Refinement algebra with dual operator

• We introduce an extension of the general refinement algebra with a dual operator.
• We introduce assertions and assumptions with simpler definitions than before.
• We defined the termination and enabledness operators in our algebra.
• We prove data refinement and Hoare rules, and we used them for an example program.
• All results are formalized in the Isabelle theorem prover.

Algebras of imperative programming languages have been successful in reasoning about programs. In general an algebra of programs is an algebraic structure with programs as elements and with program compositions (sequential composition, choice, skip) as algebra operations. Various versions of these algebras were introduced to model partial correctness, total correctness, refinement, demonic choice, and other aspects. We introduce here an algebra which can be used to model total correctness, refinement, demonic and angelic choice. The basic model of our algebra are monotonic Boolean transformers (monotonic functions from a Boolean algebra to itself).