[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84041-en":3,"doc-seo-84041-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84041,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Formalizing Scarf, Brouwer, and Nash in Lean","Formalizing in Lean 4 a complete combinatorial proof route from Scarf’s theorem to Brouwer’s fixed point theorem and to mixed Nash equilibrium existence in finite games. The development follows Ivanov’s indexed-order formulation, formalizes the room–door incidence structure and parity argument, instantiates Scarf on finite grids of the standard simplex, and applies compactness, continuity, and vanishing-diameter estimates to obtain a fixed point. It extends to finite products of simplices via an embedding–projection construction and proves Nash’s result using the Nash map, yielding BrouwerBench.","Formalizing Scarf, Brouwer, and Nash in Lean  \nYuwei Lyu 1 Kai Li 1  \narXiv :2607 .05987v 1 [ cs .LO] 7 Jul 2026  \nAbstract  \nWe 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, instantiatesthe theorem on finite grids of the standard simplex, and carries out the compactness and continuity argument needed to obtain a fixed point.  \nWe 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 Leangrounded benchmark for probing proof-structure understanding within this single formal development.  \n1. Introduction  \nBrouwer’s fixed point theorem is a fundamental existence result in topology (Brouwer, 1911) and a standard tool in analysis, economics, and game theory (Border, 1985) . A central application is Nash’s theorem for finite games (Nash, 1950): mixed strategy profiles form a finite product of simplices, and an appropriately defined continuous self-map has a fixed point exactly at a mixed Nash equilibrium. For formalization, it is useful not only to invoke Brouwer’s theorem as a black-box topological principle, but also to expose the finite combinatorial structure from which one can derive the fixed-point result. This makes the proof pipeline modular: the finite parity argument, the limiting construction, the product-of-simplices reduction, and the game-theoretic endpoint can be checked as separate for-  \n1Department of Mathematics, Xiamen University Malaysia, Sepang, Selangor, Malaysia. Correspondence to: Yuwei Lyu  \n\u003C[Lyuyuwei@gmail.com](Lyuyuwei@gmail.com)>, Kai Li \u003C[donk28510@gmail.com](donk28510@gmail.com)>.  \nProceedings of the 43 rd International Conference on Machine Learning, Seoul, South Korea. PMLR 306, 2026 . Copyright 2026 by the author(s) .  \nmal components.  \nThis paper formalizes such a route in Lean 4 . We start from Ivanov’s indexed-order formulation of Scarf’s combinatorial theorem (Scarf, 1982 ; Ivanov, 2019), where a finite set is equipped with indexed linear orders and a coloring. The proof builds a finite incidence structure of dominant sets, cells, rooms, and doors: outside doors are incident to one room, internal doors are incident to two rooms, and the resulting parity argument produces a colorful room. We then instantiate this theorem on increasingly fine grids of the standard simplex. The colorful rooms provide discrete approximations, while compactness, continuity, and vanishing-diameter estimates yield a Brouwer fixed point. Finally, we extend the standard-simplex theorem to finite products of simplices by an explicit embedding–projection construction and apply the product theorem to the Nash map for finite games.  \nThe formalization treats this route as a sequence of reusable proof components rather than only as a final fixed-point theorem. In particular, the room–door parity argument, the dominance estimates on simplex grids, and the embedding– projection reduction for products of simplices are exposed as named Lean definitions and lemmas. This makes the development inspectable as a structured proof artifact and allows the final Nash theorem to be traced back to finite combinatorial data.  \nThe development builds on Mathlib (The mathlib Community, 2020) and is written in Lean 4 (Moura & Ullrich, 2021) . Mathlib provides finite types, finite sets, real analysis, compactness, continuity, finite-dimensional spaces, and standard simplices. The development consists of 4,768 lines of Lean across five files, covering the Scarf combinatorics, the standard-simplex Brouwer proof, the productsimplex reduction, the Nash endpoint, and supporti","cbCaipIKWjrJS3Ar","https://ap.wps.com/l/cbCaipIKWjrJS3Ar","pdf",384861,2,1,18,"English","en",105,"# Introduction\n## Contributions\n# Related Work","[{\"question\":\"What core mathematical results does the Lean 4 development target?\",\"answer\":\"It connects Scarf’s combinatorial theorem to Brouwer’s fixed point theorem and then to the existence of mixed Nash equilibria for finite games.\"},{\"question\":\"How is Scarf’s theorem approached in the formalization?\",\"answer\":\"The development uses Ivanov’s indexed-order formulation, building a room–door incidence structure and applying a parity argument to obtain a colorful room.\"},{\"question\":\"How does the work handle products of simplices and the Nash equilibrium step?\",\"answer\":\"It extends from the standard simplex to finite products of simplices using an explicit embedding–projection construction, then applies the resulting product theorem to the Nash map to prove mixed Nash equilibrium 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core mathematical results does the Lean 4 development target?","Question",{"text":75,"@type":76},"It connects Scarf’s combinatorial theorem to Brouwer’s fixed point theorem and then to the existence of mixed Nash equilibria for finite games.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is Scarf’s theorem approached in the formalization?",{"text":80,"@type":76},"The development uses Ivanov’s indexed-order formulation, building a room–door incidence structure and applying a parity argument to obtain a colorful room.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the work handle products of simplices and the Nash equilibrium step?",{"text":84,"@type":76},"It extends from the standard simplex to finite products of simplices using an explicit embedding–projection construction, then applies the resulting product theorem to the Nash map to prove mixed Nash equilibrium 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