From Informal to Formal – Incorporating and Evaluating LLMs on Natural Language Requirements to Verifiable Formal Proofs

The research in AI-based formal mathematical reasoning has shown an unstoppable growth trend. These studies have excelled in mathematical competitions like IMO and have made significant progress. However, these studies intertwined multiple skills simultaneously—problem-solving, reasoning, and writing formal specifications—making it hard to precisely identify the LLMs’ strengths and weaknesses in each task. This paper focuses on formal verification, an immediate application scenario of formal reasoning, and breaks it down into sub-tasks. We constructed 18k high-quality instruction-response pairs across five mainstream formal specification languages (Coq, Lean4, Dafny, ACSL, and TLA+) in six tasks by distilling gpt-4o and evaluated against ten open-sourced LLMs, including recent popular DeepSeek-R1. We found that LLMs are good at writing proof segments when given either the code, or the detailed description of proof steps. Also, the fine-tuning brought about a nearly threefold improvement at most. Interestingly, we observed that fine-tuning with formal data also enhances mathematics, reasoning, and coding capabilities. Fine-tuned models are released to facilitate subsequent studies at https://huggingface.co/fm-universe.