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REVIEW 4 major objections 5 minor 41 references

Clarifying Before Reasoning: A Coq Prover with Structural Context

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Structured semantic context raises an LLM's Coq theorem-proving success from 21.8% to 45.8%, beating a specialized prover.

desk verdict Real engineering, but the headline result is unproven until the paper rules out 'admit' and other axiom-level shortcuts. read the letter →

arxiv 2507.02541 v1 pith:7F4MJR56 submitted 2025-07-03 cs.AI

Yanzhen Lu , Hanbin Yang , Xiaodie Wang , Ge Zhang , Biao Li , Chenxu Fu , Chao Li , Yang Yuan
show 1 more author
Andrew Chi-Chih Yao
This is my paper
classification cs.AI MSC 03B3568V15
keywords CoqtheoremprovinglargelanguagemodelsclarityscoreselectiveconceptunfoldingPlanner-Executorstructuredsemanticcontextproofsearch
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that a large language model's failures in formal theorem proving often stem from unclear task descriptions, not just weak reasoning. To test this, it measures how well a model can define the concepts in a Coq proof state, calls that the clarity score, and then enriches the prompt with three layers of semantic information: original source definitions, the compiler's elaborated internal types, and natural-language explanations. With this enrichment, clarity rises from 44.5% to 82.3% and proof success on 1,386 randomly sampled standard-library theorems rises from 21.8% to 45.8%, beating the prior Graph2Tac result of 33.2%. The authors take the near-linear relation between clarity and success as evidence that task representation is a separable lever for reasoning, independent of model scale or reinforcement learning.

What carries the argument

Selective concept unfolding with a Planner-Executor search loop. Concept unfolding recursively expands each entity referenced in a proof state by its definition; selective means the expansion stops once the model likely already knows the concept, avoiding unnecessary bloat. The system also uses a domain-specific tokenizer that maps aliases and module paths to unique semantic identifiers, and a structured prompt containing dual surface and internal representations of the proof state, related premises and tactics, and a running proof trace. These components feed a Planner that writes a strategy and an Executor that proposes tactics via beam search, with compiler feedback used to retry failed tactics.

What would settle it

Look inside every successful proof trace produced by the system and see whether any step is the 'admit' tactic or an axiom-like shortcut; then rerun the benchmark with those tactics removed from the allowed set. If a large share of the 45.8% successes vanish, the clarity-driven improvement claim collapses.

Watch

Extended reading notes

Core claim

The paper claims that task clarity can be quantified and that improving it directly improves theorem-proving performance. The central empirical result is that a Planner-Executor system built on DeepSeek-V3, with structured context inserted into the prompt, achieves a 45.8% proof success rate on a random 10% sample of the Graph2Tac test set drawn from 15 standard Coq packages, outperforming Graph2Tac's 33.2%. The same architecture also shows a 0.98 correlation between clarity scores and success rates across information configurations, which the authors read as support for the causal claim that understanding precedes reasoning. The paper also reports that a 32B fine-tuned model reaches 48.6% success, indicating the structured data itself carries much of the value.

Load-bearing premise

The claim that structured context is what improves proof success rests on the baseline comparison, and the paper does not state that the allowed tactics exclude 'admit', which can discharge any goal without a real proof; if 'admit' is allowed, the reported success rates could be inflated.

Editorial extensions

If this is right

  • General-purpose LLMs without specialized mathematical training can match or beat specialized provers when the task description is structured enough.
  • Clarity score can act as a diagnostic: measuring concept comprehension before proof search predicts downstream success and can guide which concepts to unfold.
  • Fine-tuning smaller models on structured context yields higher success than the larger general model, so data quality can substitute for parameter count.
  • The method is model-agnostic and domain-agnostic, so the same structured-description recipe should transfer to other proof assistants and to precise-reasoning tasks like verification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: the reported success rates count a goal as complete when the checker says no goals remain; if the tactic 'admit' is in the allowed tactic vocabulary, some 'solved' theorems may have been discharged by fiat, so a replication that bans admit is the real test of the comparison to Graph2Tac.
  • Extension: since the test set is a 10% sample, the 45.8% figure is an estimate; a full benchmark run or repeated sampling would show how much variance surrounds the 12.6-point lead.
  • Extension: the clarity-to-success correlation is measured over configurations that also change the prompt length and content; an ablation that holds total tokens roughly fixed would separate clarity from raw information volume.
  • Extension: the same pipeline could be built for Lean or Isabelle, where elaborated terms are also available; if the correlation holds there, the clarity lever generalizes beyond Coq.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper argues that poor task understanding, rather than insufficient reasoning, is a major cause of LLM failures in formal theorem proving. It introduces a 'clarity score' that measures how accurately a model can reproduce formal definitions from a Coq proof context, and shows that adding structured semantic context extracted from Coq's compiler—surface syntax plus kernel-elaborated representations plus natural-language intuitions—raises this score from 44.5% to 82.3%. On theorem proving, the paper reports a DeepSeek-V3 Planner-Executor system achieving 45.8% success on a subset of the Graph2Tac benchmark, outperforming Graph2Tac's 33.2%, with a fine-tuned Qwen-2.5-32B reaching 48.6%. The paper also reports a near-perfect correlation (r=0.98) between clarity and proof success.

Significance. If the main empirical claim is correct, this is a meaningful result: it suggests that semantic enrichment of proof-state inputs can give a general-purpose LLM substantial gains over a specialized GNN-based prover, and it offers a reusable recipe for extracting both surface and kernel-level information from Coq. The paper's strengths include a concrete pipeline for intercepting Coq's elaboration stages, a domain-specific tokenizer that resolves naming ambiguities, and experiments spanning multiple Coq libraries. However, the central comparison to Graph2Tac is not yet robustly supported: the success metric appears to allow axiom-level shortcuts such as 'admit', the evaluation set size is reported inconsistently, and the clarity metric is self-judged by the same model family used for proving. The broad applicability claim is also weakened by an internal contradiction in the per-library results.

major comments (4)
  1. [§7.2.2, Table 2; Algorithm 1] The central comparison to Graph2Tac is unsound as reported because the paper never states that tactic candidates containing 'admit' or other axiom-level shortcuts (e.g., 'Admitted', 'Require Import Classical', or 'intuition' with classical axioms) are excluded from the success metric. In Algorithm 1, a tactic is accepted when 'COMPILE TACTIC' returns success and 'ISGOALCOMPLETE(S')' is true, but in Coq 'admit' discharges the current goal and is accepted by the compiler as a tactic, so such a result is not evidence that the theorem has been proved. If 'admit' occurred in any successful proof, the reported 45.8% success rate and the 12.6-point gain over Graph2Tac could be inflated. The authors must state explicitly that 'admit' and similar commands are forbidden, verify that all successful proofs are axiom-free in Coq's kernel, and rerun the evaluation if any axiom-level tactic was allowed.
  2. [Abstract vs. §7.2.1] The evaluation set is reported as 1,386 theorems in the abstract and in Section 7.4, but as 1,300 theorems in Section 7.2.1 and Table 2. This discrepancy matters because the main success rate is defined over this set; the authors should reconcile the numbers and state exactly how the random 10% sample of the Graph2Tac test set was filtered to produce the final set used for the headline comparison.
  3. [§4.2 and §7.2.3] The clarity score is produced and judged by DeepSeek-V3, the same model family used as the prover, and the 'Complete Information' condition supplies the very definitions the model is asked to reproduce. The near-perfect correlation in Table 3 therefore does not establish that clarity drives proving ability; it may partly reflect that the model can copy definitions verbatim from the prompt. Please add a human-calibrated evaluation or use a held-out judge from a different model family, and report inter-annotator agreement or a similarly direct validity check for the clarity score.
  4. [§7.2.1 and §7.4] The search parameters are stated inconsistently: Section 7.2.1 says 'Maximum retries R = 3' but immediately parenthesizes 'one retry with error feedback', while Table 8 compares Retry Count 0, 3, and 1. The reader cannot tell how many retries the main 45.8% result used. Please clarify the actual retry schedule and, if R=3 was used for the main result, explain how it is compatible with the 'one retry' comment and with Table 8's conclusion that a single retry is optimal.
minor comments (5)
  1. [§7.3, Table 5] The text states that 'Both DS+DS and fine-tuned approaches show gains across all library types,' but the iris row shows DS baseline 0.10 vs. DS+DS 0.09, a decrease. Please correct the claim or explain the discrepancy.
  2. [§7.4, Table 6] The sentence before Table 6 says the information-component analysis is performed on 1,386 theorems, while Table 7 and Table 8 use 78 theorems from Ceres; it is not clear which configuration produced the 48.6% success rate in the final row of Table 6. Please state explicitly which model and dataset each ablation row uses.
  3. [Algorithm 1] Line 55 uses the inconsistent function name 'ISSUBGOAL COMPLETE' with a space; this should be 'ISSUBGOALCOMPLETE' for consistency with line 51.
  4. [Introduction and Section 1] There are several typographical and grammatical errors, including 'truely', 'general purposed model', and 'specifical trained'; the manuscript needs a careful copyediting pass.
  5. [Reproducibility] The paper does not state whether the evaluation scripts, the extracted structured dataset, or the fine-tuned checkpoints will be released; providing these would substantially strengthen reproducibility, which is especially important given the discrepancies noted above.

Circularity Check

2 steps flagged · score 6.0 of 10

Clarity metric and the r=0.98 clarity–performance correlation reduce to the definition being placed in the prompt; the 45.8% vs 33.2% Graph2Tac comparison is external and not circular.

  1. self definitional [Section 4.1 'Experimental Design' with Section 5.3 'From Extracted Data to Structured Reasoning' (Table 1)]
    "For example, when evaluating clarity in the context of structured prompts(which include concepts like nat, plus, and eq), we might ask: 'Given the following structural proof context: [full structural prompt], please provide the strict Coq definition of the concept plus.' ... For each referenced concept(examples in Appendix B): Origin: Source code definition; Internal: Kernel representation; Intuition: Natural language explanation."

    In the Complete Information condition, the structured prompt already contains the queried concept's Origin/Internal/Intuition definitions. Asking the model to 'provide the strict Coq definition of plus' after that definition has been inserted into the prompt turns the clarity probe into a copying task. The improvement from 44.5% (No Context) to 82.3% (Complete Information) is therefore built into the input rather than evidence of an independent comprehension construct; the paper's claim that structured context improves clarity is true by construction.

  2. renaming known result [Section 7.2.3 'Clarity-Performance Correlation' (Table 3)]
    "The correlation analysis reveals a remarkably strong linear relationship (correlation coefficient r = 0.98) between understanding and reasoning performance. As understanding scores increase from 44.5% to 82.3%, theorem proving success rates improve proportionally from 21% to 45%."

    The five points in Table 3 are ordered by how much definitional content is placed in the prompt. That same single manipulation mechanically raises the clarity score (answer in prompt) and also helps proof search, so the near-perfect correlation is a monotone artifact of the experimental design. The paper renames 'adding definitions/context helps' as 'clarity improves reasoning' and presents this restatement as independent confirmation of its central hypothesis.

full rationale

The main Graph2Tac comparison is external and not circular: the 45.8% success rate is measured against an external baseline (Graph2Tac 33.2%) and does not depend on the clarity metric, so the theorem-proving performance claim retains independent content. However, the paper's conceptual contribution—that a measurable 'clarity' construct improves reasoning—is partially circular. The clarity score is defined by asking the model to reproduce definitions that the enriched prompt already contains, and the r=0.98 correlation then tracks the same manipulation that drives both the clarity score and proof success. The self-citations [38,39] are present but are not the load-bearing reduction; the reduction is the answer-in-prompt design of the clarity probe. The dataset-size discrepancy (1,386 vs 1,300) and the unstated exclusion of 'admit' are correctness/soundness concerns, not circularity, and do not change this score.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claim rests on a set of hand-chosen search hyperparameters and on three unvalidated assumptions: the LLM judge's reliability for clarity, the faithfulness of the extracted kernel representation, and the implicit exclusion of 'admit' from successful proofs. No new physical or mathematical entities are introduced; the named components such as 'clarity score', 'Planner', and 'Executor' are software constructs, not entities requiring independent evidence.

free parameters (5)
  • Maximum proof depth D = 15
    Chosen based on preliminary experiments, mentioned in Section 7.2.1. It bounds the search and affects the success rate.
  • Beam width B = 3
    Hand-chosen as a balance between exploration and compute, stated in Section 7.2.1.
  • Maximum retries R = 3
    One retry with error feedback, chosen from a small comparison in Section 7.4. The value affects the search's ability to recover from failures.
  • Number of tactic candidates per state = 10
    Used in the Executor's beam search, stated in Algorithm 1. A larger value would increase the search breadth.
  • Public notebook cache size = 15
    An arbitrary fixed size for the shared insights cache, mentioned in Section 6.2. The value is not derived from data.
assumptions (3)
  • domain assumption The LLM judge accurately evaluates the semantic correctness of Coq definitions.
    The clarity score in Section 4.2 is computed by asking DeepSeek-V3 'Is the following definition semantically correct?' and taking log-probabilities. No ground-truth verification against Coq's kernel is performed, so the metric could be systematically biased.
  • domain assumption The extracted Coq kernel representation is a faithful and complete semantic ground truth for disambiguation.
    The pipeline relies on intercepting Coq's internal type-inference output to generate 'Internal' representations. This assumes that the kernel terms always carry the intended meaning and that no necessary information is lost during extraction, as described in Appendix A.1.
  • ad hoc to paper Proofs that use 'admit' or similar axiom-level shortcuts are not counted as successes, even though no such exclusion is stated.
    Algorithm 1's termination condition 'ISGOALCOMPLETE' would be satisfied by any tactic that closes all goals, including 'admit'. The paper never states that 'admit' is forbidden during search, so the success metric is undefined without this implicit assumption.

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Cite this review

Pith. "Pith review of Clarifying Before Reasoning: A Coq Prover with Structural Context." pith.science (2026). https://pith.science/paper/7F4MJR56

@misc{pith2026250702541,
  author       = {Pith},
  title        = {Pith review of: Clarifying Before Reasoning: A Coq Prover with Structural Context},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7F4MJR56}},
  note         = {Machine review of arXiv:2507.02541}
}
abstract

In this work, we investigate whether improving task clarity can enhance reasoning ability of large language models, focusing on theorem proving in Coq. We introduce a concept-level metric to evaluate task clarity and show that adding structured semantic context to the standard input used by modern LLMs, leads to a 1.85$\times$ improvement in clarity score (44.5\%~$\rightarrow$~82.3\%). Using the general-purpose model \texttt{DeepSeek-V3}, our approach leads to a 2.1$\times$ improvement in proof success (21.8\%~$\rightarrow$~45.8\%) and outperforms the previous state-of-the-art \texttt{Graph2Tac} (33.2\%). We evaluate this on 1,386 theorems randomly sampled from 15 standard Coq packages, following the same evaluation protocol as \texttt{Graph2Tac}. Furthermore, fine-tuning smaller models on our structured data can achieve even higher performance (48.6\%). Our method uses selective concept unfolding to enrich task descriptions, and employs a Planner--Executor architecture. These findings highlight the value of structured task representations in bridging the gap between understanding and reasoning.

Figures

Figures reproduced from arXiv: 2507.02541 by the authors.

Figure 1
Figure 1. Coq proof script (left) and corresponding proof state for the double-even theorem (right) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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    info ": [

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    tactic

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