Algebraic proofs of consistency and completeness

Abstract

An embedding of the relations in the predicate transformers, analogous to that of the integers in the rationals, is exploited to provide simple algebraic proofs for the consistency and completeness of a calculus of program refinement. The calculus of refinement is derived by almost direct translation of the Hoare logic inference rules, and so alternatively the proofs may be viewed as demonstrating the soundness and completeness of Hoare logic. The main attributes of the embedding used in the proofs are that it supports a weak form of inversion (i.e. Galois connection) of relations, and that it supports an operator on predicate transformers that behaves like the floor operator on rationals: the operator maps arbitrary predicate transformer down in the natural ordering to the nearest embedded relation. A more general use for the floor-like operator in extending the relational calculus is suggested by its providing decomposition of the weakest prespecification operator. A weak algebraic set theory is used as a foundation for proving all required properties of the floor-like operator.