Structuring Definitions in Mathematical Libraries

2025-09-13

AI Summary

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.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Codifying mathematical theories in a proof assistant or computer algebra system is a challenging task, of which the most difficult part is, counterintuitively, structuring definitions. This results in a steep learning curve for new users and slow progress in formalizing even undergraduate level mathematics. There are many considerations one has to make, such as level of generality, readability, and ease of use in the type system, and there are typically multiple equivalent or related definitions from which to choose. Often, a definition that is ultimately selected for formalization is settled on after a lengthy trial and error process. This process involves testing potential definitions for usability by formalizing standard theorems about them, and weeding out the definitions that are unwieldy. Inclusion of a formal definition in a centralized community-run mathematical library is typically an indication that the definition is "good." For this reason, in this survey, we make some observations about what makes a definition "good," and examine several case studies of the refining process for definitions that have ultimately been added to the Lean Theorem Prover community-run mathematical library, mathlib. We observe that some of the difficulties are shared with the design of libraries for computer algebra systems, and give examples of related issues in that context.

Problem

• Structuring mathematical definitions in proof assistants
• Addressing steep learning curve for new users
• Selecting optimal definitions from multiple equivalent options

Innovation

• Structuring definitions in proof assistants
• Refining definitions through trial process
• Evaluating definitions via theorem formalization