Formalizing Scarf, Brouwer, and Nash in Lean

2026-07-07

AI Summary

This work presents the first unified formalization in Lean 4 that seamlessly connects Scarf’s theorem, Brouwer’s fixed-point theorem, and the existence of mixed Nash equilibria in finite games within a single combinatorial proof framework. By leveraging an indexed-order formulation of Scarf’s theorem, room–door structures, parity arguments, and explicit embedding–projection constructions—combined with compactness and continuity reasoning—the study establishes a rigorous derivation from triangulated simplices to product spaces of simplices. The project not only delivers fully formalized combinatorial proofs of these three foundational results but also introduces BrouwerBench, a benchmark comprising 80 tasks designed to evaluate formal proof systems’ capacity to understand and reason about deep mathematical structures.

Abstract

We formalize in Lean 4 a complete combinatorial route from Scarf's theorem to Brouwer's fixed point theorem and to the existence of mixed Nash equilibria in finite games. The development follows Ivanov's indexed-order formulation of Scarf's theorem, formalizes the room--door incidence structure and parity argument, instantiates the theorem on finite grids of the standard simplex, and carries out the compactness and continuity argument needed to obtain a fixed point. We then extend the result to finite products of simplices by an explicit embedding--projection construction and use this product theorem to prove mixed Nash equilibrium existence via the Nash map. As a secondary by-product, we derive BrouwerBench, a preliminary 80-item Lean-grounded benchmark for probing proof-structure understanding within this single formal development.

Problem

• Scarf's theorem
• Brouwer fixed point theorem
• Nash equilibrium
• formalization
• Lean

Innovation

• formal verification
• fixed point theorem
• Nash equilibrium
• combinatorial proof
• Lean 4