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

arXiv: https://arxiv.org/abs/2606.17981 · HTML: https://mlsanity.com/paper/2606.17981/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2026.

## Built on by (most cited first, 5 of 5)

- [BlueprintRepair: Typed Local Edits for Failed Lean Proof Blueprints](https://mlsanity.com/paper/2607.28110/) (2026)
- [P3: Joint Program-and-Proof Planning for Verified Code Generation](https://mlsanity.com/paper/2608.09277/) (2026)
- [ProsaBuddy: Assisting Mechanized Real-Time Schedulability Analysis with LLM-based Agents](https://mlsanity.com/paper/2610.03796/) (2026)
- [Cost-Efficient Theorem Proving via Agent Orchestration in Program Verification](https://mlsanity.com/paper/2610.09681/) (2026)
- [LEVER: Adaptive Cost-Aware Proof Search Over AND/OR Graphs](https://mlsanity.com/paper/2610.11862/) (2026)

## Abstract (arXiv)

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, while 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 that estimates 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% to 13% 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.