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Tf 4.24 0 TD[(y)]TJ/F7 10.91 Tf 8.17 0 TD[(+)-221(1\))]TJ/F15 7.97 Tf 20.6 4.51 TD[(2)]TJ/F4 10.91 Tf 22.9 -4.51 TD[(`)]TJ/F10 10.91 Tf 24.85 0 TD[(Y)]TJ/F15 7.97 Tf 8.75 4.51 TD[(2)]TJ/F4 10.91 Tf 7.76 -4.51 TD[(\267)]TJ ET 0.4 w 321.91 -475.84 m 347.58 -475.84 l S 0.4 w 322.11 -492.92 m 322.11 -476.04 l S BT/F11 10.91 Tf 325.3 -487.21 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(x)]TJ ET 0.44 w 333.72 -488.73 m 339.95 -488.73 l S BT/F7 10.91 Tf 339.95 -487.21 TD[(\))]TJ ET 0.4 w 347.38 -492.92 m 347.38 -476.04 l S 0.4 w 321.91 -493.12 m 347.58 -493.12 l S BT 442.32 -487.21 TD[(\(4\))]TJ -439.49 -24.48 TD[(and)]TJ/F10 10.91 Tf 75.12 -24.48 TD[(x)]TJ/F7 10.91 Tf 9.26 0 TD[(:)]TJ/F14 10.91 Tf 6.06 0 TD[(N)]TJ/F10 10.91 Tf 7.88 0 TD[(;)-444(y)]TJ/F7 10.91 Tf 16.64 0 TD[(:)]TJ/F14 10.91 Tf 6.06 0 TD[(N)]TJ/F10 10.91 Tf 7.88 0 TD[(;)-444(y)]TJ/F15 7.97 Tf 13.61 4.5 TD[(2)]TJ/F4 10.91 Tf 7.76 -4.5 TD[(\267)]TJ/F10 10.91 Tf 11.52 0 TD[(x;)-444(x)-277(<)]TJ/F7 10.91 Tf 34.88 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(y)]TJ/F7 10.91 Tf 8.16 0 TD[(+)-222(1\))]TJ/F15 7.97 Tf 20.6 4.5 TD[(2)]TJ ET 0.44 w 178.73 -540.43 m 237.24 -540.43 l S BT/F4 10.91 Tf 255.41 -536.17 TD[(`)]TJ ET 0.4 w 280.25 -524.81 m 305.92 -524.81 l S 0.4 w 280.45 -541.89 m 280.45 -525.01 l S BT/F11 10.91 Tf 283.64 -536.17 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(x)]TJ ET 0.44 w 292.06 -537.7 m 298.29 -537.7 l S BT/F7 10.91 Tf 298.29 -536.17 TD[(\))]TJ ET 0.4 w 305.72 -541.89 m 305.72 -525.01 l S 0.4 w 280.25 -542.09 m 305.92 -542.09 l S BT/F10 10.91 Tf 308.95 -536.17 TD[(<)]TJ/F7 10.91 Tf 11.51 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(Y)]TJ/F7 10.91 Tf 11.18 0 TD[(+)-222(1\))]TJ/F15 7.97 Tf 20.6 4.5 TD[(2)]TJ/F7 10.91 Tf 85.84 -4.5 TD[(\(5\))]TJ -439.49 -24.49 TD[(During)-464(the)-464(second)-464(step,)-497(the)-464(goal)-464(is)-464(annotated)-465(in)-464(suc)27(h)-463(a)-464(w)27(a)27(y)84(,)-496(that)-464(its)-464(sk)26(eleton)-463(matc)27(hes)-463(the)]TJ 0 -14.62 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0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.12 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.12 0.26 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.05 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.26 TD[(.)]TJ 0.05 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.26 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.11 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.11 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.1 0.26 TD[(.)]TJ -0.11 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.11 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.17 0.26 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.24 0.27 TD[(.)]TJ/F13 7.97 Tf -252.38 -41.32 TD[(R)]TJ/F4 10.91 Tf -5.58 -6.18 TD[(7\241)166(!)]TJ/F10 10.91 Tf 20.6 0 TD[(x)-278(<)]TJ/F7 10.91 Tf 20.78 0 TD[(\()]TJ/F4 10.91 Tf 4.24 0 TD[(b)]TJ/F10 10.91 Tf 4.85 0 TD[(y)]TJ/F4 10.91 Tf 5.74 0 TD[(c)]TJ/F7 10.91 Tf 7.27 0 TD[(+)-221(1\))]TJ/F15 7.97 Tf 20.6 3.96 TD[(2)]TJ ET q 0.36 w 1 J 1 j 345.58 -90.73 m 345.58 -90.73 l 345.58 -90.73 l S Q BT/F17 9.96 Tf 345.58 -95.71 TD[(Y)]TJ/F7 10.91 Tf 15.08 -9.96 TD[(b)27(y)-332(wr)-334(\(6\))]TJ/F16 4.98 Tf -12.6 -7.28 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.19 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.18 0.27 TD[(.)]TJ 0.12 0.26 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.12 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.11 0.27 TD[(.)]TJ 0.04 0.26 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.05 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.05 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ 0.04 0.27 TD[(.)]TJ -0.03 0.26 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.27 TD[(.)]TJ -0.03 0.27 TD[(.)]TJ -0.04 0.26 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.11 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.11 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.11 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.1 0.27 TD[(.)]TJ -0.11 0.27 TD[(.)]TJ -0.1 0.26 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.17 0.27 TD[(.)]TJ -0.18 0.27 TD[(.)]TJ -0.24 0.26 TD[(.)]TJ/F7 10.91 Tf -267.78 -45.2 TD[(Figure)-333(3.)-1000(Rippling)-333(sequence)-334(generated)-333(b)27(y)-332(extended)-334(matc)27(hing)]TJ -77.45 -24.76 TD[(rippling.)-476(Hence)-344(the)-344(com)27(bination)-343(of)-344(the)-344(rippling)-344(and)-344(the)-344(rev)27(erse)-343(rippling)-344(strategies)-343(giv)26(es)-343(us)-344(the)]TJ 0 -14.63 TD[(\260exibilit)27(y)-332(to)-334(detect)-333(suc)27(h)-333(wholes)-333(in)-333(the)-334(pro)-27(of)-334(and)-333(\257ll)-333(them)-334(in)-333(a)-333(goal-orien)26(ted)-332(w)27(a)28(y)83(.)]TJ 16.94 -15.49 TD[(This)-391(substitution)-391(and)-392(its)-391(corresp)-28(onding)-391(rippling)-392(sequence)-391(\(cp.)-619(Figure)-392(3\))-391(constitute)-391(a)-392(suc-)]TJ -16.94 -14.63 TD[(cessful)-336(matc)27(h)-335(if)-335(the)-336(remaining)-336(subgoals)-335(are)-336(true)-336(under)-336(the)-335(found)-336(substitution)-336(\(step)-336(3)-335(in)-336(Figure)]TJ 0 -14.63 TD[(1\).)-649(W)82(e)-400(use)-402(a)-401(heuristic)-402(to)-401(c)27(hec)27(k)-400(the)-402(subgoals)-401(and)-402(syn)27(thesize)-400(case)-402(splits)-401(if)-402(necessary)82(.)-648(In)-401(order)]TJ 0 -14.63 TD[(to)-430(restrict)-430(searc)27(h)-430(space,)-454(w)27(e)-430(require)-430(that)-430(the)-430(subgoals)-430(can)-431(b)-27(e)-430(pro)26(v)28(en)-429(b)27(y)-430(standard)-430(arithmetic,)]TJ 0 -14.63 TD[(rippling)-422(and)-423(equational)-422(reasoning.)-712(If)-422(one)-422(of)-423(the)-422(remaining)-423(subgoals)-422(is)-422(not)-423(pro)27(v)56(able)-422(b)27(y)-421(these)]TJ 0 -14.63 TD[(tec)27(hniques,)-377(w)27(e)-368(use)-369(this)-370(subgoal)-369(as)-369(a)-369(constrain)27(t)-368(of)-369(the)-369(substitutions.)-552(In)-369(order)-369(to)-369(b)-28(e)-369(consisten)27(t,)]TJ 0 -14.63 TD[(w)27(e)-349(then)-350(pro)27(v)28(e)-349(all)-350(the)-350(remaining)-350(subgoals)-350(under)-350(the)-350(negated)-350(constrain)27(t.)-493(In)-350(this)-350(example,)-354(\(4\))-350(is)]TJ 0 -14.62 TD[(the)-288(only)-287(remaining)-288(subgoal.)-429(W)82(e)-287(use)-288(the)-287(subgoal)-288(\()]TJ/F11 10.91 Tf 233.78 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(y)]TJ/F7 10.91 Tf 5.74 0 TD[(\)\))]TJ/F15 7.97 Tf 8.48 3.95 TD[(2)]TJ/F4 10.91 Tf 7.76 -3.95 TD[(\267)]TJ/F11 10.91 Tf 11.51 0 TD[(s)]TJ/F7 10.91 Tf 4.19 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(x)]TJ/F7 10.91 Tf 6.23 0 TD[(\))-287(as)-288(a)-288(constrain)27(t,)-296(and)-288(therefore)-287(try)]TJ -290.35 -14.63 TD[(to)-343(pro)27(v)28(e)-342(b)-28(oth)-342(cases)-343(\(4\))-343(and)-343(\(5\))-343(for)-342(the)-343(case)]TJ/F4 10.91 Tf 211.83 0 TD[(:)]TJ/F7 10.91 Tf 7.27 0 TD[(\()]TJ/F11 10.91 Tf 4.24 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(y)]TJ/F7 10.91 Tf 5.74 0 TD[(\)\))]TJ/F15 7.97 Tf 8.48 3.95 TD[(2)]TJ/F4 10.91 Tf 7.93 -3.95 TD[(\267)]TJ/F11 10.91 Tf 11.69 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(x)]TJ/F7 10.91 Tf 6.23 0 TD[(\).)-472(W)82(e)-342(use)-343(unfolding)-342(of)-343(the)-343(successor)]TJ -280.25 -14.63 TD[(and)]TJ/F4 10.91 Tf 21.55 0 TD[(:)]TJ/F7 10.91 Tf 7.27 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(:)-166(:)-167(:)]TJ/F4 10.91 Tf 16.32 0 TD[(\267)]TJ/F10 10.91 Tf 12.08 0 TD[(:)-166(:)-167(:)]TJ/F7 10.91 Tf 14.54 0 TD[(\))-364(function)]TJ/F11 10.91 Tf 51.28 0 TD[(s)]TJ/F7 10.91 Tf 8.15 0 TD[(and)-364(normal)-365(matc)27(hing,)-371(to)-365(deriv)27(e)-364(a)-364(substitution.)-538(W)82(e)-363(start)-365(again)-364(b)26(y)]TJ -135.43 -14.63 TD[(insp)-28(ecting)-437(subgoal)-437(\(5\).)-756(Matc)26(hing)-436(b)-28(et)27(w)28(een)]TJ/F11 10.91 Tf 211.64 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(x)]TJ/F7 10.91 Tf 6.23 0 TD[(\))]TJ/F10 10.91 Tf 9.16 0 TD[(<)]TJ/F7 10.91 Tf 13.4 0 TD[(\()]TJ/F10 10.91 Tf 4.25 0 TD[(Y)]TJ/F7 10.91 Tf 11.93 0 TD[(+)-291(1\))]TJ/F15 7.97 Tf 21.36 3.96 TD[(2)]TJ/F7 10.91 Tf 9.5 -3.96 TD[(and)]TJ/F11 10.91 Tf 22.34 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(x)]TJ/F7 10.91 Tf 6.23 0 TD[(\))]TJ/F10 10.91 Tf 9.16 0 TD[(<)]TJ/F7 10.91 Tf 13.4 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(y)]TJ/F7 10.91 Tf 8.92 0 TD[(+)-291(1\))]TJ/F15 7.97 Tf 21.36 3.96 TD[(2)]TJ/F7 10.91 Tf 9.49 -3.96 TD[(returns)-437(the)]TJ -399.45 -14.63 TD[(substitution)-333([)]TJ/F10 10.91 Tf 63.74 0 TD[(y)-36(=)-55(Y)]TJ/F7 10.91 Tf 20.56 0 TD[(].)-444(The)-333(subgoal)-333(\(4\))-334(is)-333(trivially)-333(true)-334(under)-333(this)-333(substitution.)]TJ -67.36 -15.49 TD[(By)-314(com)27(bining)-314(logical)-314(pro)-28(of)-315(searc)27(h)-313(with)-315(rippling)-315(and)-314(extended)-315(matc)27(hing,)-317(w)26(e)-313(are)-315(able)-315(to)-314(gen-)]TJ -16.94 -14.63 TD[(erate)-288(automatically)-289(a)-288(pro)-28(of)-289(for)-288(the)-289(step)-288(case)-289(and)-288(to)-289(syn)27(thesize)-287(a)-289(set)-288(of)-289(conditional)-288(substitutions:)]TJ 0 -14.63 TD[([\()]TJ/F11 10.91 Tf 7.27 0 TD[(s)]TJ/F7 10.91 Tf 4.19 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(y)]TJ/F7 10.91 Tf 5.74 0 TD[(\)\))]TJ/F15 7.97 Tf 8.48 3.96 TD[(2)]TJ/F4 10.91 Tf 7.76 -3.96 TD[(\267)]TJ/F11 10.91 Tf 11.51 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(x)]TJ/F7 10.91 Tf 6.23 0 TD[(\))]TJ/F10 10.91 Tf 4.25 0 TD[(;)]TJ/F11 10.91 Tf 4.84 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.25 0 TD[(y)]TJ/F7 10.91 Tf 5.73 0 TD[(\))]TJ/F10 10.91 Tf 4.24 0 TD[(=)-55(Y)]TJ/F7 10.91 Tf 14.82 0 TD[(])]TJ/F10 10.91 Tf 3.03 0 TD[(;)]TJ/F7 10.91 Tf 4.85 0 TD[([)]TJ/F4 10.91 Tf 3.02 0 TD[(:)]TJ/F7 10.91 Tf 7.28 0 TD[(\(\()]TJ/F11 10.91 Tf 8.48 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.24 0 TD[(y)]TJ/F7 10.91 Tf 5.74 0 TD[(\)\))]TJ/F15 7.97 Tf 8.48 3.96 TD[(2)]TJ/F4 10.91 Tf 7.76 -3.96 TD[(\267)]TJ/F11 10.91 Tf 11.51 0 TD[(s)]TJ/F7 10.91 Tf 4.18 0 TD[(\()]TJ/F10 10.91 Tf 4.25 0 TD[(x)]TJ/F7 10.91 Tf 6.23 0 TD[(\)\))]TJ/F10 10.91 Tf 8.48 0 TD[(;)-166(y)-36(=)-56(Y)]TJ/F7 10.91 Tf 25.4 0 TD[(].)]TJ/F3 11.95 Tf -223.26 -35.26 TD[(3.3.)-1125(An)-375(Algorithm)-375(for)-375(Sim)30(ultanous)-374(Matc)31(hing)]TJ/F7 10.91 Tf 0 -23.35 TD[(In)-369(this)-369(section)-369(w)27(e)-368(presen)27(t)-368(a)-369(tec)26(hnical)-368(description)-369(of)-369(the)-369(steps)-369(p)-28(erformed)-369(during)-369(the)-369(automatic)]TJ 0 -14.63 TD[(instan)27(tiation)-394(of)-394(existen)27(tially)-394(quan)27(ti\257ed)-393(v)54(ariables)-393(\(see)-395(Figure)-394(1\).)-628(W)82(e)-393(use)]TJ/F8 9.96 Tf 352.97 0 TD[(NuPRL)]TJ/F7 10.91 Tf 36.38 0 TD[(-ML-notation)]TJ -389.35 -14.63 TD[(to)-491(describ)-28(e)-492(the)-491(algorithm.)-919(T)82(o)-490(automate)-491(step)-492(1)-491(standard)-492(theorem)-491(pro)26(ving)-490(metho)-28(ds)-491(can)-492(b)-28(e)]TJ 0 -14.63 TD[(used.)-480(W)82(e)-344(concen)27(trate)-344(on)-346(the)-345(sim)27(ultanous)-344(matc)26(hing)-344(pro)-28(cedure)-345(\(step)-345(2\))-345(whic)26(h)-344(is)-345(the)-345(core)-346(of)-345(the)]TJ 0 -14.62 TD[(automatic)-299(instan)27(tiation)-298(of)-299(existen)26(tially)-298(quan)27(ti\257ed)-298(v)55(ariables.)-432(The)-299(algorithmic)-300(description)-299(for)-299(the)]TJ 0 -14.63 TD[(sim)27(ultanous)-332(matc)26(h)-332(is)-333(giv)26(en)-332(in)-333(Figure)-334(4.)]TJ 16.94 -15.5 TD[(The)-338(function)]TJ/F11 10.91 Tf 65.24 0 TD[(simultaneous)]TJ ET 0.4 w 144.29 -545.83 m 147.56 -545.83 l S BT 147.56 -546.03 TD[(match)]TJ/F7 10.91 Tf 32.02 0 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