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REVIEW 3 major objections 6 minor 18 references

Natural Language Translation of Formal Proofs through Informalization of Proof Steps and Recursive Summarization along Proof Structure

T0 review · 3 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A method turns Lean formal proofs into natural-language proofs that capture 86% of the key points a human proof would contain, while keeping the logical structure intact.

desk verdict A useful informalization pipeline with a strong template effect, but the headline accuracy claim is supported only by a mention-based metric and is contradicted by a real mislabeling in a flagship example. read the letter →

arxiv 2509.09726 v1 pith:WUMASLPM submitted 2025-09-10 cs.CL

Seiji Hattori , Takuya Matsuzaki , Makoto Fujiwara This is my paper · ORCID
classification cs.CL
keywords formalprooftranslationinformalizationLeantheoremprovernaturallanguageproofsrecursivesummarizationlargemodelsstructuretactictemplates
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 tries to establish that machine-verified Lean proofs can be turned into natural-language proofs a human can read without losing the formal proof's logical content. The recipe is to have an LLM verbalize each tactic individually under strict templates, ground each step in natural-language descriptions of the premises it uses, and then summarize the resulting sentences from the bottom up along the proof's dependency tree. On 1,242 informalized steps across 38 formal proofs, the method was rated Correct 89.05% of the time when both templates and the premise library were used, and on 17 undergraduate textbook proofs the final summaries scored 0.857 for capturing key points versus 0.833 without recursive summarization. If this holds, formal proof libraries become a source of readable proof texts, and autoformalization gains a way to generate aligned training data without hand-written formal-natural pairs.

What carries the argument

The key object is the dependency tree of the proof, derived from Lean's abstract syntax tree: the root is the proposition being proved, and tactic steps that prove an intermediate goal declared by a 'have' become children of that declaration. Step-wise informalization is carried out by rule-based templates selected per tactic operation, with slots filled by the LLM using the pre- and post-proof states and premise-library explanations. The dependency tree then drives bottom-up recursive summarization, where each subtree's description is condensed into a sub-proof before being folded into the parent's summary.

What would settle it

Take the 17 generated proofs and have a mathematician mark every sentence that is not entailed by the preceding sentences; if any proof with a full key-point score contains an unwarranted inference, the claim that summaries preserve the formal proof's logical content is false.

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Extended reading notes

Core claim

The central claim is that a proof's logical content survives translation if each tactic is first rendered through a constrained slot-filling informalization grounded in the natural-language premises it references, and then condensed bottom-up along the goal-dependency tree. The method does not ask an LLM to translate whole proofs at once; it restricts each generation step to a small, semantically coherent unit. The evidence offered is that on 1,242 informalized steps from 38 formal proofs, 89.05% are judged Correct; on 17 formalized textbook proofs, recursive summarization yields a key-point recall score of 0.857 versus 0.833 without recursion. The recursive outputs showed no significant log

Load-bearing premise

The dependency tree obtained from Lean's syntax is a faithful decomposition of the proof into units a human would also see; if it groups tactics differently, the recursive summaries can be fluent yet logically off while key-point scoring still looks good.

Editorial extensions

If this is right

  • Formal proofs written without any accompanying human proof, such as those in Mathlib, can be turned into readable natural-language proofs, as illustrated by the generated proof of the infinitude of primes.
  • Recursive summarization along the proof structure is what keeps longer proofs faithful: without it, four of seventeen generated proofs contained extra reasoning or logical inconsistencies, while none of the recursive ones did.
  • The template-plus-premise-library combination is necessary for step-level accuracy: removing templates drops correct step informalization from over 80% to roughly 54%, and the premise library only helps when templates are present.
  • The generated natural-language proofs could serve as paired data for training autoformalization models, addressing the data scarcity the paper identifies.
  • Scaling the method to the whole Mathlib corpus is currently limited by the manual construction of tactic templates and few-shot examples, a bottleneck the paper explicitly acknowledges.

Reading between the lines

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

  • The 0.857 key-point score likely understates the advantage of recursive summarization on longer proofs, since the seventeen evaluation proofs are short and the criteria were drawn from a human proof rather than from an independent check of logical validity.
  • A natural, testable extension is to run the method in reverse: use its generated natural-language proofs as training pairs for autoformalization systems, closing the data loop the paper identifies as the main obstacle.
  • The dependency tree's reliability could be probed by flattening 'have' blocks in a formal proof and seeing whether the generated proof's logical coherence degrades even when key-point recall stays high.
  • The fixed templates impose a uniform prose style, so an open question the paper leaves implicit is whether the method can be adapted to produce proofs that mimic a particular author's exposition.
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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

3 major / 6 minor

Summary. The paper proposes an LLM-based pipeline for translating Lean 4 formal proofs into natural-language proofs. It builds a premise library of natural-language explanations of Mathlib definitions/theorems, informalizes each tactic step using manually designed templates and few-shot examples, constructs a dependency tree from Lean's AST, and then recursively summarizes the informalized steps along that tree. The method is evaluated on 17 proofs (plus 21 helper lemmas) that the authors formalized from an undergraduate calculus textbook, reporting 89.05% 'Correct' step-wise informalization with templates+premise library and a weighted key-point recall of 0.857 for recursive summarization. The paper also shows qualitative examples of translations of Mathlib proofs (e.g., infinitely many primes, Heron's formula, IMO 1972 Q5).

Significance. If the method works as described, it is a useful step toward readable auto-informalization and could supply paired formal/informal data for autoformalization training. The combination of template-constrained step informalization with structure-aware recursive summarization is sensible, and the public release of code and outputs is a strength. The use of machine-verified Lean proofs as source material anchors the pipeline in verified content. The ablation using McNemar's test gives some evidence that templates and premise library matter. However, the quantitative claims of 'accuracy' are weakened by the evaluation design: the summarization metric tests mention of key points rather than logical validity, and all scoring was done by the first author with criteria and data created in the same group. The paper's strongest evidence is therefore qualitative; the 'accurate' claim needs stronger support before the headline numbers can be accepted.

major comments (3)
  1. [§5.3, Appendix D, Appendix E.2] The summarization evaluation is a key-point recall test, not a logical-validity test. The rubric in Appendix D checks whether variables, sub-lemma statements, proof methods, and theorem references are mentioned; it does not check whether the reasoning is valid or whether variables are assigned correctly. Appendix E.2 provides a concrete failure: the generated Heron proof says 'Let a, b, and c be the lengths of the sides opposite points p1, p3, and p2, respectively,' but the formal proof defines a = dist p1 p2, b = dist p3 p2, c = dist p1 p3. This misassignment is a substantive mathematical error that the key-point rubric would not penalize, because it only checks that a, b, c are introduced and that the Law of Cosines is used. The additional claim in §5.3 that recursive summarization produced 'no logical breakdowns' is supported only by the first author's 'close inspection,' not by a mea
  2. [§4.3, §5.1–§5.3] The evaluation data and protocol are largely author-constructed in a closed loop. The 17 formal proofs were manually created by the authors 'according to the structure of the original proofs' (§5.1), with no independent check that the formalization preserves the textbook proof's content; the few-shot examples in §4.3 are extracted from these self-created proofs and Mathlib; the summarization criteria in §5.3 are manually set by the authors from the same original proofs; and the scoring is performed by the first author alone, with no inter-annotator agreement, no blind setup, and no second opinion on borderline cases. This makes the headline numbers (89.05%, 0.857) hard to interpret as objective evidence. The paper should report at least two independent annotators with agreement statistics, use held-out proofs not involved in template/few-shot development, and provide the annotation instr
  3. [§4.4–§4.5, Table 2] Recursive summarization's correctness depends on the dependency tree built from Lean's AST, but the tree is not validated as a faithful semantic decomposition of the proof. The paper says a 'have' tactic's subtree consists of the steps proving the intermediate goal (§4.4), yet for tactics that restructure goals or introduce local definitions (by_contra, rcases, set, let, calc chains, etc.) the attribution of proof steps to the correct subtree is not discussed or checked. If the tree mis-groups steps, the bottom-up summaries can be fluent but logically wrong while the key-point evaluation would still pass. Since the central claim is that recursive summarization preserves the logical content of the formal proof, this structural assumption is load-bearing. I recommend a validation experiment: for a set of proofs with varied tactic structure, compare the tree's segmentation against a human-p
minor comments (6)
  1. [§1, §5.2] Typographical errors: 'autformalization' in §1, 'allwing' in §5.2, 'supression' in §5.2. A proofreading pass is needed.
  2. [§5.3, Table 2] The three evaluation symbols in Table 2 are rendered as blank parentheses; use explicit labels (Correctly captured / Partially captured / Not captured). Also, the conclusion 'capture 86% of the key points' should be reported as a weighted score with partial credit; the raw fully-captured rate is 87/108 ≈ 80.6%.
  3. [§5.3, Table 2] The comparison 'with recursive' vs 'without recursive' reports only a 0.024 score difference with no significance test or per-proof breakdown. This is too weak to support the claim that recursive summarization is the cause of improved accuracy.
  4. [§5.1, §4.3] No sensitivity analysis is reported for the two free parameters (generation temperature 0.4 vs 1.0, number of few-shot examples). These may affect reproducibility.
  5. [§5.4, Figure 9] There is an apparent typo: 'n! + 1|1' should likely be 'n! + 1 ≠ 1' or 'p | 1'. The surrounding text suggests the intended mathematical statement is about divisibility by p or non-equality of n! + 1 to 1.
  6. [§4.1] The premise library's LLM-generated explanations are not checked for correctness. An incorrect premise explanation could silently propagate into the generated proof; a small validation set for premise descriptions would strengthen the pipeline.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pipeline and its evaluation are not defined in terms of the target outputs.

full rationale

The paper's claimed derivation is a concrete pipeline: Lean tactic information is extracted, step-wise informalizations are generated using manually prepared templates, a premise library, and few-shot examples; these informalizations are placed on a dependency tree extracted from Lean's AST; and sub-proofs are recursively summarized bottom-up. None of these steps is defined in terms of the final natural-language proof, and no parameter is fitted to the evaluation data. The evaluation uses manual classification of step-level outputs and a key-point recall score computed against criteria extracted from the original textbook proofs; these are measurements, not quantities forced by construction. There is no self-citation chain: the references are to external systems (Lean, LeanDojo, Herald), the textbook, and unrelated prior work, not to the authors' own earlier results. No uniqueness theorem is imported from the authors, and no ansatz is smuggled in through a citation. The premise library is generated by the same LLM used later, but that is a component-reuse choice, not a circular reduction. The appendix honestly documents limitations in template coverage and in summarizing simple proofs, which are practical limitations rather than circular steps. The Heron's formula example in Appendix E.2 contains a genuine side-labeling error, and the key-point metric does not check logical validity; however, that is a correctness/validity weakness of the evaluation, not evidence that the derivation reduces to its inputs. The central method remains an implemented translation system whose outputs are compared against human-authored proofs, so the paper does not exhibit definitional, fitted, or self-citation circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The ledger is light on mathematical free parameters because the paper makes no mathematical derivation, its claim is empirical. Hand-chosen hyperparameters (temperatures, few-shot counts) and the scoring convention (0.5 for partial capture) directly affect the reported headline numbers. The load-bearing assumptions are all domain assumptions about the fidelity of the pipeline's representations (AST tree semantics, template coverage, LLM-generated premise library accuracy, author-built formal proofs as ground truth), each explicit or implicit in Sections 4.1-4.5, 5.1, and Appendix A.1.

free parameters (3)
  • LLM sampling temperature = 0.4 (informalization), 1.0 (summarization)
    Hand-chosen generation settings (Section 5.1); no sensitivity analysis reported, yet they affect output determinism and the reported quality.
  • Few-shot example count = 3 per task
    Three manually created examples per prompt type (Sections 4.1, 4.3, 4.5); chosen by hand without reported experiments on count.
  • Partial-capture weight in summary score = 0.5
    The summarization score weights 'partially captured' items at 0.5 (Section 5.3); this scoring convention directly determines the headline 0.857 figure.
assumptions (4)
  • domain assumption Every tactic used in the target corpora can be covered by a finite set of hand-written templates with slot types.
    Section 4.3 and Appendix A.1: the method 'only becomes effective when a unique template is prepared for every possible operation executable by a tactic,' acknowledged as extremely time-consuming.
  • domain assumption The AST-derived dependency tree (root = top proposition; children of a 'have' step = tactics proving that goal) is a faithful decomposition of the proof's logical structure for summarization.
    Sections 4.4-4.5; no validation that tree semantics match human proof organization, especially for goal-restructuring tactics.
  • domain assumption LLM-generated premise-library descriptions of Mathlib theorems and definitions are accurate and complete.
    Section 4.1: explanations are generated by the same LLM used downstream, with no independent verification step; errors would propagate into informalization.
  • ad hoc to paper The 17 manually formalized textbook proofs faithfully preserve the structure and content of the original Japanese calculus proofs.
    Section 5.1: 'manually formalized according to the structure of the original proofs.' These author-built formal proofs are the substrate of both evaluations, and the original natural-language proofs are not quoted in the paper.

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Cite this review

Pith. "Pith review of Natural Language Translation of Formal Proofs through Informalization of Proof Steps and Recursive Summarization along Proof Structure." pith.science (2026). https://pith.science/paper/WUMASLPM

@misc{pith2026250909726,
  author       = {Pith},
  title        = {Pith review of: Natural Language Translation of Formal Proofs through Informalization of Proof Steps and Recursive Summarization along Proof Structure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WUMASLPM}},
  note         = {Machine review of arXiv:2509.09726}
}
read the original abstract

This paper proposes a natural language translation method for machine-verifiable formal proofs that leverages the informalization (verbalization of formal language proof steps) and summarization capabilities of LLMs. For evaluation, it was applied to formal proof data created in accordance with natural language proofs taken from an undergraduate-level textbook, and the quality of the generated natural language proofs was analyzed in comparison with the original natural language proofs. Furthermore, we will demonstrate that this method can output highly readable and accurate natural language proofs by applying it to existing formal proof library of the Lean proof assistant.

Figures

Figures reproduced from arXiv: 2509.09726 by the authors.

Figure 1
Figure 1. Method Overview translated into a natural language proof step. It is generated by a hybrid method that combines a rule-based approach using templates prepared for each tactic type, and slot-filling by an LLM that extracts corresponding information based on slot types that specify the content to be included. Summarization of Proof Steps We generate a nat￾ural language proof by progressively summarizing the descriptio… view at source ↗
Figure 2
Figure 2. Formal proof of "the sum of two even numbers [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Informalize task input example We first extract a tactic applied in a proof step from the tactic information, as well as the proof state before and after its application (all expressed in the form of hypotheses ⊢ goal). Next, an appropriate generation template is re￾trieved. One or more template are manually pre￾pared for each tactic; each template corresponds to different usage of the tactic. Let’s take the rw tact… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Dependencies among proof steps for the proof steps of that proposition become the children of the proof step that declares the interme￾diate goal [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Evaluation Procedure summary of the whole proof. 5 Experiment 5.1 Experiment Settings For the step-wise informalization and summariza￾tion, we used OpenAI’s GPT-4.1-mini (gpt-4.1- mini-2025-04-14). In the informalization task, the generation temperature was set to 0.4 …
Figure 6
Figure 6. Figure 6: The proof of “inf{x ∈ R|x > 0} = 0“ gener￾ated with using recursive summarization. contrast, among the 17 outputs generated without recursive summarization, four included reasoning not in the original proof and/or contained substan￾tial logical inconsistencies. These i…
Figure 8
Figure 8. Figure 8: The formal proof of “there are infinitely many [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: The generated proof of “there are infinitely [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

Discussion (0). Continue with ORCID to comment.

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