[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"Vzeg51uBCs":3},"# DeGiorgi\n\nA large-scale Lean 4 formalization of elliptic De Giorgi-Nash-Moser theory, built on top of Mathlib.\n\nThis repository formalizes the De Giorgi/Moser weak-solution theory for uniformly elliptic divergence-form equations with general bounded measurable uniformly elliptic coefficients, proving local boundedness, weak Harnack, Harnack, and Hölder regularity. The development is fully formalized, sorry-free, and axiom-free beyond Lean and Mathlib, and is organized around the namespace `DeGiorgi`.\n\nSpanning about 56,000 lines of Lean 4, the project brings a major chapter of elliptic regularity into the formal setting in dimension `d ≥ 3`, with the formal statements presented in the normalized and rescaled forms natural for the proofs. \n\nThe code was written in a roughly two-week interval with extensive use of subscription large-language models (LLMs). The authors provided detailed proof blueprints and supervised the code generated by the LLMs, and intervened when they got stuck. \n\nTo our knowledge, this development represents the first proof-assistant formalization of Sobolev spaces built from weak derivatives at this level of generality, and the first large-scale formalization of a modern PDE regularity theorem.\n\n**Authors:**\n\n- Scott Armstrong — CNRS & Laboratoire Jacques-Louis Lions, Sorbonne Université and Courant Institute, School of Mathematics, Computing, and Data Science, New York University. (scottnarmstrong@gmail.com)\n- Julia Kempe — Courant Institute, School of Mathematics, Computing, and Data Science, New York University. (kempe@nyu.edu)\n\n## Main import\n\n```lean\nimport DeGiorgi\n```\n\nThe root module re-exports the main headline results from:\n\n```lean\nimport DeGiorgi.DeGiorgiTheory\n```\n\n## Headline theorems\n\nThe root import exposes theorem names including:\n- `linfty_subsolution_DeGiorgi_normalized`\n- `weak_harnack`\n- `weak_harnack_on_ball`\n- `harnack`\n- `harnack_of_homogeneousWeakSolution`\n- `holder_Moser`\n- `holder_Moser_of_homogeneousWeakSolution`\n\n## Companion document\n\nThe repository also includes\n`DeGiorgi_Lean_to_Tex.tex`, a machine-generated natural-language rendering of\nthe Lean development. It is intended as a reading companion to the formalized\nproofs in the Lean source files.\n\n## Build\n\nThis project uses the Lean compiler and dependency versions pinned in `lean-toolchain` and `lake-manifest.json`.\n\nAssuming `elan` and `lake` are installed:\n\n```bash\nlake exe cache get\nlake build\n```\n\nTo build the root import directly:\n\n```bash\nlake build DeGiorgi\n```\n\n## Repository layout\n\n- `DeGiorgi/` contains the Lean source files\n- `DeGiorgi.lean` is the root import\n- `DeGiorgi_Lean_to_Tex.tex` is a machine-generated natural-language companion\n  to the Lean development\n- `manifest.json` records the main extracted theorem targets\n\n## License\n\nThis project is released under the Apache 2.0 license. See [`LICENSE`](LICENSE) for details.\n\n## Acknowledgements\n\nS.A. acknowledges support from the European Research Council (ERC) under the European Union's Horizon Europe research and innovation programme, grant agreement number 101200828.\n\nJ.K. thanks the Simons Foundation for support through the Collaborative Grant “The Physics of Learning and Neural Computation”.\n\n## Citation\n\nIf you use this development, please cite the accompanying paper:  \n**Formalization of De Giorgi--Nash--Moser Theory in Lean**  \nScott Armstrong and Julia Kempe  \n[arXiv:2604.05984](https://arxiv.org/abs/2604.05984)\n\nThe software repository can be cited using the metadata in [`CITATION.cff`](CITATION.cff).\n",1780113382231]