Refutation of Sallé's Longstanding Conjecture

The ⁄-calculus possesses a strong notion of extensionality, called “the Ê-rule”, which has been the subject of many investigations. It is a longstanding open problem whether the equivalence obtained by closing the theory of Böhm trees under the Ê-rule is strictly included in Morris’s original observational theory, as conjectured by Sallé in the seventies. In a recent work, Breuvart et al. have shown that Morris’s theory satisfies the Ê-rule. In this paper we demonstrate that the two aforementioned theories actually coincide, thus disproving Sallé’s conjecture. 1998 ACM Subject Classification F.4.1 Mathematical Logic