number-theory

8C The search algorithm 121Note. This .W('t, 0) trivially satisfies 80(i)-(ii). (The algorithm may be seen asbuilding approximations to an unknown term Mt; Vt is the weakest approximationand represents the fact that at this stage we know nothing at all about Mt otherthan its type.)Step d + 1. Assume sl(r,d) has been defined and satisfies 8C5(i)-(ii).Substep I. If C(T,d) = 0 or no member of .c/(r,d) contains meta-variables thenstop. (In this case .W(r,d+ 1) is undefined and the algorithm's output is just thefinite sequence d(i, 0), ... , d ('C, d).)Substep II. Otherwise, begin the construction of d(,r,d+ 1) by listing the propernf-schemes in d (?, d) and applying IIa-IIb below to each one.Subsubstep IIa. Given any proper Xt E d(r,d), list the meta-variables in Xt;say they areVi'..., Vyq (q ?1),and apply Ilal-M2 to each one.Part IIal. Given any meta-variable VP in an Xt E Qf(T,d), say(1) p = (m>0);first list all i < m for which Tail(aj) = a = Tail(p). (If there are none orm = 0, go direct to IIa2.) For each such i, ai has formDefine(ni > 0).(3) yip Ax1'...xm'(xiil1...V aini'')awhere the x's and V's are distinct new variables and meta-variables. (Yifis called a suitable replacement for VP.)Part IIa2. List the abstractors that cover the (unique) occurrence of VP inXt, in the order they occur in XT from left to right; say they are(4) Azi', ... , ?[` (t > 0).List all j < t (if any) such that Tail (Cj) = a. For each such j,1 j has form(5) j(hj >_0).Define(6) Zj Ax" Vii'...Vjhj i)awhere the x's and V's are distinct and new.replacement. for VP.)(Ze is called a suitable