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1Fix the driver behind crashes, sound loss and screen glitches2Clear out junk files and repair common Windows errors3Scan for outdated or missing drivers - takes under a minuteWhen Lean rejects an AI-generated proof, the failure does not by itself show that the mathematical claim is false. The code may have a syntax or type error, may leave goals unsolved, may rely on a lemma unavailable in the project, or may be proving a proposition different from the intended theorem. Start by reading Lean’s first meaningful diagnostic and the exact proof state. If Lean accepts the proof, that establishes a narrower result: its proof term checks against the proposition Lean elaborated in the current project context—not that the proposition faithfully captures the informal mathematics.
What Lean verification establishes—and what it does not
Lean checks a proof term against a formal proposition after elaborating the code. That proposition is interpreted in the context of the file and its imports, including definitions, notation, type classes, and assumptions. A successful check is meaningful evidence that the formal statement has a proof under those conditions.
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It does not establish that the formal statement says what the author intended in ordinary mathematics. The Lean Reference Manual puts the distinction plainly: “Furthermore it is important to distinguish the question ‘does the theorem have a valid proof’ from ‘what does the theorem statement mean’.” A theorem can be validly proved and still be the wrong formalization—for example, because a hypothesis is missing, a quantifier is misplaced, or a definition does not match the intended concept. Lean’s guide to validating proofs explains both the checking process and its limits.
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Random freezes, missing sound and display glitches usually trace back to one bad driver. Find and replace yours safely.Free scan · under a minuteSo treat Lean as a precise but bounded oracle: it answers whether the elaborated proposition has a checked proof in the project, not whether the original English claim was translated correctly. Review the statement and the proof as separate tasks.
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Why an AI-generated proof may fail
Lean’s errors identify problems in the formal code or its environment; they do not automatically diagnose the underlying mathematics. Classifying the first failure helps you choose the right repair.
| Failure type | What it often looks like | What to check |
|---|---|---|
| Syntax or elaboration | Lean cannot parse an expression, infer a type, or resolve a name. | The earliest diagnostic, spelling, imports, types, and implicit arguments. |
| Tactic failure or open goal | A tactic fails, or Lean reports goals remaining after the proof. | The current target, local hypotheses, and any unsolved branches or cases. |
| Library mismatch | A claimed lemma is unknown or does not apply. | Whether the declaration exists in this project version and whether its type matches the use. |
| Formalization mismatch | Lean accepts the code, but the result seems wrong for the intended English claim. | The statement’s types, domains, definitions, quantifiers, and hypotheses. |
| Incomplete dependency or axiom | A theorem depends on an admitted proof or an assumption that was not intended. | The theorem’s dependencies and printed axioms, including possible sorryAx. |
| Long proof or search failure | The generated approach stalls on many steps or a complex formalization. | Whether the proof strategy needs restructuring; failure is not evidence that the theorem is false. |
AI-generated code is especially vulnerable to plausible-looking names and arguments that do not fit the installed library. A model may suggest a lemma that does not exist, or one whose hypotheses differ from the current goal. FormalProofBench discusses nonexistent-lemma errors among the problems that can arise in natural-language arguments; such a mismatch is a tooling or formal-code issue, not a refutation of the mathematics. The benchmark paper also illustrates why difficult formalizations should not be treated as routine code completion.
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How to debug a Lean proof, step by step
- Locate the first useful error. Open the indicated file and line, then read the earliest diagnostic that explains the failure. Later errors may be consequences of an earlier syntax or elaboration problem. Decide whether Lean is reporting a parse or type issue, a tactic failure, an unresolved goal, a missing name, or a project/build problem. Do not infer that the mathematical idea is false from a compiler message.
- Inspect the proof state at the failure point. Record the target and all local hypotheses exactly as Lean displays them. The goal shown by the editor may differ from the prompt’s apparent theorem because elaboration, implicit arguments, coercions, or earlier tactics changed the context. Check which variables, assumptions, and cases are actually in scope.
- Reduce the obligation. If the generated proof is a long tactic block, split it into shorter steps or introduce an intermediate
havestatement. Re-check after each focused change. This makes it easier to see which inference or transformation fails instead of debugging the whole block at once. - Verify every name and the project context. Check imports and search the installed project or library for the lemma the proof invokes. Confirm its actual declaration and hypotheses rather than trusting a plausible name. Also check that the Lean and Mathlib versions match the project: a valid declaration in another version may be absent or different here.
- Review the theorem statement before polishing the proof. Compare its types, domains, quantifiers, hypotheses, and definitions against the intended informal claim. A proof tactic cannot repair a proposition that expresses the wrong claim. Pay particular attention to notation and type-class assumptions, which can influence how the statement is interpreted.
- Audit assumptions and dependencies when trust matters. In Lean, run
#print axioms theoremNamefor the theorem you want to inspect. Investigate unexpected axioms or dependencies such assorryAx; an accepted theorem is only as assumption-free as its dependency chain allows. The exact interpretation of the output is covered in the Lean Reference Manual. - Re-check the project after the repair. Use the normal project build, such as
lake build, to check the result in its project context. If the consequence of an incorrect proof would be serious, consider the additional checking options below.
Lean’s interactive proof state is part of the debugging interface, not just a final pass/fail message. The official tutorial describes developing tactics incrementally and emphasizes that formalization is a programming activity with a learning curve: “Formalization can be seen as a kind of computer programming: we will write mathematical definitions, theorems, and proofs in a regimented language, like a programming language, that Lean can understand.” See Mathematics in Lean’s introduction for its explanation of proof states, tactics, and formalization.
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How much confidence should a successful check give you?
Match the validation effort to the stakes. For ordinary development, successful checking in the project and a passing lake build are the practical baseline. For greater assurance, Lean documents replaying declarations with lean4checker --fresh. In higher-risk or adversarial settings, its documented sandboxed lake comparator workflow uses external checkers. These are stronger checks, not magic: they still depend on assumptions that include the correctness of the challenge statement and the checkers involved. The options and their limits are described in Lean’s validation documentation.
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Regardless of the checking level, keep three questions distinct: did the code check, does the formal proposition express the intended theorem, and are the dependencies and assumptions acceptable for this use? Compilation addresses the first; human review of the formalization addresses the second; dependency and axiom inspection help with the third.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What recent AI-proof results do—and do not—show
Published evaluations show both progress and persistent difficulty, but their metrics describe different tasks and should not be compared as if they were one universal proof-success rate.
- FormalProofBench: Ravi et al. report 33.5% accuracy for the best-performing foundation model in their evaluation of 200 advanced undergraduate and graduate-level formally specified problems, using the paper’s stated setup. This is a result for that benchmark and harness, not a general rate for all theorem proving or AI-generated proofs. Read the FormalProofBench paper.
- LeanProgress: Huang, Song, George, and Anandkumar report 75.1% accuracy on a task predicting proof progress or remaining steps. They also report a 3.8% improvement over a 41.2% baseline in one best-first-search integration on Mathlib4. These are progress-prediction and search results in the paper’s experimental setting—not direct theorem-proof success rates comparable to FormalProofBench. Read the LeanProgress paper.
These results are reasons to inspect generated formalizations and proof states carefully, especially for long proofs; they are not evidence that a particular failed attempt disproves its theorem, nor that one model or workflow is generally superior.
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