We develop a compositional, algebraic theory of skipping refinement, as well as local proof methods to effectively analyze the correctness of optimized reactive systems. A verification methodology based on refinement involves showing that any infinite behavior of an optimized low-level implementation is a behavior of the high-level abstract specification. Skipping refinement is a recently introduced notion to reason about the correctness of optimized implementations that run faster than their specifications, i.e., a step in the implementation can skip multiple steps of the specification. For the class of systems that exhibit bounded skipping, existing proof methods have been shown to be amenable to mechanized verification using theorem provers and model-checkers. However, reasoning about the correctness of reactive systems that exhibit unbounded skipping using these proof methods requires reachability analysis, significantly increasing the verification effort. In this paper, we develop two new sound and complete proof methods for skipping refinement. Even in presence of unbounded skipping, these proof methods require only local reasoning and, therefore, are amenable to mechanized verification. We also show that skipping refinement is compositional, so it can be used in a stepwise refinement methodology. Finally, we illustrate the utility of the theory of skipping refinement by proving the correctness of an optimized event processing system.