For a step-by-step audit, divide the generated proof into intermediate mathematical claims, formalize those claims in Lean, and seek a proof for each.
The ACL 2025 paper on SAFE describes this retrospective, step-aware approach: articulate mathematical claims in Lean 4 and supply formal proofs. It reports FormalStep, a benchmark containing 30,809 formal statements. That figure is the benchmark’s size, not a success rate or evidence that every natural-language proof can be formalized automatically. The translation of each prose step into a precise claim remains part of the verification work. Read the SAFE paper.
The paper also contrasts step-aware evidence with opaque verifier scores, which do not themselves expose checkable proof evidence. That is the authors’ research framing, not a guarantee that any particular step-verification system will catch every error.
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Why a fluent proof can still be hard to verify
Generating a formal proof is not simply choosing from a small set of moves. A system may need to select tactics and construct mathematical objects such as witnesses or intermediate lemmas. OpenAI’s article on formal math describes this as an infinite action-space challenge. That helps explain why producing a candidate proof and verifying it are separate tasks: fluency alone does not establish correctness, and a candidate may fail to formalize.
What to include when reporting a verification
A useful verification report lets someone else understand what was checked and what remains assumed. State:
- The exact formal statement and its relationship to the original claim.
- The Lean and library context used for the check.
- Whether the module compiled and whether a stronger replay check was run.
- Which dependencies and axioms were inspected, including any incomplete proofs or custom axioms.
- Any remaining gap between the formalization and the intended mathematical meaning.
Do not describe kernel acceptance as proof that the prose is faithful unless that correspondence was separately reviewed.
Learn enough Lean to check proofs yourself
Lean’s Learn page introduces it as a functional programming language and theorem prover for formalizing mathematics and formal verification. It points beginners to the Natural Number Game and to Theorem Proving in Lean and Mathematics in Lean.
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