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Contextual refinement (CR) is one of the standard notions of specifying open programs. CR has two main advantages: (i) (horizontal and vertical) compositionality that allows us to decompose a large contextual refinement into many smaller ones enabling modular and incremental verification, and (ii) no restriction on programming features thereby allowing, e.g., mutually recursive, pointer-value passing, and higher-order functions. However, CR has a downside that it cannot impose conditions on the context since it quantifies over all contexts, which indeed plays a key role in support of full compositionality and programming features. In this paper, we address the problem of finding a notion of refinement that satisfies all three requirements: support of full compositionality, full (sequential) programming features, and rich conditions on the context. As a solution, we propose a new theory of refinement, called CCR (Conditional Contextual Refinement), and develop a verification framework based on it, which allows us to modularly and incrementally verify a concrete module against an abstract module under separation-logic-style pre and post conditions about external modules. It is fully formalized in Coq and provides a proof mode that combines (i) simulation reasoning about preservation of sideffects such as IO events and termination and (ii) propositional reasoning about pre and post conditions. Also, the verification results are combined with CompCert, so that we formally establish behavioral refinement from top-level abstract programs, all the way down to their assembly code.