level: research
a mathematician and an ai system collaborated to formalize a result about the nonlinear vlasov equation in the lean 4 proof assistant. the work was framed as a formalization game, where the goal was to turn a latex document into a complete lean development. winning required the code to compile without errors, contain no sorry statements, and have all target theorems verified to rest only on lean's foundational axioms. a secondary check measured reusability: whether the development formed a self-contained layer of general mathematics that could be absorbed by the wider library.
the case study covered well-posedness for the nonlinear vlasov equation using dobrushin's mean-field approach. this included proofs of existence, uniqueness, a stability estimate, the mean-field limit, and a short-window superposition principle showing weak solutions are lagrangian. the human's role was to direct the ai, not to write proofs. they scoped definitions, steered proof decompositions, and identified gaps in the library that needed filling.
the approach treats formalization as a strategic interaction between human and machine. the human sets high-level goals and makes decisions about structure, while the ai handles the detailed proof steps. this division of labor aims to make formal verification more efficient and accessible. the resulting formalization is axiom-clean, meaning it does not rely on any unproven assumptions, and is designed to be reusable for other projects in analysis and partial differential equations.
why it matters: this method could speed up the formal verification of complex mathematical results, making ai-assisted proofs more practical for researchers in analysis and related fields.