Proving Ground Confluence and Inductive Validity in Constructor Based Equational Specifications

1 I n t r o d u c t i o n Equational specifications can be considered as the programs of an applicat ive programming language with rewriting being its computat ion mechanism. Semantics is assigned to such specification programs using canonical term algebras. Usually all ground germs of the given signature contribute to the construction of the carrier set. Sometimes however it is convenient to introduce so-cMled constructors in order to capture the intui t ion of the specifier. In that case the operat ions given by the signature are spli t into two groups: those which are used to construct the domain of computa t ion the constructors and those which do not contr ibute to this construction, but which are to be (possibly part ia l ly) defined over the domain of interest the defined operators. To consider an example, let 0 ("zero") and s ("successor") be the constructor symbols tha t introduce the natflral nulnbers. Then the following equations ( R 1 ) . . . (R4) define addi t ion and subtract ion over the natural numbers. Note that subtract ion is not a total ly defined operat ion with respect to the constructors, as there is no constructor term that is equivMcnt to (e.g.) 0 ~(0). ( n l ) ~ + 0 = . ( m ) ~ 0 = (n2) ~ + ~ ( v ) = ~ ( ~ + v ) (n~) ~ ( ~ ) 4 v ) = ~ v