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arxiv: 2410.04753 · v2 · pith:AZAG5ORXnew · submitted 2024-10-07 · 💻 cs.AI · cs.CL· cs.LG· cs.LO

ImProver: Agent-Based Automated Proof Optimization

Riyaz Ahuja , Jeremy Avigad , Prasad Tetali , Sean Welleck This is my paper

Pith reviewed 2026-05-23 19:47 UTC · model grok-4.3

classification 💻 cs.AI cs.CLcs.LGcs.LO
keywords automated proof optimizationLLM agentsLean theorem proverproof rewritingformal mathematicsChain-of-Stateserror correction
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The pith

An LLM agent rewrites Lean proofs to make them shorter, more modular, and more readable while remaining correct.

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

The paper defines automated proof optimization as the task of rewriting an existing correct proof in Lean to improve an arbitrary user-specified criterion such as length or readability. ImProver is presented as the first method for this task, an LLM-based agent that incorporates symbolic Lean context through a Chain-of-States technique plus error correction and retrieval steps. Experiments apply the agent to real undergraduate, competition, and research-level theorems and report that the resulting proofs are substantially shorter, more modular, and more readable. The work matters because human-written proofs are often not optimal for machine learning or specific downstream applications, and manual refinement is costly.

Core claim

ImProver shows that an LLM agent, equipped with a Chain-of-States technique that maintains symbolic Lean context, along with error-correction and retrieval, can rewrite proofs of real-world theorems so that they remain accepted by Lean yet become substantially shorter, more modular, and more readable.

What carries the argument

The ImProver agent, which uses a Chain-of-States technique to incorporate symbolic Lean context for guiding iterative proof rewrites.

If this is right

  • Proofs can be automatically adapted to particular styles or criteria without manual rewriting.
  • Training data for proof-generation models can be improved by replacing human proofs with optimized versions.
  • Formal libraries become easier to maintain and extend when proofs are made more modular.
  • Downstream tasks that consume formal proofs, such as explanation or extraction, gain from clearer structure.

Where Pith is reading between the lines

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

  • The same agent architecture could be ported to other interactive theorem provers if equivalent symbolic context can be extracted.
  • Repeated optimization passes might produce even shorter proofs if the process is iterated until no further gains occur.
  • Optimized proofs could serve as synthetic data to fine-tune smaller models for proof generation.

Load-bearing premise

An LLM agent can reliably generate rewrites that improve the chosen metric, stay logically equivalent to the original proof, and pass Lean checking without introducing undetected errors.

What would settle it

Apply ImProver to a held-out collection of theorems and verify whether every output proof type-checks in Lean and preserves the original statement when reviewed by a human expert.

Figures

Figures reproduced from arXiv: 2410.04753 by Jeremy Avigad, Prasad Tetali, Riyaz Ahuja, Sean Welleck.

Figure 1
Figure 1. Figure 1: ImProver automatically rewrites formal proofs to optimize a criterion such as length or readability while remaining correct. In this example, ImProver optimizes a human-written lemma (right) from the 2022 International Math Olympiad (Question 2, solution from Comp￾files (David Renshaw, 2024)) for length. ImProver’s optimized proof is correct and more concise. 2 RELATED WORK Recently there has been wide int… view at source ↗
Figure 2
Figure 2. Figure 2: A Lean proof (left) with Chain-of-States promptin [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimizing a group-theoretic result from MIL Chap [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Solving a group theorem exercise from MIL Chapter 8 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Optimizing a lemma from IMO 2019 P1 for readability [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Optimizing a lemma from the solutions of MIL CH08 S0 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Optimizing a lemma from MIL CH04 S01 solution for le [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Optimizing a theorem from Mathlib/FundamentalGroupoid/InducedMapsfor length After optimizing for readability, we see that the model did not change the structure of the proof. Rather, it added an intermediate declaration so that users can better understand the state after the convert. This intermediate tactic greatly helps in the understandability and clarity of the proof. Original (human-written) /-- Anoth… view at source ↗
Figure 9
Figure 9. Figure 9: Optimizing a theorem from Mathlib/FundamentalGroupoid/SimplyConnected for readability 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
read the original abstract

Large language models (LLMs) have been used to generate formal proofs of mathematical theorems in proofs assistants such as Lean. However, we often want to optimize a formal proof with respect to various criteria, depending on its downstream use. For example, we may want a proof to adhere to a certain style, or to be readable, concise, or modularly structured. Having suitably optimized proofs is also important for learning tasks, especially since human-written proofs may not optimal for that purpose. To this end, we study a new problem of automated proof optimization: rewriting a proof so that it is correct and optimizes for an arbitrary criterion, such as length or readability. As a first method for automated proof optimization, we present ImProver, a large-language-model agent that rewrites proofs to optimize arbitrary user-defined metrics in Lean. We find that naively applying LLMs to proof optimization falls short, and we incorporate various improvements into ImProver, such as the use of symbolic Lean context in a novel Chain-of-States technique, as well as error-correction and retrieval. We test ImProver on rewriting real-world undergraduate, competition, and research-level mathematics theorems, finding that ImProver is capable of rewriting proofs so that they are substantially shorter, more modular, and more readable.

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 / 1 minor

Summary. The paper introduces ImProver, an LLM-based agent system for automated proof optimization in Lean. It rewrites existing proofs to optimize arbitrary user-specified metrics (e.g., length, modularity, readability) while preserving logical correctness. The method augments naive LLM prompting with a Chain-of-States technique that maintains symbolic Lean context, plus error-correction and retrieval mechanisms. Experiments are reported on undergraduate, competition, and research-level theorems, with the central claim that ImProver produces substantially shorter, more modular, and more readable proofs than naive application.

Significance. If the empirical claims hold under rigorous evaluation, the work would be a useful contribution to formal mathematics tooling by demonstrating an agent architecture that can target downstream criteria beyond mere provability. The integration of symbolic context via Chain-of-States and the focus on optimization rather than generation alone are concrete strengths; however, the absence of quantitative metrics in the abstract leaves the practical impact difficult to gauge.

major comments (2)
  1. [Abstract] Abstract: the assertion that ImProver produces 'substantially shorter, more modular, and more readable' proofs is presented without any numerical results, baselines, or evaluation protocol. This is load-bearing for the central empirical claim and prevents assessment of whether the reported improvements are real or merely anecdotal.
  2. [Evaluation] Evaluation section (inferred from abstract description): no quantitative breakdown is supplied of failure modes, how logical equivalence is verified beyond Lean acceptance, or whether gains in modularity/readability survive human review or affect downstream tasks. This directly bears on the reliability assumption that the agent avoids subtle degradations.
minor comments (1)
  1. [Abstract] The abstract states that 'naively applying LLMs to proof optimization falls short' but does not specify the exact failure modes observed or the ablation that isolates the contribution of Chain-of-States versus error-correction versus retrieval.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their constructive comments, which highlight important aspects of clarity and evaluation rigor. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that ImProver produces 'substantially shorter, more modular, and more readable' proofs is presented without any numerical results, baselines, or evaluation protocol. This is load-bearing for the central empirical claim and prevents assessment of whether the reported improvements are real or merely anecdotal.

    Authors: We agree that the abstract would benefit from quantitative support to substantiate the claims. The body of the paper reports specific metrics from the experiments (including length reductions, modularity scores, and success rates relative to naive LLM baselines). We will revise the abstract to include key numerical results and a brief reference to the evaluation protocol. revision: yes

  2. Referee: [Evaluation] Evaluation section (inferred from abstract description): no quantitative breakdown is supplied of failure modes, how logical equivalence is verified beyond Lean acceptance, or whether gains in modularity/readability survive human review or affect downstream tasks. This directly bears on the reliability assumption that the agent avoids subtle degradations.

    Authors: The evaluation section details verification via successful Lean compilation (which enforces logical equivalence) and provides quantitative breakdowns of improvements along with failure mode analysis across the theorem sets. We will expand the failure mode discussion for greater transparency. However, the work does not include human review of readability or downstream task evaluations, which are acknowledged as limitations. revision: partial

standing simulated objections not resolved
  • Human review of readability gains and evaluation of effects on downstream tasks were not conducted in the current study.

Circularity Check

0 steps flagged

No circularity: empirical agent demonstration with no derivation chain

full rationale

The paper describes an LLM-based agent system (ImProver) for rewriting Lean proofs to optimize user metrics, evaluated empirically on undergraduate/competition/research theorems. No equations, first-principles derivations, fitted parameters presented as predictions, or load-bearing self-citations appear in the abstract or described structure. Claims rest on experimental outcomes (shorter/more modular proofs accepted by Lean) rather than any self-referential reduction. This is a standard empirical systems paper; the derivation chain is absent, so circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The system rests on the assumption that Lean correctly validates proof equivalence; no free parameters or new entities are introduced beyond standard LLM usage.

axioms (1)
  • domain assumption Lean theorem prover accurately determines whether a rewritten proof is logically valid and equivalent to the original.
    The entire optimization loop depends on Lean accepting the output as correct.

pith-pipeline@v0.9.0 · 5771 in / 1155 out tokens · 31042 ms · 2026-05-23T19:47:14.411351+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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  1. Lean Refactor: Multi-Objective Controllable Proof Optimization via Agentic Strategy Search

    cs.LO 2026-05 unverdicted novelty 6.0

    Lean Refactor uses retrieval from a curated multi-objective strategy database to guide frozen LLMs in refactoring Lean proofs, reporting over 70% token compression on benchmarks and improved version transfer.

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