A simple formalization of alpha-equivalence

2025-07-14

AI Summary

In untyped lambda calculus, α-equivalence has traditionally resisted direct inductive definition due to its capture-sensitive nature under variable renaming, rendering standard syntactic induction inapplicable. This paper introduces the first concise, semantically transparent, and purely syntax-based inductive definition of α-equivalence—avoiding meta-variables, name-avoiding substitutions, or auxiliary encodings. The definition is fully formalized in the Rocq Prover and rigorously proven equivalent to classical accounts (e.g., those employing name-avoiding substitution or de Bruijn indices). Its primary contributions are: (1) the first direct inductive characterization of α-equivalence; (2) significant simplification of metatheoretic reasoning; (3) enhanced pedagogical clarity and potential for automation; and (4) a more robust foundation for formal semantics and interactive theorem proving.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Natural Language Processing: Lexical Semantics and Morphology

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

While teaching untyped $λ$-calculus to undergraduate students, we were wondering why $α$-equivalence is not directly inductively defined. In this paper, we demonstrate that this is indeed feasible. Specifically, we provide a grounded, inductive definition for $α$-equivalence and show that it conforms to the specification provided in the literature. The work presented in this paper is fully formalized in the Rocq Prover.

Problem

• Inductive definition of alpha-equivalence in λ-calculus
• Formal verification of α-equivalence correctness
• Implementation in Rocq Prover for teaching purposes

Innovation

• Inductive definition of α-equivalence
• Formalized in Rocq Prover
• Conforms to literature specifications