Explain the while-loop rule duality twist in partial reverse Hoare logic

Determine the theoretical reason that, in the ordinary proof system for partial reverse Hoare logic, the while-loop inference rule is the dual of the partial Hoare logic rule rather than the dual of the total Hoare logic rule, despite the semantics of partial reverse Hoare logic being the dual of total Hoare logic. Clarify the underlying principles or structural properties that cause this mismatch between semantic duality and proof-rule duality.

Background

The paper introduces ordinary and cyclic proof systems for partial reverse Hoare logic (also called partial incorrectness logic). Semantically, partial reverse Hoare logic is presented as the dual of total Hoare logic, yet the authors’ ordinary proof system employs a while-loop rule that is the dual of the partial Hoare logic rule (rather than the total Hoare logic rule).

This mismatch is highlighted as an intriguing twist. Understanding why the proof-rule duality aligns with partial Hoare logic in the while-loop case, even though the overall semantics are dual to total Hoare logic, would deepen the theoretical foundations of the logic and may inform the design of proof rules for related Hoare-style logics.