How do we benefit from ground confluent specifications? Most of the advantages follow from Thm. 6.5: If (SIG, AX) is ground confluent, then and only then directed expansions yield all ground AX-solutions. Sects. 7.1 and 7.2 deal with refinements of directed expansion: strategic expansion and narrowing. Sect. 7.3 presents syntactic criteria for a set of terms to be a set of constructors (cf. Sect. 2.3). The results obtained in Sects. 7.2 and 7.3 provide the failure rule and the clash rule that check goals for unsolvability and thus help to shorten every kind of expansion proof (see the final remarks of Sect. 5.4).

Sect. 7.4 deals with the proof of a set CS of inductive theorems by showing that (SIG,AX∪CS) is consistent w.r.t. (SIG,AX) (cf. Sect. 3.4). Using consequences of the basic equivalence between consistency and inductive validity (Lemma 7.9) we come up with reductive expansion, which combines goal reduction and subreductive expansion (cf. Sect. 6.4) into a method for proving inductive theorems. While inductive expansion is always sound, the correctness of reductive expansion depends on ground confluence and strong termination of (SIG,AX). Under these conditions, an inductive expansion can always be turned into a reductive expansion (Thm 7.18). Conversely, a reductive expansion can be transformed in such a way that most of its “boundary conditions” hold true automatically (Thm. 7.19).

The chapter will close with a deduction-oriented concept for specification refinements, or algebraic implementations.