Today's Internet services are often expected to stay available and render high responsiveness even in the face of site crashes and network partitions. Theoretical results state that causal consistency is one of the strongest consistency guarantees that is possible under these requirements, and many practical systems provide causally consistent key-value stores. In this paper, we present modular verification of causal consistency for replicated key-value store algorithms and their client programs. Specifically, we formulate separate correctness conditions for key-value store algorithms and for their clients. The interface between the two is a novel operational semantics for causal consistency. We have verified the causal consistency of two key-value store algorithms from the literature using a novel proof technique. We have also implemented a simple automatic model checker for the correctness of client programs. The two independently verified results for the algorithms and clients can be composed to conclude correctness of any of the programs when executed with any of the algorithms. We have developed and checked our framework in Coq, extracted it to OCaml, and built executable stores.