The theory of classification part 5: axioms, assertions and subtyping

The algebraic calculus for reasoning about the complete behavior of object types and the effects of axioms upon subtyping were analyzed. The translation of pure algebra into a piecemeal treatment in terms of variants, pre-, and post conditions was studied. The existing object subtyping rules were applied to derive subtyping rules governing the strengthening, or weakening of the assertions as there was a direct relationship between axiom strengthening, and subtyping. It was found that weaker preconditions co-existed with stronger invariants, and the same system satisfied the stronger of the two.