AI helps formalize Vlasov equation in Lean 4 as a 'strategy game'

Joseph K. Miller presents a novel approach to formalizing mathematical results in the Lean 4 proof assistant by framing the process as a strategy game where a mathematician directs an AI system. The objective: convert a LaTeX document into a complete, axiom-clean Lean development. The case study is the well-posedness of the nonlinear Vlasov equation via Dobrushin's mean-field route, covering existence, uniqueness, stability, mean-field limit, and a short-window superposition principle. The human's role was to define scope, steer decompositions, and identify missing library pieces, while the AI agent executed the proof writing. The final development contains 299 declarations, of which 49 (about a sixth) form a self-contained layer of optimal-transport machinery (Wasserstein-1 metric, Kantorovich-Rubinstein duality) that compiles against Mathlib alone via a 22-declaration interface with no reverse dependencies.

Key results: headline theorems were verified in about a week, the full development in about a month. The formalization is not intended as a general law but as an observed game outcome. The methodology is tool-agnostic: the game rules name no particular system, so the framing is designed to outlast any single AI assistant. Crucially, the mathematician remains the judge of whether the written statement matches the intended theorem. This work demonstrates a practical division of labor: humans handle high-level strategy and validation, AI handles the tactical proof steps. It also showcases how formalization can yield reusable mathematics—the separated library layer is a tangible contribution to Mathlib.