TLDR: A new AI framework called Decoupled Reasoning and Proving (DRP) addresses the gap between informal reasoning and formal proving in Automated Theorem Proving (ATP). By separating high-level strategic reasoning (done by a general-purpose LLM Reasoner) from low-level proof generation (done by a specialized Prover), DRP overcomes limitations of tightly coupled systems. This approach successfully solved 5 previously unsolved International Mathematical Olympiad (IMO) problems, demonstrating a significant advance in automated reasoning for complex mathematics and highlighting how specialized provers can degrade general reasoning abilities.
Automated Theorem Proving (ATP) is a fundamental challenge in artificial intelligence, aiming to automatically generate formal proofs for mathematical or logical statements. While large language models (LLMs) have significantly advanced informal reasoning, a notable gap persists between their strong informal capabilities and their weaker formal proving performance. For instance, LLMs can achieve over 80% accuracy in informal solutions on benchmarks like PutnamBench, but the best formal provers struggle to reach even 8% success on the same problems.
Researchers argue that this disparity arises because current state-of-the-art provers, by tightly integrating reasoning and proving, are trained in ways that inadvertently penalize deep reasoning in favor of shallow, tactic-based strategies. This limitation prevents them from fully leveraging the powerful informal reasoning abilities of LLMs, especially when tackling exceptionally difficult mathematical challenges like those found in the International Mathematical Olympiad (IMO).
A Novel Decoupled Framework
To address this fundamental gap, a new framework called Decoupled Reasoning and Proving (DRP) has been proposed. This innovative approach separates high-level reasoning from low-level proof generation, utilizing two distinct, specialized models. A powerful, general-purpose LLM acts as a dedicated Reasoner, generating diverse and strategic subgoal lemmas. A separate, efficient ATP model serves as the Prover, rigorously verifying these lemmas.
This modular design liberates the model’s full reasoning potential, bypassing the pitfalls of traditional end-to-end training. The DRP pipeline involves three main stages: first, the Reasoner proposes formal statements of potential lemmas (subgoals); second, a Prover module verifies these proposed lemmas, filtering out unprovable ones; and finally, the Prover tackles the main theorem, guided by the set of verified lemmas.
Demonstrated Success on Challenging Problems
The DRP framework was evaluated on a challenging set of non-geometry IMO problems from 2000 to 2024. Remarkably, no prior open-source prover had reported success on any problem from this set. The decoupled framework successfully solved 5 of these problems: IMO 2000 Problem 2, IMO 2005 Problem 3, IMO 2011 Problem 3, IMO 2019 Problem 1, and IMO 2020 Problem 2. This demonstrates a significant step towards automated reasoning on exceptionally difficult mathematical challenges.
The research highlights that the Reasoner’s strategic decomposition for problems like IMO 2019 Problem 1 leads to a structured and principled solution, unlike the often brittle or degenerate behavior observed in prover-only models. The Reasoner identifies fundamental properties, uncovers additive structures, characterizes function forms, constrains parameters, and verifies candidate solutions, mirroring human mathematical insight.
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Addressing Degradation in Specialized Provers
A key finding of this work is that the prevailing reinforcement learning with verifiable rewards (RLVR) paradigm, while effective for optimizing success rates on specific ATP benchmarks, can inadvertently degrade a model’s intrinsic mathematical reasoning capabilities. Experiments comparing a specialized prover (Kimina-Prover) with its general-purpose base model (Qwen2.5-Math-7B-Instruct) on standard mathematical reasoning benchmarks (MATH and AIME) showed a marked decline in the prover’s performance. This supports the decoupled approach, preserving the full intellectual capacity of a dedicated reasoning model.
To foster future research, the team has released a comprehensive dataset of generated and verified lemmas for a wide range of IMO problems, available at their project website. This resource aims to offer novel perspectives for mathematicians and serve as a challenging new benchmark for the ATP community. For more details, you can read the full paper here.
