Let the Library Speak: Self-Advertised Method Selection for Formal Proving

2026-10-07

AI Summary

This study addresses the limitation that unreliable retrieval relevance undermines tactic applicability in LLM-based formal theorem proving. To this end, we propose a self-advertising mechanism that organizes mathematical tactics into contracts comprising anchors and examples. Large language models are leveraged to batch-generate proposals for explicitly evaluating premise-goal feasibility, while semantic matching against Mathlib anchors is integrated for error suppression, thereby achieving applicability-aware candidate ranking. Experimental results demonstrate that the proposed framework attains Hit@5 scores of 95.0% on the Putnam benchmark and 91.7% on IMO ProofBench. These outcomes significantly surpass existing baselines, effectively enhancing shortlist coverage for tactic selection in automated formal verification.

Abstract

LLM-based formal provers can retrieve relevant lemmas and prior proofs, but relevance alone does not say whether a mathematical method can be used on the current theorem. A method has prerequisites, a target, an intended action, and obligations that its use leaves to prove. Methods that look equally related to a theorem may therefore differ substantially in whether they offer a plausible next step. We formulate this as an applicability-aware method-selection problem and introduce self-advertisement: before candidates are ranked, a model generates a problem-specific proposal for each one, stating what part of the goal it targets, what action it would take, and what conditions that action requires. We organize 82 reusable methods from Putnam 2000-2014 as Method Contracts, which pair applicability descriptions with Mathlib anchors, a checked example or scaffold, and expected proof obligations. A single batched call elicits proposals across the library; vague or unsupported proposals are demoted, yielding a ranked shortlist accompanied by inspectable claims about each candidate's use. We analyze when similarity-based representations cannot distinguish methods with different applicability, how errors in applicability estimates affect shortlist quality, and what a checked scaffold guarantees under its stated assumptions. Against lexical, embedding, and embedding-plus-LLM reranking baselines, self-advertisement achieves 95.0% hit@5 on Putnam 2015-2025, compared with 84.2% for the strongest reranker. On IMO ProofBench, it achieves 91.7% compared with 88.3%. These results indicate improved coverage of annotated methods in the retrieved shortlists, particularly on Putnam.

Problem

• formal proving
• method selection
• applicability
• LLM
• lemma retrieval

Innovation

• Self-Advertisement
• Formal Proving
• Method Contracts
• Applicability-Aware Selection
• Large Language Models