Data Reenement with Stuttering

We prove a number of general rules for data reenement of initialised loops. In addition to ordinary (forwards) data reenement we give rules for backwards data reenement, which can be used in certain situations when the ordinary rules are not applicable. We also extend the rules to permit stuttering. This makes the rules applicable in situations where the reened loop performs computation steps which do not have a counterpart in the original loop. The proofs of the rules are algebraic and they are done on the command level, without referring to the predicate transformer semantics of the commands. We also show how the rules can be expressed in a form which makes them practically useful.