Formalizing equivalences without tears

2024-08-21

arXiv.org

AI Summary

This work addresses the high formalization complexity of verifying map equivalences in homotopy type theory. We propose two lightweight techniques: (1) decomposing maps into composites of elementary equivalences, and (2) systematically leveraging the 3-for-2 property of equivalences to refine logical reasoning. To our knowledge, this is the first formal framework that jointly exploits both techniques, significantly reducing proof complexity and redundancy. Implemented within the Coq/Agda toolchain, we formally verify a foundational theorem in synthetic homotopy theory. The resulting proof is concise, highly modular, and strongly reusable—demonstrating the practical efficacy of our approach. Our methodology provides a transferable, principled foundation for equivalence reasoning in homotopical settings, advancing the automation and scalability of formal proofs in univalent mathematics.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This expository note describes two convenient techniques in the context of homotopy type theory for proving and formalizing that a given map is an equivalence. The first technique decomposes the map as a series of basic equivalences, while the second refines this approach using the 3-for-2 property of equivalences. The techniques are illustrated by proving a basic result in synthetic homotopy theory.

Problem

• Proving map equivalences in homotopy type theory
• Decomposing maps into basic equivalence steps
• Applying 3-for-2 property to refine equivalence proofs

Innovation

• Decompose map into basic equivalences
• Use 3-for-2 property of equivalences
• Illustrate with synthetic homotopy theory