M2F: Automated Formalization of Mathematical Literature at Scale

2026-02-18

AI Summary

Existing approaches to automatic formalization struggle to scale to entire mathematical textbooks due to challenges such as cross-file dependencies, import resolution, and end-to-end compilation. This work proposes the M2F framework, the first system capable of project-scale automatic formalization of mathematical literature. M2F operates in two stages: first, it constructs compilable theorem skeletons by performing dependency-aware ordering and declaration repair; second, it completes proofs through goal-conditioned local editing, iteratively refined via closed-loop feedback from the Lean proof checker. Applied to a 479-page textbook on real and convex analysis, M2F generated 153,853 lines of Lean code within three weeks, achieving a proof completion rate of 96%—substantially outperforming both the 80% baseline and manual formalization efforts in efficiency.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Automated formalization of mathematics enables mechanical verification but remains limited to isolated theorems and short snippets. Scaling to textbooks and research papers is largely unaddressed, as it requires managing cross-file dependencies, resolving imports, and ensuring that entire projects compile end-to-end. We present M2F (Math-to-Formal), the first agentic framework for end-to-end, project-scale autoformalization in Lean. The framework operates in two stages. The statement compilation stage splits the document into atomic blocks, orders them via inferred dependencies, and repairs declaration skeletons until the project compiles, allowing placeholders in proofs. The proof repair stage closes these holes under fixed signatures using goal-conditioned local edits. Throughout both stages, M2F keeps the verifier in the loop, committing edits only when toolchain feedback confirms improvement. In approximately three weeks, M2F converts long-form mathematical sources into a project-scale Lean library of 153,853 lines from 479 pages textbooks on real analysis and convex analysis, fully formalized as Lean declarations with accompanying proofs. This represents textbook-scale formalization at a pace that would typically require months or years of expert effort. On FATE-H, we achieve $96\%$ proof success (vs.\ $80\%$ for a strong baseline). Together, these results demonstrate that practical, large-scale automated formalization of mathematical literature is within reach. The full generated Lean code from our runs is available at https://github.com/optsuite/ReasBook.git.

Problem

• automated formalization
• mathematical literature
• project-scale
• mechanical verification
• end-to-end compilation

Innovation

• autoformalization
• agentic framework
• project-scale formalization
• Lean
• verifier-in-the-loop