Communication protocols often rely on stateful mechanisms to ensure certain security properties. For example, counters and timestamps can be used to ensure authentication, or the security of communication can depend on whether a particular key is registered to a server or it has been revoked. ProVerif, like other state of the art tools for protocol analysis, achieves good performance by converting a formal protocol specification into a set of Horn clauses, that represent a monotonically growing set of facts that a Dolev-Yao attacker can derive from the system. Since this set of facts is not state-dependent, the category of protocols of our interest cannot be precisely analysed by such tools, as they would report false attacks due to the over-approximation. In this paper we present Set-p, an extension of the Applied p-calculus that includes primitives for handling databases of objects, and propose a translation from Set-p into Horn clauses that employs the set-membership abstraction to capture the non-monotonicity of the state. Furthermore, we give a characterisation of authentication properties in terms of the set properties in the language, and prove the correctness of our approach. Finally we showcase our method with three examples, a simple authentication protocol based on counters, a key registration protocol, and a model of the Yubikey security device.