The Global Sequence Protocol (GSP) is an operational model for replicated data stores, in which updates propagate asynchronously. We introduce the GSP-calculus as a formal model for GSP. We give a formal account for its proposed implementation, which addresses communication failures and compact representation of data, and use simulation to prove that the implementation is correct. Then, we use the GSP-calculus to reason about execution histories and prove ordering guarantees, such as read my writes, monotonic reads, causality and consistent prefix. We also prove that GSP extended with synchronous updates provides strong consistency guarantees.