DreamProver: Evolving Transferable Lemma Libraries... | AI Research

DreamProver: Evolving Transferable Lemma Libraries via a Wake-Sleep Theorem-Proving Agent
DreamProver is an agentic framework designed to improve automated theorem proving by enabling AI to learn and reuse mathematical knowledge over time. While many existing systems treat every mathematical problem as a unique challenge, DreamProver mimics human mathematical practice by systematically building a library of reusable lemmas—intermediate proofs or "shortcuts"—that can be applied to solve future, unseen problems. By evolving this library through an iterative cycle, the system becomes more efficient and capable at solving complex mathematical tasks.

The Wake-Sleep Learning Cycle

The core of DreamProver is a two-stage "wake-sleep" process inspired by cognitive learning models. During the "wake" stage, the agent attempts to solve a set of training theorems. If it encounters a difficult problem, it uses recursive decomposition to break the theorem into smaller, manageable sub-problems. It then collects these successful intermediate steps as potential learning material.
In the "sleep" stage, the system processes these collected experiences. It clusters similar intermediate theorems based on their semantic meaning and abstracts them into generalized, reusable lemmas. To keep the library effective, the system prunes redundant or rarely used lemmas and verifies the new ones to ensure they are mathematically sound. This cycle allows the library to grow into a compact, high-quality repository of transferable knowledge.

Improving Efficiency and Performance

By relying on this evolving library rather than generating problem-specific solutions from scratch every time, DreamProver significantly outperforms existing state-of-the-art theorem provers. Experimental results across diverse fields—including number theory, inequalities, and combinatorics—show that DreamProver improves proof success rates by an average of 61% compared to previous agentic systems.
Beyond just solving more problems, the system also improves the quality of the proofs themselves. Because the agent can utilize its library of lemmas, it produces proofs that are more concise, reducing proof length by an average of 50%. This approach also lowers the computational cost, as the system requires 48% fewer tokens to generate its solutions, making it a more efficient tool for complex mathematical reasoning.

Key Advantages of the Approach

DreamProver addresses a major limitation in current AI theorem proving: the lack of persistent, transferable knowledge. By moving away from instance-specific proof generation, the framework creates a system that learns from its own history. The "forgetting" mechanism, which removes rarely used lemmas, ensures that the library remains compact and focused, preventing the system from becoming overwhelmed by irrelevant information. This combination of abstraction, verification, and strategic pruning allows DreamProver to maintain a high level of performance across both well-represented and underrepresented mathematical domains.