Formalizing chip-firing and Riemann--Roch for graphs in Lean 4

Abstract

The Riemann--Roch theorem for graphs, due to Baker and Norine, is a foundational result establishing a powerful analogy between finite graphs and algebraic curves. We describe a complete formal proof of this theorem implemented in the Lean 4 theorem prover. Our formalization includes the existence and uniqueness of q-reduced divisors, a modified form of Dhar's burning algorithm, the bijection between acyclic orientations with unique source and maximal superstable configurations, and Clifford's theorem. We also include several challenges for future formalization.