[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"kd79Ipe8mW":3},"# LiveLeanTriathlon\n\n[![License: Apache 2.0](https://img.shields.io/badge/License-Apache_2.0-lightblue.svg)](https://opensource.org/licenses/Apache-2.0)\n\nThis repository contains a benchmark suite of mathematical theorems\nformalized in Lean 4.\n\nIt has been adapted from the [Lean Project Template repository](https://github.com/pitmonticone/LeanProject).\n\n*Note: This version of the repository has been stripped of the informal and formal proofs to avoid data leakage. Full version is available upon request.*\n\n## Overview\n\nLiveLeanTriathlon is a benchmark\nfor automated theorem proving and autoformalization in Lean 4.\nIt consists of a collection of mathematical theorems formalized in Lean 4,\nand formalizations of supporting lemmas used in the proofs of these theorems,\nalong with informal descriptions of these theorems, lemmas, and their proofs\nin the form of LaTeX blueprints.\nThese theorems come from a variety of sources,\nincluding the [1000+ theorems project](https://1000-plus.github.io/),\nextensions of projects by individual contributors,\nand other mathematical literature. They cover many different mathematical fields.\n\nLiveLeanTriathlon has a few goals:\n\n* To assess the capabilities of automated theorem proving systems\n  in Lean 4 on a set of theorems that is more diverse than previous benchmarks\n  (which largely focus on competition math) and hopefully more similar to research level mathematics.\n* To provide a testbed for autoformalization systems that can take informal descriptions of theorems\n  and lemmas and produce formal statements in Lean 4.\n* To explore ways of using human-AI collaboration to assist larger scale formalization projects.\n\nEventually, we hope that LiveLeanTriathlon can help contribute to upstream formalization efforts like\nmathlib directly by providing first drafts of formal proofs of important theorems and lemmas\nthat the community is interested in.\n\n### Repository Layout\n\nThe repository is organized as follows (listing the main folders and files):\n\n- ~~`blueprint/src/theorems/`: Contains the LaTeX blueprints for the theorems in the benchmark.~~\n- ~~`blueprint/src/content.tex`: The main LaTeX file that includes the individual theorem blueprints as `\\input`s.~~\n- `LiveLeanTriathlonSorry/\u003CTheoremName>/`: Each theorem has its own directory containing:\n  - `All.lean`: The sorried main theorem statement.\n  - ~~`MainTheorem.lean`: The main theorem statement and proof.~~\n  - ~~`BackgroundLemmas.lean`: Supporting lemmas needed for the main theorem.~~\n- `LiveLeanTriathlonSorry/Mathlib/`: Contains any lemmas that are needed but not present in mathlib.\n\n## Licensing\n\nCopyright 2025 Project Numina. All software is licensed under the Apache License,\nVersion 2.0 (Apache-2.0); you may not use this file except in compliance with\nthe Apache 2.0 license. You may obtain a copy of the Apache 2.0 license at:\nhttps://www.apache.org/licenses/LICENSE-2.0\n\nThe content may be based on third party sources and may in some cases include\nthird party content. The original source for each theorem is indicated by a\nURL within the source file. Third party content may be subject to different\nlicensing requirements. In particular:\n\n-   Material from Wikipedia articles and MathOverflow is released under the\n    Creative Commons Attribution-Share-Alike License 4.0.\n-   Material from The Stacks Project is released under the GNU Free\n    Documentation License.\n-   Material from arXiv is used under the licence applicable to the relevant\n    paper, as indicated at the URL within the source file.\n\nUnless required by applicable law or agreed to in writing, all software and\nmaterials distributed here under the Apache 2.0 license are distributed on an\n\"AS IS\" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or\nimplied. See the license for the specific language governing permissions and\nlimitations under the license.\n\n## Acknowledgements\n\nWe would like to thank Kim Morrison and Kevin Buzzard for identifying formalization errors in this development.\n",1782661966257]