Planning to Hammer: Difficulty-Aware Decomposition for Automating Rocq Proofs

As AI-generated code proliferates, formal verification—particularly through interactive theorem provers such as Rocq and Isabelle—becomes increasingly important for ensuring software correctness. However, producing machine-checked proofs in such provers remains a bottleneck. Existing solutions bring complementary strengths to proof automation: large language models (LLMs) can propose high-level proof strategies but lack local rigor; automated tactics such as CoqHammer can reliably discharge many local goals, but lack long-range planning capabilities. To combine the best of both worlds, we present Quarry, a planning-based proof synthesis framework that separates proof planning from proof execution. Specifically, Quarry asks an LLM to actively propose multiple proof decompositions with arbitrary sublemmas, type-checks them in Rocq under temporarily admitted sublemmas, and ranks candidates using a proof-state-based difficulty model estimating hammer solvability. It then recursively proves sublemmas within a bounded budget, effectively turning long proofs into sequences of hammer-solvable obligations. We implement Quarry on top of SerAPI and CoqHammer and evaluate it using multiple frontier LLMs across multiple benchmarks. The experimental results show that planning-based decomposition with solvability-aware ranking substantially improves automation while maintaining predictable cost. Under a uniform 10-minute wall-clock budget, Quarry improves over the strongest baseline by 7–13 percentage points in success rate across three Rocq benchmarks. These results demonstrate that reliable proof automation can be achieved by coordinating neural planning with symbolic execution rather than replacing either.