TLDR: A new research paper introduces a method to significantly improve Large Language Models’ (LLMs) ability to perform complex theorem proving in Coq. By providing LLMs with structured semantic context, which includes internal type-theoretic information and a Planner-Executor architecture, the models’ understanding of mathematical tasks dramatically increases. This enhanced clarity leads to a substantial improvement in theorem proving success rates, outperforming previous state-of-the-art methods and demonstrating a strong correlation between task understanding and reasoning performance.
Large Language Models (LLMs) have shown remarkable capabilities in various domains, but when it comes to complex tasks like theorem proving in formal systems such as Coq, they often struggle. A key reason for this struggle isn’t necessarily a lack of reasoning ability, but rather a fundamental misunderstanding of the task itself. Imagine trying to solve a complex math problem without fully grasping the definitions of the symbols and concepts involved; LLMs face a similar challenge when presented with raw code without its underlying semantic context.
A new research paper, “Clarifying Before Reasoning: A Coq Prover with Structural Context,” introduces a novel approach to bridge this gap. The core idea is to significantly improve the clarity of the task for LLMs by providing them with structured semantic information, much like a human expert would understand the nuances of a mathematical proof. This goes beyond just the surface-level code and delves into the internal, type-theoretic information that Coq’s compiler uses.
The researchers developed a “Clarity Score” to quantitatively measure how well a model understands Coq concepts. They found that under standard conditions, LLMs only achieved about 44.5% understanding of basic mathematical concepts. This highlights a significant “semantic gap” where crucial information, such as implicit arguments, module aliases, and notation meanings, remains hidden from the models when they only process raw source code.
To address this, the paper proposes a sophisticated “Coq Data Processing Pipeline.” This pipeline intercepts Coq’s compilation process to extract rich, internal type-theoretic information. This includes detailed entity definitions (variables, functions, theorems), complete proof state transitions (hypotheses and goals before and after each step), and a domain-specific tokenizer that assigns unique identifiers to Coq entities, ensuring semantic consistency regardless of naming variations. By providing this structured context, the task transforms from parsing ambiguous syntax to reasoning with precise mathematical semantics.
Furthermore, the research introduces a “Planner-Executor Architecture.” This architecture leverages the enhanced clarity by separating high-level strategic reasoning (the Planner) from low-level tactic generation (the Executor). The Planner analyzes the structured proof state to identify core concepts, relevant theorems, and suitable proof techniques, while the Executor translates these strategies into concrete Coq tactics. This hierarchical decomposition allows for a more systematic and effective proof construction process.
The experimental results are compelling. The structured semantic approach dramatically improved the clarity score from 44.5% to 82.3%. More importantly, this enhanced understanding directly translated into superior theorem proving performance. The Planner-Executor system achieved a 45.8% success rate, significantly outperforming the previous state-of-the-art Graph2Tac (33.2%). The study also revealed a strong positive correlation (r = 0.98) between clarity scores and theorem proving success rates, conclusively demonstrating that improving task understanding directly enhances reasoning ability.
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The findings suggest that for general-purpose LLMs like DeepSeek-V3, performance gains come not just from model scale, but crucially from principled task representations. The research also shows that fine-tuning smaller models on this structured data can achieve competitive results with significantly fewer parameters, offering a path towards more efficient and powerful theorem provers. This approach, while developed for formal mathematics, holds promise for other domains requiring precise reasoning, such as formal verification, clinical decision-making, and software engineering. You can read the full paper here: Clarifying Before Reasoning: A Coq Prover with Structural Context.
