Learned Interventions in Lean 4 grind

RL-trained Lean theorem provers mode-collapse at inference time: on miniF2F-test with DeepSeek-Prover-V1.5-RL, doubling the i.i.d.\ sampling budget from $k{=}32$ to $k{=}64$ produces zero additional solved theorems (42/244 in both cases). A fixed schedule of 15 tactic skeletons breaks this plateau and recovers a $+45$ relative improvement at $k{=}16$ (mean $Δ= +12.3 \pm 4.2$ theorems across $n{=}3$ seeds, sign preserved in every seed). A controlled diversity ablation rules out the prompt-diversity confound: tactic skeletons help, paraphrases match the baseline, and irrelevant Lean comments actively degrade. A leave-one-out formalization-difficulty stratification reveals a structural-content gradient across the three perturbations. The phenomenon is RL-specific: V1.5-Base proves zero theorems regardless of intervention, identifying RL as the stage that creates the proof capability which subsequently collapses; extending to two additional 7B Lean provers, RL-trained DeepSeek-Prover-V2-7B contributes $+3$ frontier solves no i.i.d. baseline can reach despite a flat aggregate, while SFT-trained Goedel-Prover does not ( $-10.0$ $\pm 4.4$ theorems, $n{=}3$, sign preserved every seed). Inference-time structural diversity is a cheap, complementary axis for RL-trained provers, orthogonal to scaling model size or training compute.

MerLean-Prover is an end-to-end Lean4 theorem prover that replaces sorry declarations with kernel-checkable proofs. It is built from three agent types (Planning, Check, and Lean) composed by a recursive outer loop whose unit of revision is the proof plan itself, and uses no fine-tuning, no custom RL objective, and no theorem-specific scaffolding. On FormalQualBench, a benchmark of 23 PhD-qualifying-exam theorems, MerLean-Prover solves 10/23, surpassing the strongest published open-source baseline (OpenGauss, 8/23). On Putnam2025, the same harness closes 12/12 with substantially lower total wall-clock than the next-best system that closes the full set. The harness also transfers to smaller models: Sonnet closes all four tested FormalQualBench problems, and Haiku closes the two short ones. These results suggest that harness design is a central factor in end-to-end Lean4 theorem proving, alongside raw model capability, and that a relatively simple harness can already be effective.

Automated theorem proving with large language models in Lean 4 is commonly approached through either step-level tactic prediction with tree search or whole-proof generation. These two paradigms represent opposite granularities for constructing supervised training data: the former provides dense local signals but may fragment coherent proof processes, while the latter preserves global structure but requires complex end-to-end generation. In this paper, we revisit supervision granularity as a training set construction problem over proof trajectories and propose segment-level supervision, a training data construction strategy that extracts locally coherent proof segments for training policy models. We further reuse the same strategy at inference time to trigger short rollouts for existing step-level models. When trained with segment-level supervision on STP, LeanWorkbook, and NuminaMath-LEAN, the resulting policy models achieve proof success rates of 64.84%, 60.90%, and 66.31% on miniF2F, respectively, consistently outperforming both step-level and whole-proof baselines. Goal-aware rollout further improves existing step-level provers while reducing inference costs. It increases the proof success rate of BFS-Prover-V2-7B from 68.77% to 70.74% and that of InternLM2.5-StepProver from 59.59% to 60.33%, showing that appropriate supervision granularity better aligns model learning with proof structure and search. Code and models are available at https://github.com/NJUDeepEngine/SEG-ATP.