wp Is wlp

Using only a simple transition relation one cannot model commands that may or may not terminate in a given state. In a more general approach commands are relations enriched with termination vectors. We reconstruct this model in modal Kleene algebra. This links the recursive definition of the do od loop with a combination of the Kleene star and a convergence operator. Moreover, the standard wp operator coincides with the wlp operator in the modal Kleene algebra of commands. Therefore our earlier general soundness and relative completeness proof for Hoare logic in modal Kleene algebra can be re-used for wp. Although the definition of the loop semantics is motivated via the standard Egli-Milner ordering, the actual construction does not depend on Egli-Milner-isotony of the constructs involved.