Just as imaginary numbers extend real numbers and simplify certain mathematical proofs, we introduce the concept of imaginary specifications to enhance program verification. In mathematics, imaginary numbers enable expressing intermediate steps that cannot be captured using real numbers alone, offering natural proof decomposition that reduces complex proofs into simpler, more manageable steps. Similarly, our work introduces imaginary specifications in program verification through CRIS (Contextual Refinement with Imaginary Specification), our novel verification tool.

CRIS with imaginary specifications provides a unified framework to inherently marry two fundamental approaches to program verification: separation logic with pre/post conditions as specifications, and program refinement with abstract programs as specifications. This unification not only enables proof simplification via proof decomposition but also enables elegant expression of hard-to-express properties, such as separation logic conditions involving IO events and logical atomicity—properties that traditionally require intricate mechanisms or are difficult to specify.