Visored: A Controlled-Natural-Language Prover for LLM-Generated Mathematics

2026-06-16

AI Summary

This work addresses the semantic gap between mathematical proofs generated by large language models (LLMs) and formal verification systems such as Lean by introducing a novel prover grounded in dependent type theory. The proposed system integrates controlled natural language with rule-driven automated reasoning, enabling it to automatically complete routine inference steps and accurately translate verified proofs into Lean syntax—without requiring any specialized fine-tuning of the LLM. As the first formal verification framework to combine controlled natural language with automated reasoning, this approach significantly enhances the verifiability of LLM-generated proofs on the miniF2F benchmark, achieving a reliable end-to-end translation from informal natural language to formalized proofs.

Abstract

We present a dependent-type-based prover designed around the way LLMs (and humans) tend to write mathematics, complementing existing systems such as Lean and Rocq. Its core design choices are a surface that imitates mathematical natural language and a rule-driven automation layer that closes the routine steps a textbook would omit, so that an accepted proof can be re-emitted as a checked Lean file. Early experiments suggest that, even without any prover-specific training data, LLMs can learn to use it effectively on the miniF2F benchmark. Lean output excerpts: https://github.com/xiyuzhai-husky-lang/visored/

Problem

• controlled natural language
• LLM-generated mathematics
• dependent type
• theorem proving
• formal verification

Innovation

• controlled natural language
• dependent type
• LLM-generated mathematics
• automated theorem proving
• formal verification