The lattice of data refinement

We define a general notion of data refinement that comprises the traditional notion of data refinement as a special case. Using the concepts of duals and adjoints, we define converse commands and find a symmetry between ordinary data refinement and a dual (backward) data refinement. We show how ordinary and backward data refinement are interpreted as simulation, and we derive rules for the piecewise data refinement of programs. Our results are valid for a general language, including demonic and angelic nondeterminism as well as nontermination and miracles.

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