Kleene Algebra with Tests and Coq tools for while programs

Abstract. We present a Coq library about Kleene algebra with tests, including a proof of their completeness over the appropriate notion of languages, a decision procedure for their equational theory, and tools for exploiting hypotheses of a certain kind in such a theory. Kleene algebra with tests make it possible to represent if-then-else state-ments and while loops in most imperative programming languages. They were actually introduced by Kozen as an alternative to propositional Hoare logic. We show how to exploit the corresponding Coq tools in the context of program verification by proving equivalences of while programs, correct-ness of some standard compiler optimisations, Hoare rules for partial cor-rectness, and a particularly challenging equivalence of flowchart schemes