Distributed systems adopt weak consistency to ensure high availability and low latency, but state convergence is hard to guarantee due to conflicts. Experts carefully design replicated data types (RDTs) that resemble sequential data types and embed conflict resolution mechanisms that ensure convergence. Designing RDTs is challenging as their correctness depends on subtleties such as the ordering of concurrent operations. Currently, researchers manually verify RDTs, either by paper proofs or using proof assistants. Unfortunately, paper proofs are subject to reasoning flaws and mechanized proofs verify a formalization instead of a real-world implementation. Furthermore, writing mechanized proofs is reserved to verification experts and is extremely time-consuming. To simplify the design, implementation, and verification of RDTs, we propose VeriFx, a specialized programming language for RDTs with \emph{automated} proof capabilities. VeriFx lets programmers implement RDTs atop functional collections and express correctness properties that are verified automatically. Verified RDTs can be transpiled to mainstream languages (currently Scala and JavaScript). VeriFx provides libraries for implementing and verifying Conflict-free Replicated Data Types (CRDTs) and Operational Transformation (OT) functions. These libraries implement the general execution model of those approaches and define their correctness properties. We use the libraries to implement and verify an extensive portfolio of 51 CRDTs, 16 of which are used in industrial databases, and reproduce a study on the correctness of OT functions.