Step 6 Synthesis of the Process and Compose

An instantiation of the Process predicate-variable transforms, in the sub-case of the recursive case, the heads HX of the induction parameter X into a vector HY of heads of the other parameter Y. An instantiation of the Compose predicate-variable computes, in the 1 sub-case of the recursive case, the parameter Y from its heads HY (obtained by processing the HX) and tails TY (obtained by recursion on the TX). 

The objective at Step 6 is to instantiate the predicate-variables Processj and Compose of the divide-and-conquer schema. The number w of sub-cases of the recursive case must also be found. This amounts to transforming LA r) into LA r) such that it is covered by the following schema  

We merge the Processj HXJiY) with the Cpmposej HYJY,Y) into PwcCompj iHX, TYJ), so that their instances are synthesized simultaneously. If we isolated the ProcesSf iHXJHY), then their only correct instantiations would be defined as  

This would mean that HY would have to be initialized to a vector of different variables, and without knowing their number h Moreover, w is yet unknown, so we wouldn t know how many of these instances to create. All this would of course 

We first hypothesize that all instances of the ProcCompi are defined in terms of the =/2 primitive only. So we may apply the MSG Method (see Chapter 10). The idea is to initialize the admissibility parameters a and b to h and 1, respectively, as Y is necessarily constructed from TY (which is of length 1), and optionally from HX (which is of length h). The synthesis of the instances procCompj /hH+l of the Proc-Comp] lh+t+ may thus be performed by the following sequence of tasks  

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