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

2026-04-29

AI Summary

Existing theorem-proving approaches rely on static lemma libraries or generate task-specific lemmas with limited generalization, hindering cross-task transferability. This work proposes DreamProver, a novel framework introducing a “wake–sleep” iterative paradigm that integrates agent-based program induction with a lemma evolution mechanism. During the wake phase, concrete lemmas are discovered through proof attempts; in the subsequent sleep phase, these lemmas undergo abstraction, compression, and optimization to dynamically construct a compact yet expressive higher-order lemma library. This dual-phase process substantially improves cross-domain theorem-proving success rates while yielding more concise proofs and reducing computational overhead.

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.

Problem

• theorem proving
• lemma reuse
• transferability
• formal verification
• program induction

Innovation

• wake-sleep paradigm
• transferable lemmas
• program induction
• theorem proving
• lemma library evolution