# Degrees of extensionality in the theory of B\"ohm trees and Sall\'e's conjecture

## TL;DR

This paper proves that the equivalence of B"ohm trees under the omega-rule coincides with Morris's observational theory H+, resolving a long-standing conjecture and offering a new characterization of extensional equality.

## Contribution

It demonstrates that B-omega and H+ theories are equal, disproving Sallé's conjecture, and introduces a new bounded eta-expansion characterization of extensional equality.

## Key findings

- B-omega equals H+ theory, disproving Sallé's conjecture.
- New characterization of extensional equality using bounded eta-expansions.
- Provides a taxonomy of degrees of extensionality in B"ohm tree theory.

## Abstract

The main observational equivalences of the untyped lambda-calculus have been characterized in terms of extensional equalities between B\"ohm trees. It is well known that the lambda-theory H*, arising by taking as observables the head normal forms, equates two lambda-terms whenever their B\"ohm trees are equal up to countably many possibly infinite eta-expansions. Similarly, two lambda-terms are equal in Morris's original observational theory H+, generated by considering as observable the beta-normal forms, whenever their B\"ohm trees are equal up to countably many finite eta-expansions. The lambda-calculus also possesses a strong notion of extensionality called "the omega-rule", which has been the subject of many investigations. It is a longstanding open problem whether the equivalence B-omega obtained by closing the theory of B\"ohm trees under the omega-rule is strictly included in H+, as conjectured by Sall\'e in the seventies. In this paper we demonstrate that the two aforementioned theories actually coincide, thus disproving Sall\'e's conjecture. The proof technique we develop for proving the latter inclusion is general enough to provide as a byproduct a new characterization, based on bounded eta-expansions, of the least extensional equality between B\"ohm trees. Together, these results provide a taxonomy of the different degrees of extensionality in the theory of B\"ohm trees.