Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction

Abstract

A proof tableau of Hoare logic is an annotated program with pre-and post-conditions, which corresponds to an inference tree of Hoare logic. In this paper, we show that a proof tableau for partial correctness can be transformed into an inference sequence of rewriting induction for constrained rewriting. We also show that the resulting sequence is a valid proof for an inductive theorem corresponding to the Hoare triple if the constrained rewriting system obtained from the program is terminating. Such a valid proof with termination of the constrained rewriting system implies total correctness of the program w.r.t. the Hoare triple. The transformation enables us to apply techniques for proving termination of constrained rewriting to proving total correctness of programs together with proof tableaux for partial correctness.Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction corresponding to the Hoare triple for the proof tableau if the LCTRS obtained from the program is terminating.Given a while program P and a proof tableau T P of a Hoare triple {ϕ P } P {ψ P } for partial correctness, we proceed as follows:1. We transform P into an equivalent LCTRS R P , and we prove termination of the LCTRS R P .2. We prepare rewrite rules R check to verify the post-condition ψ P in the proof tableau.3. We prepare a constrained equation e P corresponding to the Hoare triple {ϕ P } P {ψ P }.4. Starting with the equation e P , we transform the proof tableau into an inference sequence ({e P }, / 0) RI · · · RI ( / 0, H) of RI in a top-down fashion, where we do not prove termination in constructing the inference sequence of RI.