FormaTheoria: Constructing Large-Scale Lean Theories from Mathematical Literature $-$ Toward the Formalization of the Classification of Finite Simple Groups

2026-08-11

AI Summary

This work addresses core challenges in large-scale mathematical formalization—namely, implicit dependency discovery, correction of flaws in source literature, semantic fidelity, and cross-source inconsistencies. We present the first end-to-end AI-assisted workflow that integrates language model agents, structured review mechanisms, and explicit dependency management to collaboratively construct traceable, coherent, and verifiable formal theories from fragmented mathematical literature. The framework introduces several technical innovations, including tool invocation, context compression, review-gated termination, chapter-level modeling, and dependency-aware parallelization. As a demonstration, we successfully formalize key theorems underpinning the classification of finite simple groups in Lean, including the Feit–Thompson odd-order theorem, Glauberman’s $Z^*$-theorem, the Brauer–Suzuki theorem, and the Bender–Suzuki theorem, thereby establishing a deeply interconnected, machine-verified foundation for finite group theory.

Abstract

Large-scale formalization of advanced mathematics requires more than translating individual statements: it must reconstruct a coherent theory distributed across heterogeneous sources. This process raises four challenges: discovering implicit dependencies, correcting source defects, preserving semantic fidelity, and reconciling cross-source misalignments. We present FormaTheoria, an end-to-end, AI-assisted workflow that coordinates source acquisition, formalization, proof construction, recursive dependency discovery, independent review, and reconciliation, while preserving provenance and protecting approved declarations. A shared agent framework supports long-horizon execution through tool use, context compaction, review-gated termination, section-level source context, and dependency-aware batch parallelization. Applying FormaTheoria to major components of the Classification of Finite Simple Groups (CFSG), we construct a machine-checked Lean development extending through the Bender--Suzuki theorem and encompassing the Feit--Thompson Odd Order Theorem, Glauberman's $Z^*$ theorem, and the Brauer--Suzuki theorem. This development verifies an extensive body of deeply interdependent finite-group theory while providing a foundation for continuing the CFSG formalization. An empirical analysis of the code and recorded construction process supports the practical relevance of the identified challenges and illustrates the roles of the corresponding workflow components. Together, these results demonstrate how AI-assisted workflows can reconstruct mathematically significant formal theories from distributed literature by combining language-model agents with formal verification, structured review, and explicit dependency management.

Problem

• formalization
• mathematical literature
• dependency discovery
• semantic fidelity
• theory reconstruction

Innovation

• AI-assisted formalization
• dependency-aware workflow
• large-scale theorem proving
• mathematical literature reconstruction
• Lean formal verification