theorem-prover formalization

Defining mathematical concepts formally remains a critical bottleneck in interactive theorem proving: steep learning curves hinder newcomers, and undergraduate-level formalization progresses slowly. This paper investigates the generality, readability, and type-system compatibility of definitions, using Lean’s mathlib as an empirical foundation. We systematically analyze hundreds of equivalent definitions across diverse mathematical domains, evaluating them via usability metrics—theorem verification success rate, proof conciseness, and interface orthogonality. We identify three key determinants of definition quality: abstraction level, constructive strength, and interface granularity; from these, we distill reusable design principles. Furthermore, we contrast definition strategies in computer algebra systems (CAS) and, for the first time, establish a cross-system formal definition design guide. Our framework significantly improves the efficiency of standardized knowledge construction and long-term collaborative sustainability in libraries such as mathlib.

Mathematical knowledge is fragmented across literature databases and formal proof repositories, lacking a unified interoperability mechanism. This work constructs a relational bridging database that systematically connects scholarly publications with formalized mathematical results—such as those in Lean’s mathlib—through cross-document alignment techniques, establishing the first large-scale linkage between the informal and formal mathematical ecosystems. The study introduces a novel metric, “paper-level formalization coverage,” and develops a scalable estimation framework to support comprehensive coverage analysis at scale. This infrastructure lays the foundation for a knowledge graph that seamlessly integrates academic publications with machine-verifiable proofs, enabling new avenues for discovery, validation, and synthesis in mathematical research.

Mathlib faces sustainability challenges—including escalating maintenance overhead, difficulty managing breaking changes, and declining collaboration efficiency—due to rapid growth in scale and complexity. To address these, we propose a systematic governance framework comprising: (1) a declarative deprecation mechanism enabling gradual API evolution; (2) a customized toolchain integrating static analysis (linters), dependency-aware contribution routing, compilation performance diagnostics, and automated refactoring; and (3) a modular library rearchitecture coupled with quantitative technical debt management. Experimental evaluation demonstrates that our approach reduces average compilation time by 37%, shortens PR review cycles by 42%, and achieves a deprecation migration success rate exceeding 95%. These results significantly enhance maintainability, collaborative throughput, and evolutionary resilience of large-scale formalized mathematics libraries.

This work addresses the lack of a systematic, formalized knowledge base for computer science in Lean, which has hindered its adoption in education, research, and large-scale verification. To bridge this gap, we present CSLib—the first open-source library of formalized theorems and data structures specifically designed for computer science, built upon the Lean proof assistant and dependent type theory. CSLib establishes a reusable and composable formal infrastructure that significantly expands Lean’s foundational knowledge base in computer science. By providing a comprehensive and extensible collection of verified components, CSLib enables effective collaboration between human developers and AI systems in constructing large-scale formally verified software, thereby advancing the broader application and accessibility of formal methods within the field.

To address the difficulty users face in retrieving theorems from mathlib4 due to unfamiliarity with naming conventions and documentation, this paper introduces LeanSearch—the first semantic search engine tailored for the Lean mathematical library. Methodologically: (1) we construct the first evaluable cross-lingual semantic search benchmark mapping natural-language queries to formal theorems; (2) we propose a joint encoding strategy for theorems and their associated docstrings to build a customized dense semantic index over mathlib4; and (3) we implement an end-to-end embedded retrieval system. Our contributions include establishing the first reproducible, evaluable semantic search infrastructure for mathlib4; deploying a publicly accessible service (leansearch.net); and achieving significant improvements in retrieval accuracy and onboarding experience for novice users—thereby facilitating collaborative formalization within the Lean community.

This work addresses the critical challenge of reliably integrating automated reasoning tools—such as theorem provers, SAT/SMT solvers, and termination analyzers—with proof assistants to build highly trustworthy systems. It presents a systematic survey and comparative analysis of two principal technical approaches: certification and formal verification. The study examines core methodologies including logical encoding, result replay and checking, and integration mechanisms within proof assistants. By elucidating the respective strengths and limitations of these methods and illustrating them through multiple successful case studies, the paper offers clear methodological guidance for constructing high-assurance automated reasoning systems, thereby substantially enhancing the verifiability and trustworthiness of their outputs.

This work addresses the challenge of subtle errors in mathematical reasoning by large language models through a novel multi-agent framework built upon general-purpose code-oriented large language models. The framework employs a coordinator to dynamically orchestrate a customized pipeline for automatically formalizing research-level mathematical theorems in Lean 4. Its key innovation lies in the ability to dynamically extend type definitions and verify auxiliary lemmas without introducing additional axioms. The approach successfully formalizes the core theorems of five STOC papers—two of which rely solely on the Lean kernel—and produces machine-verified proofs for 32 problems on PutnamBench. All formalizations have been expert-reviewed and are publicly released.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.

This work proposes a systematic formalization of all published mathematical knowledge into a machine-verifiable, continuously evolving structured knowledge base, addressing the challenges of scalability and organization. Taking dilatations of categories in categorical algebra as the first case study, the project integrates interactive theorem proving, dependent type theory, and category theory to frame the complete formalization of mathematics as a universal reasoning benchmark. By constructing a formal prototype of category dilatations, the study demonstrates the feasibility of this approach in expressing complex algebraic structures, thereby establishing both an architectural foundation and a practical pathway toward a large-scale, interconnected, and extensible database of formalized mathematics.

This work addresses the lack of effective evaluation benchmarks for large language models (LLMs) in formal theorem proving involving advanced mathematical reasoning, particularly in analysis. It introduces MA-ProofBench, the first dual-tier formal benchmark focused on mathematical analysis, comprising 200 theorems across six core topics—including measure theory, complex analysis, and functional analysis—organized into 27 subcategories and stratified by difficulty into undergraduate and doctoral qualifying levels. Through a human-led, LLM-assisted formalization pipeline and independent expert validation, the benchmark ensures mathematical fidelity. Systematic evaluation of leading models and provers reveals that even the strongest model (GPT-5.5) achieves only a 16% Pass@8 rate at the undergraduate level, dropping to 5% at the doctoral level, highlighting a substantial gap between informal and formal reasoning capabilities, primarily due to Mathlib hallucinations and incomplete proofs.