On the Coincidence of Observational Equivalence and Labeled Bisimilarity in Applied Pi Calculus

However, this problem can be fixed by requiring active substitutions be defined on the base sort only (see, for instance, [2]). The purpose of this note is to supply a proof for the theorem. In the original semantics in [1], the use of structural equivalence introduces the nondeterminism, which makes it difficult to write a rigorous proof. To overcome the difficulty we shall use intermediate semantics, originally proposed in [2] for symbolic semantics, as a bridge. Intermediate representation is aim to simply the rules for structural equivalence as much as possible, while keep the bisimilarity. Four equivalences will be discussed: