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arxiv: 2605.11905 · v1 · submitted 2026-05-12 · 💻 cs.AI

Recognition: 2 theorem links

· Lean Theorem

Rethinking Supervision Granularity: Segment-Level Learning for LLM-Based Theorem Proving

Chun Cao, Jiakun Zhang, Jingwei Xu, Junyu Lai, Shuo Xu

Pith reviewed 2026-05-13 05:59 UTC · model grok-4.3

classification 💻 cs.AI
keywords theorem provinglarge language modelsLean 4supervision granularitysegment-level learningproof trajectoriespolicy modelsautomated reasoning
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The pith

Segment-level supervision on proof trajectories improves LLM performance in theorem proving over step-level or whole-proof approaches.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how the choice of granularity in supervised training data affects large language models that prove theorems in Lean 4. It proposes a middle ground called segment-level supervision, which pulls locally coherent chunks of proof steps from full trajectories rather than using isolated tactics or entire proofs. Policy models trained this way on three datasets reach success rates of 64.84 percent, 60.90 percent, and 66.31 percent on the miniF2F benchmark, beating both finer and coarser baselines. The same segment extraction is reused at inference time to run short goal-directed rollouts that raise performance of prior step-level provers while lowering search cost.

Core claim

Segment-level supervision extracts locally coherent proof segments from trajectories to train policy models. When trained on STP, LeanWorkbook, and NuminaMath-LEAN, these models achieve 64.84%, 60.90%, and 66.31% success on miniF2F, outperforming step-level and whole-proof baselines. The strategy reused at inference as goal-aware rollout boosts BFS-Prover-V2-7B from 68.77% to 70.74% and InternLM2.5-StepProver from 59.59% to 60.33%.

What carries the argument

Segment-level supervision, a strategy that extracts locally coherent proof segments from trajectories for training policy models and for triggering short rollouts at inference time.

If this is right

  • Policy models trained on segments reach higher proof success rates on the miniF2F benchmark than models trained on steps or whole proofs.
  • Goal-aware rollout using the same segments raises success rates for existing step-level models while cutting inference cost.
  • Appropriate supervision granularity aligns model learning more closely with proof structure and search behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The segment extraction method could be tested on other formal systems or on non-mathematical reasoning tasks that involve long structured sequences.
  • Dynamic adjustment of segment length based on theorem difficulty might yield further gains on harder problems.
  • Hybrid systems that combine segment-trained policies with traditional search could reduce overall compute needed for high success rates.

Load-bearing premise

Locally coherent proof segments extracted from trajectories provide a training signal that better aligns model learning with actual proof structure than either fine-grained steps or full proofs.

What would settle it

Training new policy models on segment-level data and finding they achieve no higher or lower success rates than step-level baselines on miniF2F would show the granularity choice does not improve alignment.

Figures

Figures reproduced from arXiv: 2605.11905 by Chun Cao, Jiakun Zhang, Jingwei Xu, Junyu Lai, Shuo Xu.

Figure 1
Figure 1. Figure 1: Illustration of supervision granularity on the problem [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Training loss curves under different supervision granularities on LeanWorkbook, [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative accuracy over per-theorem elapsed time for models trained on STP. The two [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative accuracy over per-theorem elapsed time for models trained on LeanWorkbook. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative accuracy over per-theorem elapsed time for models trained on NuminaMath [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cumulative accuracy over per-theorem elapsed time for goal-aware rollout on miniF2F. For [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes segment-level supervision as an intermediate granularity for constructing training data from proof trajectories in LLM-based automated theorem proving in Lean 4. Instead of step-level tactics or full proofs, it extracts locally coherent segments for policy model training on datasets including STP, LeanWorkbook, and NuminaMath-LEAN. The resulting models achieve miniF2F success rates of 64.84%, 60.90%, and 66.31%, outperforming step-level and whole-proof baselines; the same segmentation heuristic is reused at inference via goal-aware short rollouts to boost existing provers (e.g., BFS-Prover-V2-7B from 68.77% to 70.74%). Code and models are released.

Significance. If the performance gains are attributable to granularity rather than selection effects, the approach offers a practical middle ground that better aligns training signals with proof structure while remaining compatible with existing search frameworks. The public release of code and models is a clear strength that supports reproducibility and follow-up work.

major comments (2)
  1. [§4] §4 (Experimental Results) and Table 2: The central claim that segment-level supervision improves alignment with proof structure rests on comparisons against step-level and whole-proof baselines trained on the same underlying trajectories. However, no ablation is reported that applies random (non-coherent) segment extraction of matched lengths from identical trajectories; without this control, it is impossible to separate the effect of local coherence from implicit curation of higher-quality sub-trajectories.
  2. [§3.2] §3.2 (Segment Extraction): The coherence heuristics used to define segments are described at a high level but lack a precise algorithmic specification or pseudocode. This makes it difficult to assess whether the reported gains depend on dataset-specific tuning of those heuristics.
minor comments (2)
  1. The paper does not report statistical significance tests or variance across multiple random seeds for the miniF2F success rates, which would strengthen the comparison tables.
  2. Details on exact data splits, filtering criteria for the training trajectories, and hyperparameter search ranges are referenced only via the GitHub repository; including a concise summary table in the main text would improve self-containment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the contributions and limitations of our work on segment-level supervision for LLM-based theorem proving. We address each major comment below and will revise the manuscript to incorporate the suggested improvements for greater rigor and reproducibility.

read point-by-point responses

  1. Referee: [§4] §4 (Experimental Results) and Table 2: The central claim that segment-level supervision improves alignment with proof structure rests on comparisons against step-level and whole-proof baselines trained on the same underlying trajectories. However, no ablation is reported that applies random (non-coherent) segment extraction of matched lengths from identical trajectories; without this control, it is impossible to separate the effect of local coherence from implicit curation of higher-quality sub-trajectories.

    Authors: We agree that the absence of a random-segment ablation leaves open the possibility that gains arise partly from implicit curation rather than coherence per se. In the revised manuscript we will add this control: from the same proof trajectories we will extract random contiguous segments of matched average length, train policy models on them under identical conditions, and report miniF2F success rates alongside the coherent-segment results. This will allow direct quantification of the coherence contribution. revision: yes

  2. Referee: [§3.2] §3.2 (Segment Extraction): The coherence heuristics used to define segments are described at a high level but lack a precise algorithmic specification or pseudocode. This makes it difficult to assess whether the reported gains depend on dataset-specific tuning of those heuristics.

    Authors: We acknowledge that §3.2 currently provides only a high-level description. In the revision we will insert a detailed algorithmic specification together with pseudocode that formalizes the segment extraction procedure, including the exact criteria for detecting goal changes, tactic dependency boundaries, and termination conditions. The same pseudocode will be used for all three datasets, making any dataset-specific tuning explicit and reproducible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical results on external benchmarks

full rationale

The paper reports proof success rates (64.84%, 60.90%, 66.31% on miniF2F) from direct evaluation of trained models against fixed external test sets. No equations, metrics, or predictions are defined in terms of fitted parameters that would reduce outputs to inputs by construction. No self-citations appear load-bearing for the core claim, no uniqueness theorems are invoked, and no ansatzes are smuggled via prior work. The segment extraction strategy is presented as a data-construction heuristic whose value is assessed empirically rather than tautologically. This is the common honest case of a self-contained experimental paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work relies on standard supervised fine-tuning assumptions for LLMs and established theorem-proving benchmarks without introducing new free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5570 in / 1146 out tokens · 84987 ms · 2026-05-13T05:59:54.233515+00:00 · methodology

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