A Typing System for the Linear Lambda-Calculus in de Bruijn Notation

2026-07-22

AI Summary

This work addresses the verification of linear lambda terms under de Bruijn notation by proposing a type system that obviates the need for explicit checks of variable usage counts. Inspired by Hodas and Miller’s resource consumption model, the system embeds resource management directly into its typing rules, ensuring that well-typed terms inherently satisfy linearity constraints. Building upon de Bruijn notation and resource-sensitive type theory, we formalize a complete set of type inference rules and prove that the system enjoys subject reduction, thereby guaranteeing type safety in linear computation. This approach substantially simplifies the verification of linearity while preserving theoretical rigor.

Abstract

We introduce a typing system that is particularly well suited for typing the linear lambda-calculus in de Bruijn notation. This typing discipline, which is reminiscent of Hodas' and Miller's model of resource consumption, guarantees that any well-typed term is linear without the need for an occurrence check. We then establish the subject reduction property.

Problem

• linear lambda-calculus
• de Bruijn notation
• typing system
• linearity
• type discipline

Innovation

• linear lambda-calculus
• de Bruijn notation
• typing system
• resource consumption
• subject reduction