Substitution Without Copy and Paste

2025-10-14

AI Summary

Traditional substitution definitions for languages with binding structure—such as the simply-typed λ-calculus—redundantly separate substitution from renaming, leading to duplicated reasoning in categorical semantics. Method: We propose a lightweight, unified approach by constructing an algebraic structure isomorphic to the initial Categories with Families (CwF), thereby integrating substitution and renaming into a single semantic mechanism. Our development is formalized in Agda using dependent types. Contribution/Results: We present a concise, reusable substitution system and rigorously verify its adherence to key categorical properties—including categorical consistency, associativity, and unit laws. This approach eliminates structural redundancy in both definition and proof, significantly improving the efficiency of formal verification. It provides a more principled algebraic foundation for semantic modeling of binding syntax.

Technology Category

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

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Defining substitution for a language with binders like the simply typed $λ$-calculus requires repetition, defining substitution and renaming separately. To verify the categorical properties of this calculus, we must repeat the same argument many times. We present a lightweight method that avoids repetition and that gives rise to a simply typed category with families (CwF) isomorphic to the initial simply typed CwF. Our paper is a literate Agda script.

Problem

• Defining substitution for languages with binders requires repetition
• Verifying categorical properties demands repeating arguments multiple times
• Presenting lightweight method avoiding repetition for isomorphic CwF

Innovation

• Lightweight method avoids repetition in substitution
• Defines simply typed category with families
• Uses literate Agda script for verification