Lambda Congruences and Extensionality

TL;DR

This paper offers new formulations of lambda and extensional theories that avoid substitution and variable set concepts, clarifying the roles of alpha-renaming and eta-extensionality as properties of extensionality.

Contribution

It introduces alternative lambda and extensional theories that do not rely on substitution or variable sets, enhancing conceptual understanding.

Findings

Reformulation of lambda and extensional theories without substitution

Clarification of alpha-renaming and eta-extensionality roles

Better conceptual understanding of extensionality in lambda calculus

Abstract

In this work we provide alternative formulations of the concepts of lambda theory and extensional theory without introducing the notion of substitution and the sets of all, free and bound variables occurring in a term. We also clarify the actual role of $\alpha$-renaming and $\eta$-extensionality in the lambda calculus: both of them can be described as properties of extensionality for certain classes of terms.