number-theory

166 Answers to starred exercisesTo do this, the algorithm first applies 7C4 to obtain a term R such thatPT(R) = (f-*a1)-*(g-*b)->(al->b-->a2)-+(f-*g-*a2),and then defines N - (Ax'Rxx)I(I,.K).8A12.1 The eight regions contain the following terms in order from left to right.Top row: Axy'xy, Axyz'x(xyyy)z, 2xyzu'x(xyyy)zu, Axyzu'xyzu;bottom row: Ax-x, Axyz'xz(xyyy), Axyzu'xu(xyyy)z, Axyzu'uxyz.8B7 For items 1, 6, 8, 11 in Table 8B7a see 8B3-8B6. For the rest, see the answer to8C6.4. (In item 12, PO Nprinc(T) since the PT of Po is ((a--+a)--+b)-+b by the PTalgorithm, 3E1.)8C6.4 For rows 6, 8 and 11 of Table 8B7a see Examples 8C6.1-3. The other rowsare dealt with below. (For ease of reading, types are omitted and x'M is used forx(x(... (xM) ...)) with d x's.)1. d(T,0) = {V}, ,q/(T,1) = 0.2. &/('r,0) _ {V}, .sJI(T,1) _ {Ax1'x1}.3. .Q/' (T,0) _ {V}, Ql(T,1) _ {Ax1x2'xl}.4. d(T,0) _ {V}, d(T,1) _ {Ax1x2x3'x1 VI 1, .9/(T,2) = {Ax1x2x3'x1(x2V2)},,SV(T, 3) = {Ax1x2x3'x1(x2x3)}5. .Ql(T,0) = {V}, Q/(T,1) = {Ax1x2x3'x1 V1 V2}, JV(T,2) = {Axlx2x3'xlx3x2}.7. crl(T,0) = {V}, d(T,1) = {Axix2'x1 V1 V2}, 4(T,2)= {Ax1x2'x1x2x2}.9. 5z/(T,O) = {V}, .21(T, 1) = {Axix2'xl V1, Axtx2'x2}srl(T,d) = {2x1x2'xiVd, Axlx2'xi-1x2} for all d > 2.10. Al(T,O) = {V}, .sal(T, 1) = V1}, sl(T,2) = {Ax1x2'xlx2}.12. d(T,0) = {V}, s.V/(T,1) = {Ax'xVl}d(T,2) =Ax'x(Ayryi)}d(T,3) =2x'x(2y1'x(2y2'xV3))2x'x(AY1'x(2Y2'Y1)), Ax'x(2Y1'x(2Y2'Y2))},(V(T,d) _ {2x'x(Ayl'x(... (2yd-1'xVd)...)),Ax'x(Ay1'x(... (AYd-2'x(AYd-1'Y1))...)), ...2x'x(Ay1'x(... (AYd-2'x(AYd-1'Yd-1))...))} for all d 4.8E7.3 For (iii), use (ii) and the fact that #(szl(T,0)) = 1 (since d(T,0) = {VT} byStep 0 of Algorithm 8C6).For (i), use induction on d. The basis is trivial since d(T, 0) = {V' J.For the induction step (d to d+ 1), let XT E d(T, d) contain q metavariables where1 < q < ITId, and let VP be one of these.Consider Part Hal of Step d + 1 of Algorithm 8C6: using the notation of IIal,note that each suitable replacement YfP generated by IIal for VP contains < n1metavariables, where n1 is the arity of a1. But o1 occurs in p which occurs in T by8E7.1, soni < jail - 1 < ITI - 1 < ITI.