TLDR: ByteDance Seed AI4Math introduces Seed-Prover, an AI model using lemma-style proving and iterative refinement for automated theorem proving in Lean 4, and Seed-Geometry, a fast and powerful geometry reasoning engine. These systems achieved significant success, proving 5 out of 6 problems in IMO 2025 and setting new state-of-the-art benchmarks in formal mathematics and geometry problem-solving.
A new research paper introduces Seed-Prover and Seed-Geometry, two advanced AI systems developed by ByteDance Seed AI4Math, designed to significantly enhance automated mathematical reasoning and theorem proving. These systems have demonstrated remarkable capabilities, including successfully proving problems at the International Mathematical Olympiad (IMO) level.
Seed-Prover: A New Approach to Theorem Proving
Seed-Prover is a large language model specifically tailored for formal reasoning within the Lean 4 formal language. Unlike previous models that generate entire proofs in one go, Seed-Prover employs a unique “lemma-style proving” paradigm. This means it first generates useful intermediate lemmas (smaller, provable statements) before tackling the main theorem. This modular approach allows for clearer tracking of progress, independent compilation and storage of lemmas, and the ability to combine lemmas from different reasoning paths to solve more complex problems.
The system also features iterative proof refinement, where it continuously improves its proofs based on feedback from the Lean compiler, previously proved lemmas, and self-summarization. This iterative process allows Seed-Prover to fix syntax errors and refine initial proof sketches, sometimes completely altering its reasoning trajectory to find a solution. For challenging problems, Seed-Prover uses a “conjecture proposing” module, which explores a wide range of potential properties of a problem without committing to a single approach, similar to how human mathematicians might brainstorm. This broad exploration helps in discovering crucial properties needed for a proof.
Seed-Prover’s training involves a multi-stage, multi-task reinforcement learning approach, rewarding successful proofs and penalizing formatting errors. It incorporates diverse inputs during training, including natural language hints, proofs, similar lemmas, and compiler feedback, making it highly adaptable.
Seed-Geometry: Mastering Geometric Reasoning
Recognizing the limitations of Lean in geometry, the researchers also developed Seed-Geometry, a dedicated geometry reasoning engine. This system builds upon prior work but introduces significant upgrades. It features an extended domain-specific language that simplifies complex geometric constructions, making them more concise for AI processing. A key improvement is its extremely fast reasoning engine, rewritten in C++ for a roughly 100-fold speed increase compared to previous Python implementations. This speed is critical for deep searches within the reasoning process.
Seed-Geometry utilizes a high-performing large language model from the Seed family, extensively pre-trained on coding and mathematics datasets. When faced with a new problem, Seed-Geometry first attempts a direct solution. If unsuccessful, it initiates an extensive search process using beam search, generating proposals for auxiliary constructions that might lead to a proof. This distributed setup allows multiple GPU processes to work in parallel, significantly enhancing its efficiency and scalability.
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Performance and Impact
The combined power of Seed-Prover and Seed-Geometry led to impressive results. In the IMO 2025 contest, the systems successfully proved 5 out of 6 problems. Seed-Geometry solved one problem instantly, while Seed-Prover tackled others using its medium and heavy inference settings. Across a curated dataset of 155 past IMO problems, Seed-Prover achieved an overall success rate of 78.1%. It also saturated the MiniF2F benchmark, proving 99.6% of problems, and significantly outperformed previous state-of-the-art models on PutnamBench and CombiBench, demonstrating its capability across various mathematical domains, including algebra, number theory, and combinatorics.
Seed-Geometry has also set a new state-of-the-art in automated geometry problem solving, surpassing AlphaGeometry 2 by solving more problems in both standard IMO and the more challenging IMO shortlist benchmarks. For a deeper dive into the technical details, you can refer to the full research paper here.
This work represents a significant leap forward in automated mathematical reasoning, showcasing the effectiveness of integrating formal verification with advanced large language models. The ability of these systems to rapidly verify proofs in formal languages like Lean offers a cost-effective and reliable alternative to human experts, paving the way for tackling open conjectures in the future.
