DreamProver: Evolving Transferable Lemma Libraries via a Wake-Sleep Theorem-Proving Agent

Chuqin Geng; Hangrui Bi; Jialiang Sun; Wenjie Ma; Xujie Si; Youyuan Zhang; Zhaoyu Li

arxiv: 2604.26311 · v1 · submitted 2026-04-29 · cs.AI

DreamProver: Evolving Transferable Lemma Libraries via a Wake-Sleep Theorem-Proving Agent

Youyuan Zhang , Jialiang Sun , Hangrui Bi , Chuqin Geng , Wenjie Ma , Zhaoyu Li , Xujie Si This is my paper

Reviewed by Pith2026-05-07 13:11 UTCgrok-4.3open to challenge →

classification cs.AI

keywords theorem provinglemma librarieswake-sleep learningformal verificationagentic systemsprogram inductionreusable lemmas

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The pith

DreamProver uses repeated wake-sleep cycles to evolve compact libraries of reusable lemmas that improve theorem proving on new problems.

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

The paper introduces an agent that alternates between attempting proofs with its current lemmas and proposing fresh candidates, then abstracts and consolidates those candidates into a smaller set of higher-level lemmas. This cycle repeats so the library grows more general and transferable rather than staying fixed or becoming overly specific to single theorems. A reader would care because such libraries could let automated provers handle wider ranges of mathematics with less manual engineering and lower compute.

Core claim

DreamProver runs a wake stage that tries to prove training theorems using the existing lemma library while emitting new candidate lemmas for unsolved cases, followed by a sleep stage that abstracts, refines, and merges the candidates into a compressed library of high-level, reusable lemmas. Iterating these stages produces a library that raises proof success rates on diverse benchmarks, shortens the proofs it finds, and lowers overall computation.

What carries the argument

The wake-sleep cycle that generates candidate lemmas during proof search and then compresses them into a compact, transferable library.

If this is right

Proof success rates rise across a range of mathematical benchmarks.
The proofs the system finds become shorter on average.
Computational cost per successful proof drops.
The same library works on theorems never seen in training.

Where Pith is reading between the lines

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

The same cycle might discover lemmas that correspond to recognized mathematical patterns rather than just local tricks.
The method could transfer to other formal tasks such as program verification or constraint solving.
Over many iterations the library might stabilize around a core set of lemmas that human mathematicians already treat as fundamental.

Load-bearing premise

Candidate lemmas proposed during proof attempts can be turned into a small set of high-level lemmas that still help prove theorems outside the original training set.

What would settle it

A controlled run on held-out theorems in which the final evolved library produces no gain in success rate, proof length, or runtime compared with a static baseline library.

Figures

Figures reproduced from arXiv: 2604.26311 by Chuqin Geng, Hangrui Bi, Jialiang Sun, Wenjie Ma, Xujie Si, Youyuan Zhang, Zhaoyu Li.

**Figure 1.** Figure 1: Overview of DreamProver as an iterative wake–sleep framework for learning view at source ↗

**Figure 2.** Figure 2: Total number of output tokens (millions) per sample on three well-represented view at source ↗

**Figure 3.** Figure 3: Proof length statistics on three well-represented domain benchmarks. view at source ↗

read the original abstract

We introduce DreamProver, an agentic framework that leverages a "wake-sleep" program induction paradigm to discover reusable lemmas for formal theorem proving. Existing approaches either rely on fixed lemma libraries, which limit adaptability, or synthesize highly specific intermediate lemmas tailored to individual theorems, thereby lacking generality. DreamProver addresses this gap through an iterative two-stage process. In the wake stage, DreamProver attempts to prove theorems from a training set using the current lemma library while proposing new candidate lemmas. In the "sleep" stage, it abstracts, refines, and consolidates these candidates to compress and optimize the library. Through this alternating cycle, DreamProver progressively evolves a compact set of high-level, transferable lemmas that can be effectively used to prove unseen theorems in related domains. Experimental results demonstrate that DreamProver substantially improves proof success rates across a diverse set of mathematical benchmarks, while also producing more concise proofs and reducing computational cost.

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.

Desk Editor's Note private letter to a colleague

The wake-sleep lemma evolution is a reasonable idea for closing the gap between fixed libraries and per-theorem synthesis, but the abstract supplies no numbers or transfer evidence to back the claimed gains.

read the letter

The main thing here is the iterative wake-sleep loop that proposes candidate lemmas during proof attempts and then abstracts and consolidates them into a smaller library meant to transfer. That framing directly targets the stated limitation of prior work, and the two-stage cycle is a clean way to operationalize library evolution without hand-crafted rules each time. The paper earns credit for spelling out why fixed libraries are too rigid and why one-off lemmas do not scale, then offering a single mechanism that tries to do both discovery and compression in alternation. That part reads as a genuine attempt to make progress on reusable strategies rather than another incremental search tweak. The experimental claims, however, sit on thin ground. The abstract says the method raises success rates, shortens proofs, and cuts cost across benchmarks, yet it gives no baselines, no success percentages, no ablation on the sleep stage, and no held-out transfer numbers. Without those, it is impossible to tell whether the gains come from better lemmas or simply from running more search steps. The stress-test point about the sleep stage is fair: the description of abstracting and consolidating candidates does not include a soundness argument or a utility metric that would show the resulting lemmas are used more on unseen theorems than on the training set. If the consolidation is mostly frequency-based or clustering, the library could easily memorize patterns from the training distribution instead of learning general ones. Minor implementation details like how lemmas are represented or how the prover is called in the wake phase are also missing from the summary, which makes it hard to judge reproducibility. This work is aimed at researchers in automated theorem proving who already follow library-learning or program-induction lines. Someone building on Lean or Isabelle tactics might find the wake-sleep template useful as a starting point even if the current results need tightening. It is worth sending to peer review because the core loop is well-motivated and the gap it names is real; referees can push for the missing metrics and transfer tests without the paper being fundamentally broken.

Referee Report

2 major / 1 minor

Summary. The paper introduces DreamProver, an agentic framework using a wake-sleep program induction paradigm to evolve reusable lemma libraries for formal theorem proving. In the wake stage, the system attempts to prove theorems from a training set using the current library while proposing new candidate lemmas; in the sleep stage, these candidates are abstracted, refined, and consolidated into a compact set of high-level, transferable lemmas. The central claim is that this iterative cycle produces lemmas that substantially improve proof success rates, yield more concise proofs, and reduce computational cost across diverse mathematical benchmarks.

Significance. If the results hold, the work would be significant for automated theorem proving by providing a mechanism to discover generalizable lemmas automatically, rather than relying on fixed libraries or overly specific per-theorem lemmas. The wake-sleep alternation is a natural extension of program induction ideas to formal reasoning and could scale to larger domains if the consolidation step reliably produces transferable lemmas.

major comments (2)

Sleep stage description: the consolidation process is described only at a high level as abstracting/refining/consolidating candidates, with no formal soundness argument for the abstraction step, no ablation isolating the sleep stage's contribution, and no transfer metric (e.g., lemma usage frequency on held-out vs. training theorems). Without these, it remains possible that any success-rate gains arise from increased search budget or domain overlap rather than evolved transferable lemmas.
Experimental results: the abstract asserts substantial improvements in success rates, conciseness, and cost but supplies no quantitative metrics, baselines, error bars, or experimental protocol. This prevents evaluation of whether the claimed gains are real or reproducible.

minor comments (1)

The abstract would be strengthened by including at least one key numerical result and a baseline comparison to convey the magnitude of the claimed improvements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. We address each major comment below, clarifying aspects of the work and committing to revisions that strengthen the presentation without altering the core contributions.

read point-by-point responses

Referee: Sleep stage description: the consolidation process is described only at a high level as abstracting/refining/consolidating candidates, with no formal soundness argument for the abstraction step, no ablation isolating the sleep stage's contribution, and no transfer metric (e.g., lemma usage frequency on held-out vs. training theorems). Without these, it remains possible that any success-rate gains arise from increased search budget or domain overlap rather than evolved transferable lemmas.

Authors: We appreciate the referee's emphasis on rigor for the sleep stage. The current manuscript presents the consolidation process at a conceptual level to highlight the overall wake-sleep framework. In the revised version, we will expand this section to include a formal soundness argument for the abstraction step, based on equivalence-preserving transformations that maintain logical validity. We will also add an ablation study comparing the full system against a variant without the sleep-stage consolidation, and introduce a transfer metric that reports lemma usage frequency on held-out theorems relative to the training set. These changes will directly address the possibility of confounding factors such as search budget or domain overlap. revision: yes
Referee: Experimental results: the abstract asserts substantial improvements in success rates, conciseness, and cost but supplies no quantitative metrics, baselines, error bars, or experimental protocol. This prevents evaluation of whether the claimed gains are real or reproducible.

Authors: The referee correctly notes that the abstract is qualitative. The full manuscript already contains the quantitative results, including success rates on multiple benchmarks, comparisons against baselines (e.g., theorem provers using fixed or per-theorem lemmas), error bars from repeated runs, and the complete experimental protocol, all reported in the Experiments section. To facilitate evaluation, we will revise the abstract to incorporate concise quantitative highlights drawn from those results while keeping the abstract within length limits. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes an iterative algorithmic process (wake stage for theorem proving and candidate lemma proposal; sleep stage for abstraction, refinement, and consolidation) whose central claims rest on experimental improvements in proof success rates, conciseness, and cost across benchmarks. No mathematical derivations, equations, or first-principles results are present that reduce to fitted parameters or self-referential definitions. The framework makes no load-bearing self-citations, uniqueness theorems, or ansatzes; outcomes are externally validated rather than forced by construction from inputs. This is the normal case of a self-contained empirical system description.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no details on specific free parameters, axioms, or invented entities; the approach builds on standard wake-sleep concepts from machine learning and existing theorem-proving infrastructure.

pith-pipeline@v0.9.0 · 5482 in / 1090 out tokens · 75750 ms · 2026-05-07T13:11:13.469763+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

URL https://proceedings.neurips.cc/paper_files/ paper/2024/file/1f0832859514e53a0e4f229fc9b3a4a2-Paper-Conference.pdf

doi: 10.52202/079017-0549. URL https://proceedings.neurips.cc/paper_files/ paper/2024/file/1f0832859514e53a0e4f229fc9b3a4a2-Paper-Conference.pdf. Chenyi Li, Ziyu Wang, Wanyi He, Yuxuan Wu, Shengyang Xu, and Zaiwen Wen. Formaliza- tion of complexity analysis of the first-order algorithms for convex optimization. arXiv preprint arXiv:2403.11437, 2024a. Yang...

work page doi:10.52202/079017-0549 2024
[2]

Jialin Lu, Kye Emond, Kaiyu Yang, Swarat Chaudhuri, Weiran Sun, and Wuyang Chen

URL https://openreview.net/forum?id=3UWuFoksGb. Jialin Lu, Kye Emond, Kaiyu Yang, Swarat Chaudhuri, Weiran Sun, and Wuyang Chen. Lean finder: Semantic search for mathlib that understands user intents. arXiv preprint arXiv:2510.15940, 2025. The mathlib Community. The Lean Mathematical Library. In Proceedings of the ACM SIGPLAN International Conference on C...

work page doi:10.52202/079017-0133 2025
[3]

Under review

+ 1 := by <proof> theorem Finset.card_eq_of_mem_powersetCard { α : Type*} {X : Finset α} {r : N} {s : Finset α} : s ∈ X.powersetCard r → s.card = r := by <proof> theorem Finset.card_compl_of_fintype { α : Type*} [Fintype α] (s : Finset α) : sc.card = Fintype.card α - s.card := by <proof> theorem card_perm_fixing_finset { α : Type*} [Fintype α] [DecidableE...

work page
[4]

Keep the description short, within 1 to 5 sentences

work page
[5]

Introduce the name if it is some well-known theorem or one of its variants

work page
[6]

Prompt for Cluster-based Lemma Abstraction You are a mathematician and an expert in Lean

Enclose the description in one pair of <description> tags. Prompt for Cluster-based Lemma Abstraction You are a mathematician and an expert in Lean. Below are a set of theorems that you proved: {theorem_section} Please propose new general theorems that are closely related to the above theorems. Propose only theorems in the {domain} domain. The proposed th...

work page
[7]

Do NOT add any definitions

Do NOT include any imports or open statements. Do NOT add any definitions. ONLY include the theorem statement.,→

work page
[8]

Use a separate Lean 4 ``lean`` block for each theorem

work page
[9]

:= by sorry

Do NOT prove any theorem. Do not include ":= by sorry" but only the statement

work page
[10]

Use distinct theorem names for the different theorems

Do NOT duplicate theorem names. Use distinct theorem names for the different theorems

work page
[11]

Do Not use general types, align types and number of variables with the ones in the provided theorems

work page
[12]

MAKE SURE that the proposed theorems are all syntactically and logically correct. 30

work page

[1] [1]

URL https://proceedings.neurips.cc/paper_files/ paper/2024/file/1f0832859514e53a0e4f229fc9b3a4a2-Paper-Conference.pdf

doi: 10.52202/079017-0549. URL https://proceedings.neurips.cc/paper_files/ paper/2024/file/1f0832859514e53a0e4f229fc9b3a4a2-Paper-Conference.pdf. Chenyi Li, Ziyu Wang, Wanyi He, Yuxuan Wu, Shengyang Xu, and Zaiwen Wen. Formaliza- tion of complexity analysis of the first-order algorithms for convex optimization. arXiv preprint arXiv:2403.11437, 2024a. Yang...

work page doi:10.52202/079017-0549 2024

[2] [2]

Jialin Lu, Kye Emond, Kaiyu Yang, Swarat Chaudhuri, Weiran Sun, and Wuyang Chen

URL https://openreview.net/forum?id=3UWuFoksGb. Jialin Lu, Kye Emond, Kaiyu Yang, Swarat Chaudhuri, Weiran Sun, and Wuyang Chen. Lean finder: Semantic search for mathlib that understands user intents. arXiv preprint arXiv:2510.15940, 2025. The mathlib Community. The Lean Mathematical Library. In Proceedings of the ACM SIGPLAN International Conference on C...

work page doi:10.52202/079017-0133 2025

[3] [3]

Under review

+ 1 := by <proof> theorem Finset.card_eq_of_mem_powersetCard { α : Type*} {X : Finset α} {r : N} {s : Finset α} : s ∈ X.powersetCard r → s.card = r := by <proof> theorem Finset.card_compl_of_fintype { α : Type*} [Fintype α] (s : Finset α) : sc.card = Fintype.card α - s.card := by <proof> theorem card_perm_fixing_finset { α : Type*} [Fintype α] [DecidableE...

work page

[4] [4]

Keep the description short, within 1 to 5 sentences

work page

[5] [5]

Introduce the name if it is some well-known theorem or one of its variants

work page

[6] [6]

Prompt for Cluster-based Lemma Abstraction You are a mathematician and an expert in Lean

Enclose the description in one pair of <description> tags. Prompt for Cluster-based Lemma Abstraction You are a mathematician and an expert in Lean. Below are a set of theorems that you proved: {theorem_section} Please propose new general theorems that are closely related to the above theorems. Propose only theorems in the {domain} domain. The proposed th...

work page

[7] [7]

Do NOT add any definitions

Do NOT include any imports or open statements. Do NOT add any definitions. ONLY include the theorem statement.,→

work page

[8] [8]

Use a separate Lean 4 ``lean`` block for each theorem

work page

[9] [9]

:= by sorry

Do NOT prove any theorem. Do not include ":= by sorry" but only the statement

work page

[10] [10]

Use distinct theorem names for the different theorems

Do NOT duplicate theorem names. Use distinct theorem names for the different theorems

work page

[11] [11]

Do Not use general types, align types and number of variables with the ones in the provided theorems

work page

[12] [12]

MAKE SURE that the proposed theorems are all syntactically and logically correct. 30

work page