Learning to Prove Theorems by Learning to Generate Theorems

The evolution of automated theorem proving (ATP) has transitioned from rule-based symbolic methods like resolution and SMT solvers to hybrid approaches incorporating machine learning. Early systems such as E prover and Vampire relied on handcrafted heuristics, but faced scalability issues. Recent developments include neural-guided proof search algorithms like Holophrasm, DeepHOL, and ASTactic, which leverage deep neural networks to predict proof steps and guide search. These systems have demonstrated promising results within limited formal systems like Metamath, HOL Light, and Coq. However, their dependence on large, manually curated datasets limits scalability. Formal systems like Metamath offer a simplified, highly expressive language with a single inference rule—substitution—making them ideal testbeds. Yet, extending neural proof methods to more complex systems remains a challenge due to their richer tactic languages and proof strategies.

The main challenge in automated theorem proving is the scarcity of large, high-quality proof datasets, which constrains the training of neural models. Existing methods heavily rely on human-annotated proofs, which are expensive and time-consuming to produce, especially for complex or abstract theories. This leads to limited generalization and slow progress in tackling new domains. Moreover, current models struggle to generate novel, meaningful theorems that can expand the proof corpus, thus limiting their ability to improve proof search heuristics. Addressing these issues requires methods to automatically generate valid, diverse, and human-like theorems and proofs, thereby augmenting training data and enhancing model robustness.

The core innovation of this work is the development of MetaGen—a neural network-based theorem and proof synthesizer that can generate high-quality, human-like theorems without extensive manual data. It combines proof tree grafting, a relevance network for selecting background theorems, and a substitution network for variable replacements, all trained via adversarial and reinforcement learning. Unlike prior work that solely imitates human proofs, MetaGen actively creates new theorems by sampling proof steps, guided by a discriminator that ensures similarity to human proofs. This approach enables scalable, automatic data augmentation, significantly expanding the training corpus. Additionally, the integration with Holophrasm proof search demonstrates practical improvements in proof success rates, validating the method’s effectiveness.

• �� Train a neural generator on existing proof trees to learn proof step operations, including inference rule application and theorem grafting. • Use a discriminator network to evaluate the similarity of generated theorems to human proofs, guiding adversarial training. • In scenarios lacking proof data, employ reinforcement learning with rewards based on theorem similarity scores, optimizing the generator via the Reinforce algorithm. • Incorporate a relevance network to select background theorems for proof step construction, and a substitution network to generate variable replacements, ensuring logical consistency. • Generate shallow proof trees and graft them with existing subtrees to produce longer, multi-step proofs. • Enforce constraints to avoid meaningless theorems by matching hypotheses to existing proof nodes, maintaining logical validity. • Iteratively sample, evaluate, and graft proof trees, expanding the synthetic theorem corpus for training the proof search model.

Experiments utilize the set.mm and iset.mm datasets, containing thousands of theorems formalized in Metamath. The models are trained with varying proportions of human proofs (0%, 10%, 100%) to evaluate the impact of synthetic data. Performance metrics include Top-1 accuracy, Mean Reciprocal Rank (MRR), and proof success rate. Baselines include random generation, imitation learning, and adversarial training variants. Hyperparameters such as proof tree depth, number of candidate proof trees, and discriminator thresholds are tuned for optimal results. Ablation studies compare different training strategies, demonstrating the superiority of adversarial and reinforcement learning approaches. The experiments confirm that synthetic theorem generation enhances proof search efficiency and success, especially in low-data regimes.

MetaGen-generated theorems significantly improve proof success metrics, with Top-1 accuracy reaching 53.20% and MRR 0.6474, outperforming baselines. The adversarially trained MetaGen-RL-Adv achieves the highest performance, confirming the importance of quality and diversity in synthetic data. The results show a consistent trend across datasets and proof complexities, validating the approach’s robustness. Notably, even with only 10% human proofs, the model benefits substantially from synthetic data, indicating strong data augmentation effects. The experiments also reveal that grafting longer proofs and constraining theorem invocation improve the logical coherence and practical utility of generated theorems.

The framework can be applied in formal verification of software and hardware, automated mathematical discovery, and AI reasoning systems. It enables the automatic expansion of proof libraries, reducing manual effort and accelerating proof development. The approach is adaptable to various formal systems, provided they have a well-defined inference rule set and proof structure. In industry, it can assist in verifying complex systems, generating formal specifications, and supporting theorem proving in safety-critical applications. Its ability to generate human-like theorems makes it suitable for educational tools and research in mathematical logic.

Current models are limited in handling highly abstract or advanced mathematical theories due to training data constraints. Adapting to systems with richer tactic languages (e.g., Coq, HOL) requires significant modifications. Computational costs for training and inference are high, especially for large proof trees, limiting real-time deployment. The generated theorems, while similar to human proofs, may lack deep conceptual insight, restricting their use in groundbreaking mathematical discovery. Future work must address these issues by improving model efficiency, expanding training data diversity, and integrating human feedback for better semantic coherence.