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b Fu(F)614 791 y Fp(eq)685 776 y Fv(,)g Fu(R)834 791
y Fp(eq)905 776 y Fv(,)g(and)f Fu(R)1243 734 y Fr(0)1301
776 y Fv(ar)-5 b(e)35 b(de\014ne)-5 b(d)34 b(as)g(fol)5
b(lows:)373 980 y(1.)48 b Fu(F)583 995 y Fp(eq)681 980
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b(al)r(se)p Fu(g)p Fv(,)34 b(and)h Fu(^)g Fv(is)g(assume)-5
b(d)34 b(to)h(b)-5 b(e)34 b(right-asso)-5 b(ciative.)373
1183 y(2.)48 b(The)34 b(TRS)h Fu(R)1015 1198 y Fp(eq)1121
1183 y Fv(c)-5 b(onsists)34 b(of)h(the)g(rules)g(\()p
Fu(C)41 b Fv(is)34 b(the)h(set)g(of)g(c)-5 b(onstructors)34
b(in)h Fu(R)p Fv(\))1590 1403 y Fs(c)p Fw(==)p Fs(c)83
b Fu(!)g Fs(tr)s(ue)488 b Fu(8)35 b Fw(0)p Fv(-ary)g
Fs(c)27 b Fu(2)i(C)1581 1523 y Fs(c)p Fw(==)p Fs(d)83
b Fu(!)g Fs(f)11 b(al)r(se)439 b Fu(8)p Fs(c;)17 b(d)27
b Fu(2)h(C)7 b Fs(;)17 b(c)27 b Fu(6)p Fw(=)h Fs(d)623
1644 y(c)p Fw(\()p Fs(x)758 1659 y Fq(1)797 1644 y Fs(;)17
b(:)g(:)g(:)f(;)h(x)1071 1659 y Fp(n)1118 1644 y Fw(\)==)p
Fs(c)p Fw(\()p Fs(y)1436 1659 y Fq(1)1475 1644 y Fs(;)g(:)g(:)g(:)f(;)h
(y)1742 1659 y Fp(n)1788 1644 y Fw(\))83 b Fu(!)2092
1569 y Ft(V)2175 1595 y Fp(n)2175 1673 y(i)p Fq(=1)2293
1644 y Fw(\()p Fs(x)2386 1659 y Fp(i)2414 1644 y Fw(==)p
Fs(y)2614 1659 y Fp(i)2642 1644 y Fw(\))g Fu(8)35 b Fs(n)p
Fv(-ary)g Fs(c)28 b Fu(2)g(C)594 1764 y Fs(c)p Fw(\()p
Fs(x)729 1779 y Fq(1)769 1764 y Fs(;)17 b(:)g(:)g(:)f(;)h(x)1043
1779 y Fp(n)1090 1764 y Fw(\)==)p Fs(d)p Fw(\()p Fs(y)1417
1779 y Fq(1)1455 1764 y Fs(;)g(:)g(:)g(:)f(;)h(y)1722
1779 y Fp(m)1788 1764 y Fw(\))83 b Fu(!)g Fs(f)11 b(al)r(se)439
b Fu(8)p Fs(c;)17 b(d)27 b Fu(2)h(C)7 b Fs(;)17 b(c)27
b Fu(6)p Fw(=)h Fs(d)1477 1885 y(tr)s(ue)22 b Fu(^)g
Fs(x)84 b Fu(!)f Fs(x)1427 2005 y(f)11 b(al)r(se)23 b
Fu(^)f Fs(x)84 b Fu(!)f Fs(f)11 b(al)r(se)373 2269 y
Fv(3.)48 b(and)41 b Fu(R)782 2227 y Fr(0)845 2269 y Fw(=)f
Fu(f)p Fs(l)i Fu(!)d Fs(r)k Fu(\()d Fs(s)1494 2284 y
Fq(1)1533 2269 y Fw(==)p Fs(t)1720 2284 y Fq(1)1799 2269
y Fu(!)g Fs(tr)s(ue;)17 b(:)g(:)g(:)32 b(;)17 b(s)2403
2284 y Fp(k)2445 2269 y Fw(==)p Fs(t)2632 2284 y Fp(k)2715
2269 y Fu(!)40 b Fs(tr)s(ue)17 b Fu(j)g Fs(l)40 b Fu(!)g
Fs(r)j Fu(\()501 2390 y Fs(s)547 2405 y Fq(1)614 2390
y Fw(=)28 b Fs(t)753 2405 y Fq(1)793 2390 y Fs(;)17 b(:)g(:)g(:)32
b(;)17 b(s)1074 2405 y Fp(k)1144 2390 y Fw(=)27 b Fs(t)1282
2405 y Fp(k)1353 2390 y Fu(2)h(R)q(g)257 2618 y Fv(Pr)-5
b(o)g(of)67 b Fw(Similar)30 b(to)i(the)h(pro)s(of)e(for)h
(unconditional)f(TRSs;)i(see)h([1].)645 b Fj(\003)404
2838 y Fw(F)-8 b(rom)28 b(no)m(w)j(on)f(rewrite)g(rules)g(of)f(a)h
(CTRS)h(with)e(strict)h(equalit)m(y)f(will)f(b)s(e)i(written)257
2958 y(as)j Fs(l)d Fu(!)d Fs(r)k Fu(\()c Fs(s)811 2973
y Fq(1)850 2958 y Fw(==)p Fs(t)1037 2973 y Fq(1)1077
2958 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(s)1358 2973 y
Fp(k)1401 2958 y Fw(==)o Fs(t)1587 2973 y Fp(k)1630 2958
y Fw(.)404 3079 y(In)35 b(the)h(follo)m(wing,)d(w)m(e)j(need)g(a)f
(result)g(obtained)g(b)m(y)h(Staples)f([18)o(].)51 b(An)36
b(abstract)257 3199 y(reduction)28 b(system)i(\(ARS\))e
Fu(A)1388 3214 y Fq(2)1455 3199 y Fw(=)g(\()p Fs(A;)17
b Fu(!)1814 3214 y Fq(2)1853 3199 y Fw(\))28 b(is)f(called)g(a)h
Fv(r)-5 b(e\014nement)27 b Fw(of)h(another)g(ARS)257
3319 y Fu(A)337 3334 y Fq(1)417 3319 y Fw(=)40 b(\()p
Fs(A;)17 b Fu(!)788 3334 y Fq(1)827 3319 y Fw(\))40 b(if)e
Fu(!)1101 3334 y Fq(1)1141 3319 y Fu(\022!)1318 3283
y Fr(\003)1318 3344 y Fq(2)1357 3319 y Fw(.)66 b(Suc)m(h)41
b(a)f(re\014nemen)m(t)h(is)e(called)g Fv(c)-5 b(omp)g(atible)39
b Fw(if)g(for)g(all)257 3440 y Fs(a)47 b Fu(!)455 3404
y Fr(\003)455 3465 y Fq(2)541 3440 y Fs(b)p Fw(,)g(there)e(is)e(a)g
Fs(c)k Fu(2)g Fs(A)d Fw(suc)m(h)h(that)f Fs(a)j Fu(!)2089
3404 y Fr(\003)2089 3465 y Fq(1)2175 3440 y Fs(c)c Fw(and)h
Fs(b)k Fu(!)2650 3404 y Fr(\003)2650 3465 y Fq(1)2735
3440 y Fs(c)p Fw(.)77 b(Staples')44 b(result)257 3560
y(states)38 b(that)e(a)g(compatible)f(re\014nemen)m(t)j
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y Fq(1)2252 3560 y Fw(is)g(con\015uen)m(t)i(if)d(and)i(only)f(if)f
Fu(A)3456 3575 y Fq(1)257 3681 y Fw(is)e(con\015uen)m(t.)45
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(whic)m(h)g(can)g(b)s(e)h(found)f(in)f([15].)47 b(Let)35
b Fu(A)2176 3816 y Fq(1)2246 3801 y Fw(=)30 b(\()p Fs(A;)17
b Fu(!)2607 3816 y Fq(1)2645 3801 y Fw(\))34 b(and)g
Fu(A)2988 3816 y Fq(2)3058 3801 y Fw(=)29 b(\()p Fs(A;)17
b Fu(!)3418 3816 y Fq(2)3457 3801 y Fw(\))257 3921 y(b)s(e)31
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Fu(!)2877 3936 y Fq(1)2916 3921 y Fu(\022!)3093 3885
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4006 y Fr(\003)2242 4066 y Fq(1)2321 4042 y Fs(c)p Fw(,)h
Fs(b)f Fu(!)2614 4006 y Fr(\003)2614 4066 y Fq(1)2693
4042 y Fs(d)p Fw(,)h(and)e Fs(c)g Fu(\030)h Fs(d)p Fw(.)65
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4162 y Fw(is)g(con\015uen)m(t)h(mo)s(dulo)d Fu(\030)j
Fw(\(i.e.,)f(for)g(all)e Fs(c)2559 4126 y Fr(\003)2559
4187 y Fp(i)2597 4162 y Fu( )e Fs(a)h Fu(\030)h Fs(b)f
Fu(!)3094 4126 y Fr(\003)3094 4187 y Fp(i)3165 4162 y
Fs(d)p Fw(,)j(there)257 4283 y(are)d Fs(e;)17 b(f)38
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Fs(f)1723 4246 y Fr(\003)1723 4307 y Fp(i)1757 4283 y
Fu( )f Fs(d)p Fw(\))k(and,)h(for)f(all)e Fs(a)f Fu(\030)g
Fs(b)g Fu(!)2855 4246 y Fr(\003)2855 4307 y Fq(3)p Fr(\000)p
Fp(i)3001 4283 y Fs(c)p Fw(,)j(there)i(is)d(a)257 4403
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Fp(i)1291 4403 y Fs(d)g Fu(\030)g Fs(c)p Fw(,)33 b(then)g
Fu(A)1878 4418 y Fq(3)p Fr(\000)p Fp(i)2029 4403 y Fw(is)f(con\015uen)m
(t)i(mo)s(dulo)d Fu(\030)p Fw(.)257 4736 y Fx(3)161 b(Almost)54
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b(start)f(with)g(a)g(result)g(from)e(Bergstra)j(and)f(Klop)f([5].)60
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b(Hence)40 b(these)257 5196 y(systems)34 b(are)f(lev)m(el-con\015uen)m
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%%Page: 7 7
7 6 bop 257 573 a Fo(De\014nition)36 b(3.1)49 b Fw(Let)31
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b Fw(of)h Fs(q)k Fw(in)c Fs(t)h Fw(is)f(de\014ned)i(b)m(y:)376
994 y Fs(q)t Fu(n)p Fs(A)27 b Fw(=)677 790 y Ft(8)677
879 y(<)677 1059 y(:)807 873 y Fu(f)p Fs(q)t Fu(g)922
b Fw(if)32 b Fs(q)f(<)d(p)k Fw(or)h Fs(q)j Fu(k)c Fs(p;)807
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b Fs(r)1363 1009 y Fr(j)p Fp(p)1419 1018 y Fl(3)1485
993 y Fw(=)27 b Fs(l)1617 1009 y Fr(j)p Fp(p)1673 1018
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b Fs(p)2817 1008 y Fq(1)2884 993 y Fu(2)c(V)8 b(P)h Fs(os)p
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1295 y(If)36 b Fs(Q)c Fu(\022)h(P)8 b Fs(os)p Fw(\()p
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2074 1220 y Ft(S)2157 1324 y Fp(q)r Fr(2)p Fp(Q)2314
1295 y Fs(q)t Fu(n)p Fs(A)p Fw(.)51 b(The)37 b(notion)d(of)h(descen-)
257 1415 y(dan)m(t)e(is)f(extended)j(to)d(rewrite)h(sequences)j(in)31
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1623 y Fm(n)2546 1600 y Fs(t)42 b Fw(if)e Fs(t)i Fw(can)f(b)s(e)h
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Fw(b)m(y)h Fu(!)3159 1736 y Fr(R)3219 1744 y Fm(n)3266
1721 y Fw(.)67 b(W)-8 b(e)257 1841 y(write)42 b Fs(s)32
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2573 y Fs(t)1249 2588 y Fq(3)1289 2573 y Fv(.)126 b(Mor)-5
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2708 y Fq(1)1815 2693 y Fv(\()p Fs(t)1890 2708 y Fq(2)1930
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257 3120 y(2-CTRS)g(is)f(lev)m(el-con\015uen)m(t.)43
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3666 y Fo(De\014nition)k(3.4)49 b Fw(A)29 b(3-CTRS)f
Fu(R)h Fw(with)f(strict)g(equalit)m(y)g(is)f(called)g
Fv(almost)k(functional)257 3786 y Fw(if)c(it)g(is)g(orthogonal)f(and)i
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4271 y(2.)48 b(Ev)m(ery)35 b Fs(t)817 4286 y Fp(j)853
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4648 y Ft(8)1024 4738 y(>)1024 4768 y(>)1024 4798 y(<)1024
4977 y(>)1024 5007 y(>)1024 5037 y(:)1154 4731 y Fw(0)22
b(+)g Fs(x)246 b Fu(!)83 b Fs(x)1154 4851 y(s)p Fw(\()p
Fs(x)p Fw(\))23 b(+)f Fs(y)124 b Fu(!)83 b Fs(s)p Fw(\()p
Fs(x)22 b Fw(+)g Fs(y)t Fw(\))1154 4972 y Fs(f)11 b(ib)p
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b(Let)41 b Fs(i)i(>)g Fw(0.)70 b(Ac-)257 2956 y(cording)36
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b(Since)257 3099 y Fu(V)8 b Fs(ar)s Fw(\()p Fs(s)508
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y Fp(i)1105 3340 y Fs(\033)1160 3355 y Fq(1)1200 3340
y Fu(!)1300 3303 y Fr(\003)1300 3377 y(R)1360 3354 y
Fm(d)1360 3398 y(m)p Fk(\000)p Fl(1)1501 3340 y Fs(u)1557
3355 y Fq(1)1661 3303 y Fr(\003)1661 3377 y(R)1721 3354
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y Fq(1)2119 3340 y Fw(and)39 b Fs(s)2361 3355 y Fp(i)2389
3340 y Fs(\033)2444 3355 y Fq(2)2484 3340 y Fu(!)2584
3303 y Fr(\003)2584 3377 y(R)2644 3354 y Fm(d)2644 3398
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3303 y Fr(\003)2929 3377 y(R)2989 3354 y Fm(d)2989 3398
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3503 y Fp(i)2335 3488 y Fs(\033)2390 3503 y Fq(1)2462
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3502 y Fm(d)2600 3546 y(m)p Fk(\000)p Fl(1)2741 3488
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3503 y Fp(i)2983 3488 y Fs(\033)3038 3503 y Fq(2)3110
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3502 y Fm(d)3248 3546 y(n)p Fk(\000)p Fl(1)3373 3488
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b(the)g(inductiv)m(e)g(h)m(yp)s(othesis)g(on)g Fs(`)f
Fw(that)g(the)h(t)m(w)m(o)g(ground)g(normal)e(forms)g
Fs(u)3456 3646 y Fq(1)257 3751 y Fw(and)j Fs(u)519 3766
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3766 y Fp(i)1435 3751 y Fs(\033)1490 3766 y Fq(1)1530
3751 y Fu(!)1630 3715 y Fr(\003)1630 3788 y(R)1690 3766
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3766 y Fq(1)1991 3715 y Fr(\003)1991 3788 y(R)2051 3766
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b(ariables)257 3917 y Fs(x)66 b Fu(2)f(V)8 b Fs(ar)s
Fw(\()p Fs(t)749 3932 y Fp(i)777 3917 y Fw(\))37 b Fu(n)g
Fw(\()p Fu(V)8 b Fs(ar)s Fw(\()p Fs(l)r Fw(\))37 b Fu([)1392
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Fw(that)f Fs(x\033)1427 4053 y Fq(1)1502 4038 y Fw(and)h
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Fq(2)689 4306 y Fw(.)257 4427 y(\(2\))g(Since)g(parallel)d(reduction)j
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4566 y Fr(R)2373 4547 y Fm(d)2373 4583 y(m)2435 4547
y Fs(t)2470 4562 y Fq(1)2544 4547 y Fw(and)33 b Fs(t)g
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4668 y Fm(d)3347 4703 y(n)3393 4667 y Fs(t)3428 4682
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4908 y Fm(d)1744 4944 y(m)1807 4908 y Fs(r)s(\033)k Fw(=)c
Fs(t)2108 4923 y Fq(1)2189 4908 y Fw(for)e(some)h(rule)g
Fs(l)30 b Fu(!)d Fs(r)j Fu(\()e Fs(c)42 b Fu(2)h(R)p
Fw(,)257 5029 y(con)m(taining)29 b(a)i(prop)s(er)f(subredex)j
Fs(s)d Fw(whic)m(h)h(is)f(con)m(tracted)i(to)e Fs(s)2616
4992 y Fr(0)2670 5029 y Fw(in)f(the)i(step)h Fs(t)p Fu(!)3287
5048 y Fr(R)3347 5029 y Fm(d)3347 5064 y(n)3393 5029
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5269 y(subterm)32 b(of)g Fs(x\033)t Fw(.)43 b(So)32 b
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%%Page: 10 10
10 9 bop 257 573 a Fs(t)32 b Fw(suc)m(h)g(that)f Fs(t)p
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537 y Fr(0)2728 573 y Fw(of)c Fs(q)35 b Fw(in)c Fs(t)3063
588 y Fq(1)3102 573 y Fw(,)h(w)m(e)g(ha)m(v)m(e)257 693
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b Fs(t)1005 708 y Fq(3)1072 693 y Fw(=)28 b Fs(t)1211
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b Fs(q)1712 657 y Fr(0)1762 693 y Fu(2)28 b Fs(q)t Fu(n)p
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1092 y(R)1528 1069 y Fm(d)1528 1113 y(m)p Fk(\000)p Fl(1)1669
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Fr(0)2946 1203 y Fu(k)-72 b(!)3024 1167 y Fr(\003)3024
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3851 y(R)3204 3832 y Fm(d)3204 3868 y(n)3250 3818 y Fs(u)f
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3971 y(R)452 3952 y Fm(d)452 3988 y(n)499 3938 y Fs(u)p
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%%Page: 11 11
11 10 bop 257 573 a Fs(x)39 b Fu(2)f(V)8 b Fs(ar)s Fw(\()p
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%%Page: 12 12
12 11 bop 404 573 a Fw(Let)33 b Fu(F)43 b Fw(b)s(e)33
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b Fw(b)s(e)f(the)h(set)f(of)g Fv(marke)-5 b(d)43 b(variables)p
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Fw(.)257 1295 y(The)f(set)f(of)e Fv(marke)-5 b(d)43 b(terms)f
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(the)h(usual)f(w)m(a)m(y)h(and)257 1416 y(is)36 b(denoted)h(b)m(y)g
Fu(T)25 b Fw(\()p Fu(F)1069 1374 y Fr(\003)1108 1416
y Fs(;)17 b Fu(V)1221 1374 y Fr(\003)1261 1416 y Fw(\).)53
b(The)37 b(set)g(of)f(all)e(marks)h(app)s(earing)g(in)h(a)f(mark)m(ed)i
(term)257 1536 y Fs(t)292 1500 y Fr(\003)361 1536 y Fu(2)30
b(T)25 b Fw(\()p Fu(F)656 1494 y Fr(\003)695 1536 y Fs(;)17
b Fu(V)808 1494 y Fr(\003)848 1536 y Fw(\))33 b(is)g(denoted)h(b)m(y)h
Fs(mar)s(k)s(s)p Fw(\()p Fs(t)1880 1500 y Fr(\003)1920
1536 y Fw(\).)46 b(The)34 b(set)g Fu(T)2439 1551 y Fp(w)2496
1536 y Fw(\()p Fu(F)2616 1500 y Fr(\003)2655 1536 y Fs(;)17
b Fu(V)2768 1500 y Fr(\003)2808 1536 y Fw(\))33 b(of)g
Fv(wel)5 b(l-marke)-5 b(d)257 1656 y(terms)36 b Fw(o)m(v)m(er)h
Fu(F)824 1615 y Fr(\003)899 1656 y Fw(and)f Fu(V)1162
1615 y Fr(\003)1237 1656 y Fw(is)f(the)h(subset)h(of)f
Fu(T)25 b Fw(\()p Fu(F)2125 1620 y Fr(\003)2164 1656
y Fs(;)17 b Fu(V)2277 1620 y Fr(\003)2317 1656 y Fw(\))35
b(suc)m(h)i(that)f Fs(t)2863 1620 y Fr(\003)2936 1656
y Fu(2)d(T)3089 1671 y Fp(w)3146 1656 y Fw(\()p Fu(F)3266
1620 y Fr(\003)3305 1656 y Fs(;)17 b Fu(V)3418 1620 y
Fr(\003)3457 1656 y Fw(\))257 1777 y(if)46 b(and)h(only)f(if,)k(for)c
(ev)m(ery)i(pair)e(\()p Fs(t)1650 1741 y Fr(\003)1650
1801 y Fq(1)1690 1777 y Fs(;)17 b(t)1769 1741 y Fr(\003)1769
1801 y Fq(2)1808 1777 y Fw(\))47 b(of)f(subterms)i(of)e
Fs(t)2611 1741 y Fr(\003)2651 1777 y Fw(,)k Fs(mar)s(k)s
Fw(\()p Fs(r)s(oot)p Fw(\()p Fs(t)3252 1741 y Fr(\003)3252
1801 y Fq(1)3292 1777 y Fw(\)\))h(=)257 1897 y Fs(mar)s(k)s
Fw(\()p Fs(r)s(oot)p Fw(\()p Fs(t)781 1861 y Fr(\003)781
1922 y Fq(2)821 1897 y Fw(\)\))c(implies)e Fs(t)1325
1861 y Fr(\003)1325 1922 y Fq(1)1417 1897 y Fw(=)52 b
Fs(t)1580 1861 y Fr(\003)1580 1922 y Fq(2)1620 1897 y
Fw(.)87 b(F)-8 b(or)47 b(example,)j(the)e(term)f(0)2831
1861 y Fq(1)2902 1897 y Fw(+)2978 1861 y Fq(0)3049 1897
y Fw(0)3098 1861 y Fq(1)3185 1897 y Fw(is)f(w)m(ell-)257
2017 y(mark)m(ed)35 b(but)g(0)836 1981 y Fq(1)899 2017
y Fw(+)975 1981 y Fq(1)1038 2017 y Fw(0)1087 1981 y Fq(1)1161
2017 y Fw(is)g(not.)49 b(If)35 b(a)g(term)f Fs(t)1933
1981 y Fr(\003)2007 2017 y Fw(is)h(w)m(ell-mark)m(ed)e(and)i
Fs(\026)d Fu(2)f Fs(mar)s(k)s(s)p Fw(\()p Fs(t)3390 1981
y Fr(\003)3430 2017 y Fw(\),)257 2138 y(then)h Fs(t)513
2102 y Fr(\003)552 2138 y Fu(n)p Fs(\026)e Fw(denotes)i(the)f(unique)g
(subterm)g Fs(s)1947 2102 y Fr(\003)2017 2138 y Fw(of)f
Fs(t)2161 2102 y Fr(\003)2231 2138 y Fw(for)g(whic)m(h)h
Fs(mar)s(k)s Fw(\()p Fs(r)s(oot)p Fw(\()p Fs(s)3190 2102
y Fr(\003)3229 2138 y Fw(\)\))d(=)g Fs(\026)257 2258
y Fw(holds.)50 b(W)-8 b(ell-mark)m(ed)33 b(terms)h(ha)m(v)m(e)i(an)f
(exact)g(corresp)s(ondance)h(to)f(directed)g(acyclic)257
2379 y(graphs;)k(the)e(reader)h(is)e(referred)h(to)g([12)o(])g(for)f
(details.)55 b(In)36 b(con)m(trast)i(to)e([12],)h(w)m(e)h(are)257
2499 y(solely)f(in)m(terested)i(in)e(w)m(ell-mark)m(ed)g(terms.)58
b(Th)m(us,)41 b(throughout)c(the)h(whole)g(pap)s(er,)257
2619 y(mark)m(ed)30 b(stands)h(for)f(w)m(ell-mark)m(ed.)41
b(Tw)m(o)31 b(subterms)g Fs(t)2299 2583 y Fr(\003)2299
2644 y Fq(1)2368 2619 y Fw(and)f Fs(t)2590 2583 y Fr(\003)2590
2644 y Fq(2)2659 2619 y Fw(of)g(a)f(mark)m(ed)h(term)g
Fs(t)3456 2583 y Fr(\003)257 2740 y Fw(are)j(shared)g(in)f
Fs(t)878 2704 y Fr(\003)950 2740 y Fw(if)g Fs(t)1075
2704 y Fr(\003)1075 2764 y Fq(1)1142 2740 y Fw(=)c Fs(t)1281
2704 y Fr(\003)1281 2764 y Fq(2)1320 2740 y Fw(;)33 b(e.g.)g(0)1608
2704 y Fq(1)1680 2740 y Fw(and)f(0)1918 2704 y Fq(1)1990
2740 y Fw(are)h(shared)g(in)f(0)2625 2704 y Fq(1)2686
2740 y Fw(+)2762 2704 y Fq(0)2824 2740 y Fw(0)2873 2704
y Fq(1)2912 2740 y Fw(.)404 2860 y(If)38 b Fs(t)542 2824
y Fr(\003)620 2860 y Fw(is)g(a)g(mark)m(ed)h(term,)h(then)f
Fs(e)p Fw(\()p Fs(t)1778 2824 y Fr(\003)1818 2860 y Fw(\))f(denotes)i
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2980 y Fw(and)h Fs(t)2832 2944 y Fr(\003)2914 2980 y
Fw(are)f Fv(bisimilar)3457 2944 y Fq(1)257 3101 y Fw(\(denoted)36
b(b)m(y)g Fs(s)850 3065 y Fr(\003)922 3101 y Fu(\030)c
Fs(t)1066 3065 y Fr(\003)1106 3101 y Fw(\))j(if)f(and)h(only)f(if)g
Fs(e)p Fw(\()p Fs(s)1900 3065 y Fr(\003)1940 3101 y Fw(\))e(=)g
Fs(e)p Fw(\()p Fs(t)2236 3065 y Fr(\003)2275 3101 y Fw(\).)51
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3101 y Fw(and)257 3221 y Fs(t)292 3185 y Fr(\003)376
3221 y Fw(are)45 b Fv(isomorphic)e Fw(\(denoted)i(b)m(y)h
Fs(s)1672 3185 y Fr(\003)1759 3194 y Fu(\030)1760 3225
y Fw(=)1885 3221 y Fs(t)1920 3185 y Fr(\003)1959 3221
y Fw(\))f(if)e(and)i(only)f(if)f Fs(t)2707 3185 y Fr(\003)2791
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3305 y Fr(\003)627 3342 y Fw(b)m(y)i(a)f Fv(r)-5 b(enaming)42
b Fw(of)g(marks,)k(that)d(is,)j(there)e(exists)g(a)f(bijectiv)m(e)g
(function)257 3462 y(\010)31 b(:)f Fs(mar)s(k)s(s)p Fw(\()p
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Fw(\()p Fs(t)1376 3426 y Fr(\003)1416 3462 y Fw(\))k(suc)m(h)h(that)f
(\010\()p Fs(s)2076 3426 y Fr(\003)2116 3462 y Fw(\))c(=)g
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y Fw(\()p Fu(F)607 3546 y Fr(\003)646 3582 y Fs(;)17
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661 3862 y(\010\()p Fs(s)815 3821 y Fr(\003)854 3862
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3765 y Fq(\010\()p Fp(\026)p Fq(\))2178 3801 y Fw(if)j
Fs(s)2313 3765 y Fr(\003)2380 3801 y Fw(=)d Fs(x)2539
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b Fs(;)1140 3923 y(f)1199 3887 y Fq(\010\()p Fp(\026)p
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3923 y Fw(\)\))83 b(if)31 b Fs(s)2313 3887 y Fr(\003)2380
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Fs(t)2662 3887 y Fr(\003)2662 3948 y Fq(1)2702 3923 y
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3948 y Fp(n)3003 3923 y Fw(\))257 4137 y(Note)46 b(that)e
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b Fs(t)1749 4100 y Fr(\003)1789 4137 y Fw(.)81 b(The)46
b(marks)f(of)f(a)h(mark)m(ed)g(term)g Fs(s)3281 4100
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b(esh)33 b Fw(w.r.t.)i(another)f(mark)m(ed)g(term)f Fs(t)2007
4221 y Fr(\003)2081 4257 y Fw(if)g Fs(mar)s(k)s(s)p Fw(\()p
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4257 y Fw(\))30 b(=)g Fu(;)p Fw(.)47 b(A)257 4377 y Fv(marke)-5
b(d)44 b(substitution)f Fs(\033)1199 4341 y Fr(\003)1284
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b Fw(\()p Fu(F)1854 4341 y Fr(\003)1893 4377 y Fs(;)17
b Fu(V)2006 4341 y Fr(\003)2045 4377 y Fw(\))43 b(is)f(a)h
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4462 y Fp(\026)359 4498 y Fs(\033)418 4462 y Fr(\003)501
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4462 y Fr(\003)861 4498 y Fw(for)d(all)f Fs(x)1219 4462
y Fp(\026)1266 4498 y Fs(;)17 b(x)1365 4462 y Fp(\027)1451
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4498 y Fw(\))e(with)f Fs(sy)t(mbol)r Fw(\()p Fs(x)2483
4462 y Fp(\026)2530 4498 y Fw(\))i(=)g Fs(sy)t(mbol)r
Fw(\()p Fs(x)3125 4462 y Fp(\027)3169 4498 y Fw(\).)71
b(This)257 4618 y(de\014nition)46 b(of)h(mark)m(ed)g(substitution)g
(ensures)i(that)e(the)g(unmark)m(ed)h(substitution)257
4738 y Fs(\033)e Fw(obtained)41 b(from)g Fs(\033)1067
4702 y Fr(\003)1148 4738 y Fw(b)m(y)i(erasing)e(all)f(marks)h(is)g(w)m
(ell-de\014ned)h(\(i.e.,)i Fs(\033)i Fw(really)40 b(is)h(a)257
4859 y(substitution\).)h(Let)29 b Fs(\033)1105 4823 y
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4823 y Fr(\003)1413 4883 y Fq(2)1485 4859 y Fw(b)s(e)h(mark)m(ed)g
(substitutions)f(with)h Fu(D)s Fw(\()p Fs(\033)2930 4823
y Fr(\003)2926 4883 y Fq(1)2969 4859 y Fw(\))e(=)h Fu(D)s
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4941 y Fl(1)362 5004 y Fq(1)444 4979 y Fs(;)17 b(:)g(:)g(:)f(;)h(x)718
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y Fu(g)p Fw(.)40 b Fs(\033)983 4943 y Fr(\003)979 5004
y Fq(1)1044 4979 y Fw(and)22 b Fs(\033)1282 4943 y Fr(\003)1278
5004 y Fq(2)1343 4979 y Fw(are)g Fv(isomorphic)e Fw(\()p
Fs(\033)2078 4943 y Fr(\003)2074 5004 y Fq(1)2145 4952
y Fu(\030)2146 4983 y Fw(=)2250 4979 y Fs(\033)2309 4943
y Fr(\003)2305 5004 y Fq(2)2349 4979 y Fw(\))i(if)e Fs(D)2571
4943 y Fp(\026)2618 4979 y Fw(\()p Fs(x)2711 4932 y Fp(\026)2753
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4943 y Fr(\003)2847 5004 y Fq(1)2891 4979 y Fs(;)d(:)g(:)g(:)e(;)i(x)
3164 4943 y Fp(\026)3206 4951 y Fm(n)3164 5004 y Fp(n)3254
4979 y Fs(\033)3313 4943 y Fr(\003)3309 5004 y Fq(1)3352
4979 y Fw(\))3418 4952 y Fu(\030)3419 4983 y Fw(=)257
5100 y Fs(D)341 5063 y Fp(\026)388 5100 y Fw(\()p Fs(x)481
5052 y Fp(\026)523 5061 y Fl(1)481 5124 y Fq(1)562 5100
y Fs(\033)621 5063 y Fr(\003)617 5124 y Fq(2)661 5100
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y Fm(n)935 5124 y Fp(n)1024 5100 y Fs(\033)1083 5063
y Fr(\003)1079 5124 y Fq(2)1122 5100 y Fw(\),)35 b(where)h
Fs(D)h Fw(is)e(a)f(fresh)h(sym)m(b)s(ol)f(of)g(arit)m(y)g
Fs(n)h Fw(and)g Fs(\026)f Fw(is)g(a)g(fresh)p 257 5193
1296 4 v 370 5254 a Fg(1)407 5284 y Ff(The)27 b(origin)g(of)g(the)h
(notion)g(\\bisimilarit)n(y")e(is)h(explained)g(in)h([2].)1828
5637 y Fw(12)p eop
%%Page: 13 13
13 12 bop 257 573 a Fw(mark)31 b(w.r.t.)g(ev)m(ery)h
Fs(x)1076 526 y Fp(\026)1118 536 y Fm(i)1076 598 y Fp(i)1150
573 y Fs(\033)1209 537 y Fr(\003)1205 597 y Fp(j)1248
573 y Fw(,)f(where)h(1)c Fu(\024)g Fs(i)g Fu(\024)g Fs(n)j
Fw(and)g(1)c Fu(\024)h Fs(j)34 b Fu(\024)28 b Fw(2.)43
b(The)31 b(notion)f Fv(marke)-5 b(d)257 693 y(c)g(ontext)33
b Fw(is)f(de\014ned)i(in)e(the)h(ob)m(vious)f(w)m(a)m(y)-8
b(.)404 814 y(F)g(or)34 b(instance,)j(0)1041 777 y Fq(1)1105
814 y Fw(+)1181 777 y Fq(0)1244 814 y Fw(0)1293 777 y
Fq(1)1365 786 y Fu(\030)1366 818 y Fw(=)1476 814 y(0)1525
777 y Fq(2)1588 814 y Fw(+)1664 777 y Fq(0)1728 814 y
Fw(0)1777 777 y Fq(2)1851 814 y Fw(but)f(0)2082 777 y
Fq(1)2146 814 y Fw(+)2222 777 y Fq(0)2285 814 y Fw(0)2334
777 y Fq(1)2406 814 y Fu(6)2406 786 y(\030)2407 818 y
Fw(=)2517 814 y(0)2566 777 y Fq(1)2629 814 y Fw(+)2705
777 y Fq(0)2769 814 y Fw(0)2818 777 y Fq(2)2857 814 y
Fw(.)52 b(On)36 b(the)g(other)257 934 y(hand,)d(0)577
898 y Fq(1)637 934 y Fw(+)713 898 y Fq(0)773 934 y Fw(0)822
898 y Fq(1)888 934 y Fu(\030)c Fw(0)1043 898 y Fq(1)1102
934 y Fw(+)1178 898 y Fq(0)1238 934 y Fw(0)1287 898 y
Fq(2)1327 934 y Fw(.)43 b(Moreo)m(v)m(er,)33 b Fs(\033)1910
898 y Fr(\003)1906 959 y Fq(1)1978 934 y Fw(=)27 b Fu(f)p
Fs(x)2186 949 y Fq(1)2253 934 y Fu(7!)h Fw(0)2430 898
y Fq(1)2490 934 y Fw(+)2566 898 y Fq(0)2626 934 y Fw(0)2675
898 y Fq(1)2714 934 y Fs(;)17 b(x)2813 949 y Fq(2)2880
934 y Fu(7!)27 b Fw(0)3056 898 y Fq(1)3116 934 y Fw(+)3192
898 y Fq(0)3252 934 y Fw(0)3301 898 y Fq(1)3340 934 y
Fu(g)h(6)3418 906 y(\030)3419 938 y Fw(=)257 1054 y Fu(f)p
Fs(x)362 1069 y Fq(1)430 1054 y Fu(7!)f Fw(0)606 1018
y Fq(1)667 1054 y Fw(+)743 1018 y Fq(0)805 1054 y Fw(0)854
1018 y Fq(1)893 1054 y Fs(;)17 b(x)992 1069 y Fq(2)1059
1054 y Fu(7!)28 b Fw(0)1236 1018 y Fq(3)1297 1054 y Fw(+)1373
1018 y Fq(2)1434 1054 y Fw(0)1483 1018 y Fq(3)1523 1054
y Fu(g)f Fw(=)h Fs(\033)1763 1018 y Fr(\003)1759 1079
y Fq(2)1802 1054 y Fw(.)257 1283 y Fo(De\014nition)36
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Fs(c)1775 1247 y Fr(\003)1844 1283 y Fw(is)i(a)g Fv(marke)-5
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Fw(\()p Fs(l)670 1367 y Fr(\003)710 1403 y Fw(\))k(=)f
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1403 y Fw(\))g(=)f Fs(r)s Fw(,)g Fs(e)p Fw(\()p Fs(c)1529
1367 y Fr(\003)1568 1403 y Fw(\))h(=)g Fs(c)p Fw(,)f(and,)g(for)e(all)e
Fs(x)2370 1367 y Fp(\026)2417 1403 y Fs(;)17 b(y)2513
1367 y Fp(\027)2583 1403 y Fu(2)28 b(V)8 b Fs(ar)s Fw(\()p
Fs(l)2913 1367 y Fr(\003)2980 1403 y Fu(!)28 b Fs(r)3155
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1403 y Fw(\),)257 1523 y Fs(sy)t(mbol)r Fw(\()p Fs(x)652
1487 y Fp(\026)699 1523 y Fw(\))g(=)g Fs(sy)t(mbol)r
Fw(\()p Fs(y)1261 1487 y Fp(\027)1303 1523 y Fw(\))k(if)g(and)g(only)g
(if)g Fs(mar)s(k)s Fw(\()p Fs(x)2286 1487 y Fp(\026)2333
1523 y Fw(\))c(=)f Fs(mar)s(k)s Fw(\()p Fs(y)2829 1487
y Fp(\027)2872 1523 y Fw(\).)404 1752 y(The)43 b(last)e(condition)g
(can)h(b)s(e)h(rephrased)g(as:)63 b(ev)m(ery)44 b(mark)m(ed)e(o)s
(ccurrence)i(of)d(a)257 1872 y(v)-5 b(ariable)29 b Fs(x)f
Fu(2)g(V)8 b Fs(ar)s Fw(\()p Fs(l)30 b Fu(!)e Fs(r)i
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Fr(\003)2426 3089 y Fp(k)2468 3064 y Fw(==)p Fs(t)2655
3027 y Fr(\003)2655 3089 y Fp(k)2750 3064 y Fw(of)52
b(a)g(rewrite)h(rule)257 3184 y Fs(\032)28 b Fw(:)g Fs(l)i
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Fw(==)p Fs(t)1050 3199 y Fq(1)1090 3184 y Fs(;)17 b(:)g(:)g(:)32
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(substitution)e Fs(\033)3159 3148 y Fr(\003)3244 3184
y Fw(and)h(a)257 3304 y(mark)m(ed)33 b(con)m(text)h Fs(C)1028
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Fw(])33 b(suc)m(h)h(that)403 3508 y Fu(\017)48 b Fs(s)547
3472 y Fr(\003)614 3508 y Fw(=)28 b Fs(C)795 3472 y Fr(\003)834
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(ground)g(constructor)h(terms)f Fs(u)3270 3878 y Fr(\003)3270
3939 y Fp(i)3338 3914 y Fw(and)501 4035 y Fs(v)552 3999
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4059 y Fp(i)1809 4035 y Fs(\033)1868 3999 y Fr(\003)1935
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4911 y Fs(s)303 4875 y Fr(\003)375 4911 y Fu(\))475 4875
y Fp(l)497 4851 y Fk(\003)533 4875 y Fp(\033)575 4851
y Fk(\003)475 4936 y Fr(R)535 4944 y Fm(n)648 4911 y
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Fw(13)p eop
%%Page: 14 14
14 13 bop 257 573 a Fo(De\014nition)36 b(4.3)49 b Fw(The)44
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257 814 y(additionally)24 b(required)k(that)f Fs(x\033)1488
777 y Fr(\003)1554 814 y Fw(is)g(a)g(mark)m(ed)g(ground)g(constructor)h
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Fr(R)1802 1395 y Fw(\(in)32 b(fact,)65 b Fu(\))2306 1414
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299 1605 y Fs(doubl)r(e)570 1569 y Fq(0)609 1605 y Fw(\()p
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1726 y Fw(\()p Fs(t)2620 1690 y Fr(\003)2660 1726 y Fw(\))1569
1846 y Fu(\))1669 1861 y Fr(R)1849 1846 y Fw(\(0)1936
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y Fs(s)2216 1810 y Fq(9)2255 1846 y Fw(\(0)2342 1810
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y Fw(\(0)3145 1810 y Fq(10)3220 1846 y Fw(\)\))1569 1967
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2896 y(mark)m(ed)23 b(term)e(is)h(an)g(adequate)h(mapping)d(in)h(the)i
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(is)257 3137 y(co\014nal.)42 b(Surjectivit)m(y)27 b(ensures)j(that)d
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3257 y(acyclic)g(graph)g(\(w)m(ell-mark)m(ed)g(term\).)41
b(The)30 b(normal)c(form)h(condition)g(ensures)j(that)e(a)257
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3498 y(and)i(vice)h(v)m(ersa.)43 b(Preserv)-5 b(ation)29
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3618 y(sequence)41 b(represen)m(ts)f(some)d(term)g(reduction)h
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b(Recall)34 b(that)i Fu(R)g Fw(denotes)h(an)f(almost)e(functional)g
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y Fo(Theorem)k(4.4)49 b Fv(F)-7 b(or)27 b(al)5 b(l)27
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Fr(R)1737 4321 y Fm(d)1737 4356 y(n)1846 4320 y Fv(is)27
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y(n)1623 5388 y Fs(e)p Fw(\()p Fs(t)1741 5352 y Fr(\003)1781
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%%Page: 15 15
15 14 bop 257 573 a Fv(Pr)-5 b(o)g(of)67 b Fw(W)-8 b(e)33
b(use)h(induction)e(on)g Fs(n)p Fw(.)257 693 y(\(1\))h(Surjectivit)m(y)
f(is)g(ob)m(vious.)257 814 y(\(2\))k(The)h Fv(if)f Fw(direction)f(is)h
(easily)f(sho)m(wn.)56 b(F)-8 b(or)35 b(an)i(indirect)e(pro)s(of)g(of)h
(the)g Fv(only)j(if)c Fw(di-)257 934 y(rection,)44 b(supp)s(ose)g
Fs(e)p Fw(\()p Fs(t)1116 898 y Fr(\003)1155 934 y Fw(\))g
Fu(62)g Fs(N)10 b(F)k Fw(\()p Fu(!)1650 953 y Fr(R)1710
934 y Fm(d)1710 970 y(n)1757 934 y Fw(\),)44 b(i.e.,)g
Fs(e)p Fw(\()p Fs(t)2180 898 y Fr(\003)2220 934 y Fw(\))f(=)h
Fs(C)7 b Fw([)p Fs(l)r(\033)t Fw(])p Fu(!)2742 953 y
Fr(R)2802 934 y Fm(d)2802 970 y(n)2849 934 y Fs(C)g Fw([)p
Fs(r)s(\033)t Fw(])42 b(b)m(y)g(using)257 1054 y(the)29
b(rule)e Fs(\032)h Fw(:)g Fs(l)h Fu(!)f Fs(r)i Fu(\()d
Fs(s)1177 1069 y Fq(1)1217 1054 y Fw(==)p Fs(t)1404 1069
y Fq(1)1443 1054 y Fs(;)17 b(:)g(:)g(:)33 b(;)17 b(s)1725
1069 y Fp(k)1767 1054 y Fw(==)p Fs(t)1954 1069 y Fp(k)2024
1054 y Fw(at)28 b(p)s(osition)e Fs(p)p Fw(.)41 b(So,)29
b(for)e(ev)m(ery)j Fs(s)3225 1069 y Fp(i)3253 1054 y
Fw(==)p Fs(t)3440 1069 y Fp(i)3468 1054 y Fw(,)257 1175
y(there)e(is)f(a)g(ground)g(constructor)h(term)f Fs(u)1789
1190 y Fp(i)1843 1175 y Fw(suc)m(h)i(that)e Fs(s)2310
1190 y Fp(i)2338 1175 y Fs(\033)t Fu(!)2497 1139 y Fr(\003)2497
1212 y(R)2557 1189 y Fm(d)2557 1233 y(n)p Fk(\000)p Fl(1)2682
1175 y Fs(u)2738 1190 y Fp(i)2793 1175 y Fw(and)g Fs(t)3012
1190 y Fp(i)3041 1175 y Fs(\033)t Fu(!)3200 1139 y Fr(\003)3200
1212 y(R)3260 1189 y Fm(d)3260 1233 y(n)p Fk(\000)p Fl(1)3384
1175 y Fs(u)3440 1190 y Fp(i)3468 1175 y Fw(.)257 1323
y(Let)48 b Fs(l)478 1287 y Fr(\003)565 1323 y Fw(and)f
Fs(\033)828 1287 y Fr(\003)915 1323 y Fw(b)s(e)g(mark)m(ed)h(v)m
(ersion)g(of)f Fs(l)i Fw(and)f Fs(\033)j Fw(suc)m(h)e(that)e
Fs(t)2781 1287 y Fr(\003)2820 1323 y Fu(j)2848 1338 y
Fp(p)2941 1323 y Fw(=)52 b Fs(l)3100 1287 y Fr(\003)3140
1323 y Fs(\033)3199 1287 y Fr(\003)3238 1323 y Fw(.)88
b(Let)257 1444 y Fs(l)288 1407 y Fr(\003)383 1444 y Fu(!)55
b Fs(r)585 1407 y Fr(\003)679 1444 y Fu(\()f Fs(s)879
1407 y Fr(\003)879 1468 y Fq(1)919 1444 y Fw(==)o Fs(t)1105
1407 y Fr(\003)1105 1468 y Fq(1)1145 1444 y Fs(;)17 b(:)g(:)g(:)32
b(;)17 b(s)1426 1407 y Fr(\003)1426 1469 y Fp(k)1469
1444 y Fw(==)o Fs(t)1655 1407 y Fr(\003)1655 1469 y Fp(k)1747
1444 y Fw(b)s(e)49 b(a)f(mark)m(ed)h(v)m(ersion)g(of)f
Fs(\032)h Fw(suc)m(h)h(that)e(all)257 1564 y(marks)40
b(on)g Fs(r)745 1528 y Fr(\003)784 1564 y Fw(,)i Fs(s)899
1528 y Fr(\003)899 1589 y Fp(i)938 1564 y Fw(,)g(and)e
Fs(t)1239 1528 y Fr(\003)1239 1589 y Fp(i)1318 1564 y
Fw(are)g(fresh)g(w.r.t.)h Fs(t)2034 1528 y Fr(\003)2113
1564 y Fw(and)f(m)m(utually)e(distinct.)65 b(F)-8 b(urther-)257
1684 y(more,)50 b Fs(\033)605 1648 y Fr(\003)691 1684
y Fw(is)c(extended)i(to)f Fu(E)9 b(V)e Fs(ar)s Fw(\()p
Fs(\032)p Fw(\))47 b(in)f(the)h(usual)f(w)m(a)m(y:)73
b(for)46 b(all)e Fs(z)57 b Fu(2)51 b(E)9 b(V)f Fs(ar)s
Fw(\()p Fs(\032)p Fw(\))257 1805 y(let)46 b Fs(z)t(\033)520
1769 y Fr(\003)606 1805 y Fw(b)s(e)g(a)g(mark)m(ed)g(v)m(ersion)g(of)g
Fs(z)t(\033)51 b Fw(suc)m(h)c(that)f(all)e(marks)i(are)g(m)m(utually)e
(dis-)257 1925 y(tinct)35 b(and)f(fresh)i(w.r.t.)f Fs(t)1220
1889 y Fr(\003)1260 1925 y Fw(,)g Fs(r)1369 1889 y Fr(\003)1408
1925 y Fw(,)p Fs(s)1481 1889 y Fr(\003)1481 1950 y Fp(i)1521
1925 y Fw(,)g(and)g Fs(t)1810 1889 y Fr(\003)1810 1950
y Fp(i)1849 1925 y Fw(.)50 b(Let)35 b Fs(C)2180 1889
y Fr(\003)2220 1925 y Fw([)p Fs(;)17 b(:)g(:)g(:)e(;)i
Fw(])35 b(b)s(e)g(the)g(mark)m(ed)g(con)m(text)257 2045
y(suc)m(h)49 b(that)d Fs(t)752 2009 y Fr(\003)843 2045
y Fw(=)52 b Fs(C)1048 2009 y Fr(\003)1087 2045 y Fw([)p
Fs(l)1145 2009 y Fr(\003)1185 2045 y Fs(\033)1244 2009
y Fr(\003)1284 2045 y Fs(;)17 b(:)g(:)g(:)e(;)i(l)1533
2009 y Fr(\003)1573 2045 y Fs(\033)1632 2009 y Fr(\003)1671
2045 y Fw(])47 b(and)g Fs(l)1980 2009 y Fr(\003)2019
2045 y Fs(\033)2078 2009 y Fr(\003)2165 2045 y Fw(is)f(not)g(a)h
(subterm)g(of)f Fs(C)3156 2009 y Fr(\003)3195 2045 y
Fw([)p Fs(;)17 b(:)g(:)g(:)f(;)h Fw(].)257 2166 y(Since)49
b Fs(e)p Fw(\()p Fs(s)657 2130 y Fr(\003)657 2190 y Fp(i)696
2166 y Fs(\033)755 2130 y Fr(\003)795 2166 y Fw(\))54
b(=)h Fs(s)1064 2181 y Fp(i)1092 2166 y Fs(\033)t Fu(!)1251
2130 y Fr(\003)1251 2203 y(R)1311 2180 y Fm(d)1311 2224
y(n)p Fk(\000)p Fl(1)1436 2166 y Fs(u)1492 2181 y Fp(i)1520
2166 y Fw(,)d Fs(e)p Fw(\()p Fs(t)1717 2130 y Fr(\003)1717
2190 y Fp(i)1757 2166 y Fs(\033)1816 2130 y Fr(\003)1855
2166 y Fw(\))j(=)f Fs(t)2113 2181 y Fp(i)2142 2166 y
Fs(\033)t Fu(!)2301 2130 y Fr(\003)2301 2203 y(R)2361
2180 y Fm(d)2361 2224 y(n)p Fk(\000)p Fl(1)2486 2166
y Fs(u)2542 2181 y Fp(i)2569 2166 y Fw(,)f(and)48 b Fs(u)2910
2181 y Fp(i)2986 2166 y Fw(is)g(a)g(ground)257 2309 y(constructor)h
(term,)i(it)c(follo)m(ws)f(from)g(the)i(inductiv)m(e)g(h)m(yp)s
(othesis)h(that)e(there)i(ex-)257 2429 y(ist)44 b(mark)m(ed)h(terms)g
Fs(v)1099 2393 y Fr(\003)1095 2454 y Fp(i)1182 2429 y
Fw(and)g Fs(w)1457 2393 y Fr(\003)1454 2454 y Fp(i)1540
2429 y Fw(suc)m(h)h(that)e Fs(s)2041 2393 y Fr(\003)2041
2454 y Fp(i)2081 2429 y Fs(\033)2140 2393 y Fr(\003)2212
2429 y Fu(\))2312 2393 y Fr(\003)2312 2466 y(R)2372 2443
y Fm(d)2372 2487 y(n)p Fk(\000)p Fl(1)2496 2429 y Fs(v)2547
2393 y Fr(\003)2543 2454 y Fp(i)2587 2429 y Fw(,)j Fs(t)2696
2393 y Fr(\003)2696 2454 y Fp(i)2736 2429 y Fs(\033)2795
2393 y Fr(\003)2867 2429 y Fu(\))2967 2393 y Fr(\003)2967
2466 y(R)3027 2443 y Fm(d)3027 2487 y(n)p Fk(\000)p Fl(1)3152
2429 y Fs(w)3225 2393 y Fr(\003)3222 2454 y Fp(i)3264
2429 y Fw(,)g(and)257 2577 y Fs(e)p Fw(\()p Fs(v)391
2541 y Fr(\003)387 2602 y Fp(i)431 2577 y Fw(\))27 b(=)h
Fs(u)656 2592 y Fp(i)711 2577 y Fw(=)g Fs(e)p Fw(\()p
Fs(w)971 2541 y Fr(\003)968 2602 y Fp(i)1010 2577 y Fw(\).)40
b(The)26 b(latter)d(particularly)f(implies)g(that)i Fs(v)2662
2541 y Fr(\003)2658 2602 y Fp(i)2725 2577 y Fw(and)h
Fs(w)2980 2541 y Fr(\003)2977 2602 y Fp(i)3043 2577 y
Fw(are)f(ground)257 2698 y(constructor)34 b(terms)f(and)g
Fs(v)1286 2662 y Fr(\003)1282 2722 y Fp(i)1353 2698 y
Fu(\030)c Fs(w)1532 2662 y Fr(\003)1529 2722 y Fp(i)1571
2698 y Fw(.)44 b(Therefore,)34 b Fs(t)2146 2662 y Fr(\003)2219
2698 y Fu(\))2319 2717 y Fr(R)2379 2698 y Fm(d)2379 2734
y(n)2458 2698 y Fs(C)2535 2662 y Fr(\003)2574 2698 y
Fw([)p Fs(r)2648 2662 y Fr(\003)2687 2698 y Fs(\033)2746
2662 y Fr(\003)2786 2698 y Fs(;)17 b(:)g(:)g(:)e(;)i(r)3051
2662 y Fr(\003)3090 2698 y Fs(\033)3149 2662 y Fr(\003)3189
2698 y Fw(])33 b(whic)m(h)257 2818 y(con)m(tradicts)g
Fs(t)794 2782 y Fr(\003)862 2818 y Fu(2)28 b Fs(N)10
b(F)k Fw(\()32 b Fu(\))1291 2837 y Fr(R)1351 2818 y Fm(d)1351
2854 y(n)1430 2818 y Fw(\).)257 2939 y(\(3\))55 b(W)-8
b(e)55 b(pro)s(ceed)g(b)m(y)h(induction)e(on)g(the)h(length)g
Fs(`)f Fw(of)g Fs(s)2565 2902 y Fr(\003)2637 2939 y Fu(\))2737
2902 y Fr(\003)2737 2972 y(R)2797 2953 y Fm(d)2797 2988
y(n)2876 2939 y Fs(t)2911 2902 y Fr(\003)2951 2939 y
Fw(.)110 b(The)55 b(base)257 3080 y(case)g Fs(`)65 b
Fw(=)f(0)54 b(clearly)f(holds.)108 b(Th)m(us)56 b(consider)e
Fs(s)2241 3043 y Fr(\003)2313 3080 y Fu(\))2413 3043
y Fp(l)2435 3020 y Fk(\003)2470 3043 y Fp(\033)2512 3020
y Fk(\003)2413 3113 y Fr(R)2473 3094 y Fm(d)2473 3129
y(n)2586 3080 y Fs(u)2642 3043 y Fr(\003)2713 3080 y
Fu(\))2813 3043 y Fp(`)2813 3113 y Fr(R)2873 3094 y Fm(d)2873
3129 y(n)2952 3080 y Fs(t)2987 3043 y Fr(\003)3027 3080
y Fw(.)108 b(Accord-)257 3220 y(ing)49 b(to)g(the)h(inductiv)m(e)g(h)m
(yp)s(othesis)h(on)f Fs(`)p Fw(,)k Fs(e)p Fw(\()p Fs(u)2110
3184 y Fr(\003)2149 3220 y Fw(\))p Fu(!)2287 3184 y Fr(\003)2287
3254 y(R)2347 3235 y Fm(d)2347 3270 y(n)2393 3220 y Fs(e)p
Fw(\()p Fs(t)2511 3184 y Fr(\003)2551 3220 y Fw(\).)95
b(Since)49 b Fs(s)3028 3184 y Fr(\003)3100 3220 y Fu(\))3200
3184 y Fp(l)3222 3161 y Fk(\003)3258 3184 y Fp(\033)3300
3161 y Fk(\003)3200 3254 y Fr(R)3260 3235 y Fm(d)3260
3270 y(n)3373 3220 y Fs(u)3429 3184 y Fr(\003)3468 3220
y Fw(,)257 3353 y(w)m(e)c(ha)m(v)m(e)g Fs(s)694 3317
y Fr(\003)780 3353 y Fw(=)h Fs(C)979 3317 y Fr(\003)1019
3353 y Fw([)p Fs(l)1077 3317 y Fr(\003)1117 3353 y Fs(\033)1176
3317 y Fr(\003)1215 3353 y Fs(;)17 b(:)g(:)g(:)f(;)h(l)1465
3317 y Fr(\003)1504 3353 y Fs(\033)1563 3317 y Fr(\003)1603
3353 y Fw(],)46 b Fs(l)1734 3317 y Fr(\003)1774 3353
y Fs(\033)1833 3317 y Fr(\003)1916 3353 y Fw(is)d(not)h(a)f(subterm)h
(of)f Fs(C)2892 3317 y Fr(\003)2932 3353 y Fw([)p Fs(;)17
b(:)g(:)g(:)f(;)h Fw(],)46 b Fs(u)3334 3317 y Fr(\003)3419
3353 y Fw(=)257 3474 y Fs(C)334 3438 y Fr(\003)374 3474
y Fw([)p Fs(r)448 3438 y Fr(\003)487 3474 y Fs(\033)546
3438 y Fr(\003)585 3474 y Fs(;)17 b(:)g(:)g(:)f(;)h(r)851
3438 y Fr(\003)890 3474 y Fs(\033)949 3438 y Fr(\003)988
3474 y Fw(],)28 b(and,)g(for)d(ev)m(ery)j Fs(s)1721 3438
y Fr(\003)1721 3498 y Fp(i)1760 3474 y Fw(==)p Fs(t)1947
3438 y Fr(\003)1947 3498 y Fp(i)1987 3474 y Fw(,)f(there)g(are)f(mark)m
(ed)g(ground)g(construc-)257 3594 y(tor)40 b(terms)g
Fs(u)757 3558 y Fr(\003)757 3619 y Fp(i)837 3594 y Fw(and)g
Fs(v)1085 3558 y Fr(\003)1081 3619 y Fp(i)1164 3594 y
Fw(suc)m(h)i(that)e Fs(s)1657 3558 y Fr(\003)1657 3619
y Fp(i)1697 3594 y Fs(\033)1756 3558 y Fr(\003)1828 3594
y Fu(\))1928 3558 y Fr(\003)1928 3631 y(R)1988 3609 y
Fm(d)1988 3653 y(n)p Fk(\000)p Fl(1)2112 3594 y Fs(u)2168
3558 y Fr(\003)2168 3619 y Fp(i)2207 3594 y Fw(,)j Fs(t)2312
3558 y Fr(\003)2312 3619 y Fp(i)2351 3594 y Fs(\033)2410
3558 y Fr(\003)2482 3594 y Fu(\))2582 3558 y Fr(\003)2582
3631 y(R)2642 3609 y Fm(d)2642 3653 y(n)p Fk(\000)p Fl(1)2767
3594 y Fs(v)2818 3558 y Fr(\003)2814 3619 y Fp(i)2857
3594 y Fw(,)g(and)d Fs(u)3180 3558 y Fr(\003)3180 3619
y Fp(i)3260 3594 y Fu(\030)h Fs(v)3429 3558 y Fr(\003)3425
3619 y Fp(i)3468 3594 y Fw(.)257 3743 y(Let)h Fs(\033)k
Fw(=)41 b Fs(e)p Fw(\()p Fs(\033)801 3706 y Fr(\003)841
3743 y Fw(\),)i(i.e.,)g Fs(x\033)j Fw(=)c Fs(e)p Fw(\()p
Fs(x\033)1615 3706 y Fr(\003)1655 3743 y Fw(\))f(for)f(all)f
Fs(x)k Fu(2)f(D)s Fw(\()p Fs(\033)2418 3706 y Fr(\003)2457
3743 y Fw(\).)69 b(By)41 b(the)h(inductiv)m(e)f(h)m(y-)257
3863 y(p)s(othesis)29 b(on)g Fs(n)p Fw(,)h Fs(e)p Fw(\()p
Fs(s)1007 3827 y Fr(\003)1007 3888 y Fp(i)1046 3863 y
Fw(\))p Fs(\033)t Fu(!)1243 3827 y Fr(\003)1243 3900
y(R)1303 3877 y Fm(d)1303 3921 y(n)p Fk(\000)p Fl(1)1428
3863 y Fs(e)p Fw(\()p Fs(u)1567 3827 y Fr(\003)1567 3888
y Fp(i)1606 3863 y Fw(\))e(=)f Fs(e)p Fw(\()p Fs(v)1909
3827 y Fr(\003)1905 3888 y Fp(i)1949 3863 y Fw(\))2052
3827 y Fr(\003)2052 3900 y(R)2112 3877 y Fm(d)2112 3921
y(n)p Fk(\000)p Fl(1)2165 3863 y Fu( )32 b Fs(e)p Fw(\()p
Fs(t)2415 3827 y Fr(\003)2415 3888 y Fp(i)2455 3863 y
Fw(\))p Fs(\033)t Fw(.)42 b(Hence)30 b Fs(l)r(\033)t
Fu(!)3097 3882 y Fr(R)3157 3863 y Fm(d)3157 3899 y(n)3204
3863 y Fs(r)s(\033)i Fw(and)257 4018 y Fs(e)p Fw(\()p
Fs(s)386 3982 y Fr(\003)426 4018 y Fw(\))c Fu(!)592 3977
y Fq(+)592 4053 y Fr(R)652 4034 y Fm(d)652 4070 y(n)726
4018 y Fs(e)p Fw(\()p Fs(t)844 3982 y Fr(\003)883 4018
y Fw(\).)257 4160 y(\(4\))53 b(W)-8 b(e)54 b(use)h(induction)d(on)h
(the)h(length)f Fs(`)h Fw(of)f Fs(e)p Fw(\()p Fs(s)2288
4124 y Fr(\003)2327 4160 y Fw(\))p Fu(!)2465 4124 y Fp(`)2465
4194 y Fr(R)2525 4175 y Fm(d)2525 4210 y(n)2604 4160
y Fs(u)p Fw(.)106 b(The)54 b(pro)s(of)f(is)g(il-)257
4281 y(lustrated)k(in)e(Figure)g(2.)115 b(The)57 b(case)g
Fs(`)68 b Fw(=)g(0)56 b(holds)g(v)-5 b(acuously)d(.)115
b(So)56 b(w)m(e)h(con-)257 4401 y(sider)45 b Fs(e)p Fw(\()p
Fs(s)632 4365 y Fr(\003)671 4401 y Fw(\))p Fu(!)809 4365
y Fp(`)809 4434 y Fr(R)869 4415 y Fm(d)869 4451 y(n)954
4401 y Fw(\026)-55 b Fs(u)p Fu(!)1104 4420 y Fr(R)1164
4401 y Fm(d)1164 4437 y(n)1210 4401 y Fs(u)p Fw(.)78
b(By)45 b(the)f(inductiv)m(e)h(h)m(yp)s(othesis)g(on)f
Fs(`)p Fw(,)j(there)e(exists)g(a)259 4516 y(\026)257
4534 y Fs(t)292 4498 y Fr(\003)360 4534 y Fu(2)28 b(T)508
4549 y Fp(w)565 4534 y Fw(\()p Fu(F)685 4498 y Fr(\003)724
4534 y Fs(;)17 b Fu(V)837 4498 y Fr(\003)876 4534 y Fw(\))29
b(suc)m(h)i(that)e Fs(s)1414 4498 y Fr(\003)1486 4534
y Fu(\))1586 4498 y Fr(\003)1586 4567 y(R)1646 4548 y
Fm(d)1646 4584 y(n)1726 4516 y Fw(\026)1725 4534 y Fs(t)1760
4498 y Fr(\003)1829 4534 y Fw(and)35 b(\026)-55 b Fs(u)p
Fu(!)2171 4498 y Fr(\003)2171 4567 y(R)2231 4548 y Fm(d)2231
4584 y(n)2277 4534 y Fs(e)p Fw(\()2361 4516 y(\026)2360
4534 y Fs(t)2395 4498 y Fr(\003)2435 4534 y Fw(\).)42
b(Let)2715 4516 y(\026)2714 4534 y Fs(t)27 b Fw(=)h Fs(e)p
Fw(\()2964 4516 y(\026)2963 4534 y Fs(t)2998 4498 y Fr(\003)3038
4534 y Fw(\).)42 b(Supp)s(ose)263 4667 y(\026)-55 b Fs(u)32
b Fw(=)f Fs(C)7 b Fw([)p Fs(l)r(\033)t Fw(])p Fu(!)773
4686 y Fr(R)833 4667 y Fm(d)833 4703 y(n)880 4667 y Fs(C)g
Fw([)p Fs(r)s(\033)t Fw(])31 b(=)h Fs(u)i Fw(b)m(y)i(using)f(the)g
(rule)f Fs(\032)28 b Fw(:)g Fs(l)i Fu(!)d Fs(r)k Fu(\()c
Fs(s)2676 4682 y Fq(1)2715 4667 y Fw(==)p Fs(t)2902 4682
y Fq(1)2942 4667 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(s)3223
4682 y Fp(k)3266 4667 y Fw(==)o Fs(t)3452 4682 y Fp(k)257
4787 y Fw(at)35 b(the)h(p)s(osition)e Fs(p)p Fw(,)i(i.e.,)g
Fs(C)7 b Fw([)p Fs(l)r(\033)t Fw(])p Fu(j)1474 4802 y
Fp(p)1546 4787 y Fw(=)32 b Fs(l)r(\033)t Fw(.)52 b(By)36
b(the)g(parallel)c(mo)m(v)m(es)37 b(lemma)c(for)i Fu(!)3362
4806 y Fr(R)3422 4788 y Fm(d)3422 4823 y(n)3468 4787
y Fw(,)257 4908 y(there)g(is)e(a)h Fs(v)g Fu(2)c(T)c
Fw(\()p Fu(F)10 b Fs(;)17 b Fu(V)8 b Fw(\))33 b(suc)m(h)i(that)f
Fs(u)e Fu(k)-72 b(!)1851 4871 y Fr(\003)1851 4941 y(R)1911
4922 y Fm(d)1911 4958 y(n)1957 4908 y Fs(v)38 b Fw(and)2234
4889 y(\026)2233 4908 y Fs(t)33 b Fu(k)-72 b(!)2379 4927
y Fr(R)2439 4908 y Fm(d)2439 4943 y(n)2485 4908 y Fs(v)t
Fw(.)47 b(In)34 b(particular,)f(the)h(re-)257 5041 y(dexes)h(con)m
(tracted)f(in)e(the)h(step)1491 5022 y(\026)1490 5041
y Fs(t)g Fu(k)-72 b(!)1636 5060 y Fr(R)1696 5041 y Fm(d)1696
5076 y(n)1742 5041 y Fs(v)36 b Fw(are)d(the)g(descendan)m(ts)j
Fs(p)p Fu(n)6 b Fw(\026)-55 b Fs(u)o Fu(!)2951 5004 y
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%%Page: 16 16
16 15 bop 894 665 a Fs(e)p Fw(\()p Fs(s)1023 629 y Fr(\003)1062
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Fw(Figure)32 b(2:)43 b(Pro)s(of)32 b(of)g(Theorem)h(4.4.)257
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Fw(\))26 b(and)f Fs(u)j Fu(2)g(T)d Fw(\()p Fu(F)10 b
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%%Page: 17 17
17 16 bop 373 573 a Fv(1.)48 b Fs(s)547 537 y Fr(\003)621
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b Fs(l)514 5111 y Fr(\003)512 5172 y Fq(2)643 5147 y
Fu(!)89 b Fs(r)879 5111 y Fr(\003)876 5172 y Fq(2)1007
5147 y Fu(\()g Fs(c)1238 5111 y Fr(\003)1238 5172 y Fq(2)1346
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g(rule)257 5268 y Fs(\032)64 b Fw(:)g Fs(l)30 b Fu(!)d
Fs(r)j Fu(\()d Fs(c)64 b Fu(2)g(R)54 b Fw(b)s(ecause)h
Fu(R)f Fw(is)f(orthogonal.)104 b(Apparen)m(tly)-8 b(,)60
b(the)54 b(restric-)257 5388 y(tions)k(of)g Fs(\033)718
5352 y Fr(\003)714 5413 y Fq(1)815 5388 y Fw(and)h Fs(\033)1090
5352 y Fr(\003)1086 5413 y Fq(2)1187 5388 y Fw(to)f Fu(V)8
b Fs(ar)s Fw(\()p Fs(l)r Fw(\))59 b(coincide.)120 b(So)58
b(if)f Fu(V)8 b Fs(ar)s Fw(\()p Fs(r)s Fw(\))71 b Fu(\022)h(V)8
b Fs(ar)s Fw(\()p Fs(l)r Fw(\),)65 b(then)1828 5637 y(18)p
eop
%%Page: 19 19
19 18 bop 570 503 a Fs(:)625 476 y Fu(\030)626 507 y
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451 y(n)983 437 y Fk(0)1151 478 y Fs(:)630 770 y Fr(R)690
747 y Fm(d)690 801 y(m)744 787 y Fk(0)1197 770 y Fr(R)1257
747 y Fm(d)1257 801 y(m)1311 787 y Fk(0)584 1045 y Fs(:)743
988 y Fr(R)803 964 y Fm(d)803 1018 y(n)841 1004 y Fk(0)1005
1070 y Fs(:)1060 1043 y Fu(\030)1060 1074 y Fw(=)1165
1070 y Fs(:)p 787 475 331 4 v 1035 473 a Fh(-)p 596 993
4 473 v 598 993 a(?)p 1163 539 4 19 v 1163 577 V 1163
615 V 1163 653 V 1163 691 V 1163 728 V 1163 766 V 1163
804 V 1163 842 V 1163 880 V 1163 917 V 1163 955 V 1163
993 V 1163 993 4 48 v 484 w(?)p 664 1042 19 4 v 702 1042
V 739 1042 V 777 1042 V 815 1042 V 853 1042 V 891 1042
V 928 1042 48 4 v 893 1040 a(-)1518 503 y Fs(:)1573 476
y Fu(\030)1574 507 y Fw(=)1678 503 y Fs(:)1833 421 y
Fr(R)1893 397 y Fm(d)1893 451 y(n)1931 437 y Fk(0)2099
478 y Fs(:)1578 770 y Fr(R)1638 747 y Fm(d)1638 801 y(m)1692
787 y Fk(0)1578 989 y Fr(\003)2145 770 y(R)2205 747 y
Fm(d)2205 801 y(m)2259 787 y Fk(0)2145 989 y Fr(\003)1532
1045 y Fs(:)1691 988 y Fr(R)1751 964 y Fm(d)1751 1018
y(n)1789 1004 y Fk(0)1953 1070 y Fs(:)2008 1043 y Fu(\030)2008
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473 a Fh(-)p 1544 993 4 473 v 1546 993 a(?)p 2111 539
4 19 v 2111 577 V 2111 615 V 2111 653 V 2111 691 V 2111
728 V 2111 766 V 2111 804 V 2111 842 V 2111 880 V 2111
917 V 2111 955 V 2111 993 V 2111 993 4 48 v 484 w(?)p
1612 1042 19 4 v 1650 1042 V 1687 1042 V 1725 1042 V
1763 1042 V 1801 1042 V 1838 1042 V 1876 1042 48 4 v
1841 1040 a(-)2466 503 y Fs(:)2521 476 y Fu(\030)2522
507 y Fw(=)2626 503 y Fs(:)2781 421 y Fr(R)2841 397 y
Fm(d)2841 451 y(n)2879 437 y Fk(0)3047 478 y Fs(:)2960
449 y Fu(\003)2526 770 y Fr(R)2586 747 y Fm(d)2586 801
y(m)2640 787 y Fk(0)2526 989 y Fr(\003)3093 770 y(R)3153
747 y Fm(d)3153 801 y(m)3207 787 y Fk(0)3093 989 y Fr(\003)2480
1045 y Fs(:)2639 988 y Fr(R)2699 964 y Fm(d)2699 1018
y(n)2737 1004 y Fk(0)2825 1008 y Fr(\003)2901 1070 y
Fs(:)2956 1043 y Fu(\030)2956 1074 y Fw(=)3061 1070 y
Fs(:)p 2683 475 331 4 v 2931 473 a Fh(-)p 2492 993 4
473 v 2494 993 a(?)p 3059 539 4 19 v 3059 577 V 3059
615 V 3059 653 V 3059 691 V 3059 728 V 3059 766 V 3059
804 V 3059 842 V 3059 880 V 3059 917 V 3059 955 V 3059
993 V 3059 993 4 48 v 483 w(?)p 2560 1042 19 4 v 2597
1042 V 2635 1042 V 2673 1042 V 2711 1042 V 2749 1042
V 2786 1042 V 2824 1042 48 4 v 2788 1040 a(-)776 1351
y Fw(Figure)31 b(3:)43 b(The)34 b(induction)d(h)m(yp)s(othesis)j(of)e
(Lemma)f(4.10.)257 1637 y Fs(r)304 1600 y Fr(\003)301
1661 y Fq(1)343 1637 y Fs(\033)402 1600 y Fr(\003)398
1661 y Fq(1)516 1609 y Fu(\030)517 1641 y Fw(=)667 1637
y Fs(r)714 1600 y Fr(\003)711 1661 y Fq(2)753 1637 y
Fs(\033)812 1600 y Fr(\003)808 1661 y Fq(2)911 1637 y
Fw(since)61 b Fs(r)1225 1600 y Fr(\003)1222 1661 y Fq(1)1323
1637 y Fw(and)f Fs(r)1587 1600 y Fr(\003)1584 1661 y
Fq(2)1686 1637 y Fw(are)g(freshly)g(mark)m(ed.)125 b(Supp)s(ose)61
b(otherwise)257 1757 y(that)45 b Fu(V)8 b Fs(ar)s Fw(\()p
Fs(r)s Fw(\))48 b Fu(6\022)g(V)8 b Fs(ar)s Fw(\()p Fs(l)r
Fw(\).)80 b(W)-8 b(e)45 b(sho)m(w)h(that)e(in)g(this)g(case)i
Fs(\033)2588 1721 y Fr(\003)2584 1782 y Fq(1)2675 1729
y Fu(\030)2676 1761 y Fw(=)2801 1757 y Fs(\033)2860 1721
y Fr(\003)2856 1782 y Fq(2)2944 1757 y Fw(holds.)79 b(Since)257
1877 y Fs(\033)316 1841 y Fr(\003)312 1902 y Fq(1)388
1877 y Fw(=)32 b Fs(\033)555 1841 y Fr(\003)551 1902
y Fq(2)629 1877 y Fw([)p Fu(V)8 b Fs(ar)s Fw(\()p Fs(l)r
Fw(\)],)36 b(it)e(remains)g(to)h(b)s(e)g(sho)m(wn)i(that)d
Fs(\033)2314 1841 y Fr(\003)2310 1902 y Fq(1)2386 1850
y Fu(\030)2387 1881 y Fw(=)2495 1877 y Fs(\033)2554 1841
y Fr(\003)2550 1902 y Fq(2)2629 1877 y Fw([)p Fu(E)9
b(V)f Fs(ar)s Fw(\()p Fs(\032)p Fw(\)].)51 b(W)-8 b(e)35
b(sho)m(w)257 1998 y(b)m(y)48 b(induction)f(on)g Fs(i)g
Fw(that)g Fs(\033)1371 1962 y Fr(\003)1367 2022 y Fq(1)1463
1970 y Fu(\030)1464 2002 y Fw(=)1593 1998 y Fs(\033)1652
1962 y Fr(\003)1648 2022 y Fq(2)1738 1998 y Fw([)p Fu(V)8
b Fs(ar)s Fw(\()p Fs(l)r Fw(\))33 b Fu([)2170 1923 y
Ft(S)2253 1949 y Fp(i)2253 2027 y(j)t Fq(=1)2397 1998
y Fu(V)8 b Fs(ar)s Fw(\()p Fs(t)2637 2013 y Fp(j)2673
1998 y Fw(\)].)88 b(If)47 b Fs(i)53 b Fw(=)f(0,)f(then)257
2118 y Fs(\033)316 2082 y Fr(\003)312 2143 y Fq(1)407
2118 y Fw(=)f Fs(\033)592 2082 y Fr(\003)588 2143 y Fq(2)678
2118 y Fw([)p Fu(V)8 b Fs(ar)s Fw(\()p Fs(l)r Fw(\)].)85
b(So)46 b(let)f Fs(i)51 b(>)g Fw(0.)84 b(According)46
b(to)g(the)h(inductiv)m(e)f(h)m(yp)s(othesis,)257 2238
y Fs(\033)316 2202 y Fr(\003)312 2263 y Fq(1)383 2211
y Fu(\030)384 2243 y Fw(=)489 2238 y Fs(\033)548 2202
y Fr(\003)544 2263 y Fq(2)617 2238 y Fw([)p Fu(V)8 b
Fs(ar)s Fw(\()p Fs(l)r Fw(\))18 b Fu([)1020 2164 y Ft(S)1103
2190 y Fp(i)p Fr(\000)p Fq(1)1103 2267 y Fp(j)t Fq(=1)1246
2238 y Fu(V)8 b Fs(ar)s Fw(\()p Fs(t)1486 2253 y Fp(j)1523
2238 y Fw(\)].)43 b(Since)30 b Fu(V)8 b Fs(ar)s Fw(\()p
Fs(s)2161 2253 y Fp(i)2189 2238 y Fw(\))28 b Fu(\022)g(V)8
b Fs(ar)s Fw(\()p Fs(l)r Fw(\))18 b Fu([)2736 2164 y
Ft(S)2819 2190 y Fp(i)p Fr(\000)p Fq(1)2819 2267 y Fp(j)t
Fq(=1)2962 2238 y Fu(V)8 b Fs(ar)s Fw(\()p Fs(t)3202
2253 y Fp(j)3239 2238 y Fw(\),)30 b(it)g(is)257 2359
y(su\016cien)m(t)39 b(to)e(sho)m(w)h Fs(\033)1101 2323
y Fr(\003)1097 2383 y Fq(1)1177 2331 y Fu(\030)1178 2363
y Fw(=)1290 2359 y Fs(\033)1349 2323 y Fr(\003)1345 2383
y Fq(2)1426 2359 y Fw([)p Fu(V)8 b Fs(ar)s Fw(\()p Fs(t)1693
2374 y Fp(i)1721 2359 y Fw(\)].)58 b(There)39 b(are)e(mark)m(ed)h
(ground)f(constructor)257 2479 y(terms)f Fs(u)588 2443
y Fr(\003)588 2504 y Fq(1)627 2479 y Fs(;)17 b(u)727
2443 y Fr(\003)727 2504 y Fq(2)766 2479 y Fs(;)g(v)861
2443 y Fr(\003)857 2504 y Fq(1)900 2479 y Fs(;)g(v)995
2443 y Fr(\003)991 2504 y Fq(2)1069 2479 y Fw(suc)m(h)38
b(that)d Fs(s)1553 2443 y Fr(\003)1553 2504 y Fp(i)1593
2479 y Fs(\033)1652 2443 y Fr(\003)1648 2504 y Fq(1)1724
2479 y Fu(\))1824 2443 y Fr(\003)1824 2516 y(R)1884 2494
y Fm(d)1884 2538 y(m)p Fk(\000)p Fl(1)2025 2479 y Fs(u)2081
2443 y Fr(\003)2081 2504 y Fq(1)2120 2479 y Fw(,)i Fs(t)2219
2443 y Fr(\003)2219 2504 y Fp(i)2258 2479 y Fs(\033)2317
2443 y Fr(\003)2313 2504 y Fq(1)2389 2479 y Fu(\))2489
2443 y Fr(\003)2489 2516 y(R)2549 2494 y Fm(d)2549 2538
y(m)p Fk(\000)p Fl(1)2690 2479 y Fs(u)2746 2443 y Fr(\003)2746
2504 y Fq(2)2785 2479 y Fw(,)g(where)g Fs(u)3190 2443
y Fr(\003)3190 2504 y Fq(1)3262 2479 y Fu(\030)d Fs(u)3429
2443 y Fr(\003)3429 2504 y Fq(2)3468 2479 y Fw(,)257
2622 y(and)40 b Fs(s)500 2586 y Fr(\003)500 2647 y Fp(i)539
2622 y Fs(\033)598 2586 y Fr(\003)594 2647 y Fq(2)670
2622 y Fu(\))770 2586 y Fr(\003)770 2659 y(R)830 2636
y Fm(d)830 2680 y(n)p Fk(\000)p Fl(1)955 2622 y Fs(v)1006
2586 y Fr(\003)1002 2647 y Fq(1)1045 2622 y Fw(,)h Fs(t)1148
2586 y Fr(\003)1148 2647 y Fp(i)1187 2622 y Fs(\033)1246
2586 y Fr(\003)1242 2647 y Fq(2)1318 2622 y Fu(\))1418
2586 y Fr(\003)1418 2659 y(R)1478 2636 y Fm(d)1478 2680
y(n)p Fk(\000)p Fl(1)1603 2622 y Fs(v)1654 2586 y Fr(\003)1650
2647 y Fq(2)1693 2622 y Fw(,)g(where)f Fs(v)2100 2586
y Fr(\003)2096 2647 y Fq(1)2178 2622 y Fu(\030)f Fs(v)2345
2586 y Fr(\003)2341 2647 y Fq(2)2385 2622 y Fw(.)62 b(It)39
b(no)m(w)h(follo)m(ws)e(from)g(the)257 2765 y(inductiv)m(e)k(h)m(yp)s
(othesis)h(on)f Fs(`)f Fw(in)g(com)m(bination)f(with)h
Fs(s)2371 2729 y Fr(\003)2371 2790 y Fp(i)2411 2765 y
Fs(\033)2470 2729 y Fr(\003)2466 2790 y Fq(1)2552 2737
y Fu(\030)2553 2769 y Fw(=)2673 2765 y Fs(s)2719 2729
y Fr(\003)2719 2790 y Fp(i)2758 2765 y Fs(\033)2817 2729
y Fr(\003)2813 2790 y Fq(2)2899 2765 y Fw(that)g Fs(u)3175
2729 y Fr(\003)3175 2790 y Fq(1)3257 2737 y Fu(\030)3258
2769 y Fw(=)3378 2765 y Fs(v)3429 2729 y Fr(\003)3425
2790 y Fq(1)3468 2765 y Fw(.)257 2885 y(Th)m(us)i Fs(u)569
2849 y Fr(\003)569 2910 y Fq(2)650 2885 y Fu(\030)g Fs(u)826
2849 y Fr(\003)826 2910 y Fq(1)907 2885 y Fu(\030)g Fs(v)1078
2849 y Fr(\003)1074 2910 y Fq(1)1160 2885 y Fu(\030)f
Fs(v)1330 2849 y Fr(\003)1326 2910 y Fq(2)1369 2885 y
Fw(.)70 b(As)41 b(in)g(the)g(pro)s(of)g(of)f(Lemma)g(3.7,)j(for)e(ev)m
(ery)i(extra)257 3006 y(v)-5 b(ariable)34 b Fs(x)p Fw(,)j(there)f(are)g
(mark)m(ed)g(ground)f(constructor)i(terms)e Fs(u)2695
2969 y Fr(\003)2695 3030 y Fp(x)2774 3006 y Fw(and)h
Fs(v)3018 2969 y Fr(\003)3014 3030 y Fp(x)3093 3006 y
Fw(suc)m(h)h(that)257 3126 y Fs(x\033)371 3090 y Fr(\003)367
3151 y Fq(1)452 3126 y Fw(=)k Fs(u)625 3090 y Fr(\003)625
3151 y Fp(x)668 3126 y Fw(,)h Fs(x\033)851 3090 y Fr(\003)847
3151 y Fq(2)932 3126 y Fw(=)e Fs(v)1099 3090 y Fr(\003)1095
3151 y Fp(x)1139 3126 y Fw(,)j(and)d Fs(u)1462 3090 y
Fr(\003)1462 3151 y Fp(x)1546 3126 y Fu(\030)h Fs(v)1715
3090 y Fr(\003)1711 3151 y Fp(x)1755 3126 y Fw(.)67 b(Since)40
b(the)h(marks)f(on)g(instan)m(tiated)g(extra)257 3246
y(v)-5 b(ariables)29 b(are)i(fresh)g(and)f(m)m(utually)e(distinct,)i(w)
m(e)i(\014nally)d(deriv)m(e)i Fs(\033)2807 3210 y Fr(\003)2803
3271 y Fq(1)2874 3219 y Fu(\030)2875 3251 y Fw(=)2979
3246 y Fs(\033)3038 3210 y Fr(\003)3034 3271 y Fq(2)3108
3246 y Fw([)p Fu(V)8 b Fs(ar)s Fw(\()p Fs(t)3375 3261
y Fp(i)3403 3246 y Fw(\)].)257 3381 y(\(2\))62 b(By)h(Lemma)d(4.9,)69
b(it)61 b(is)h(su\016cien)m(t)h(to)f(pro)m(v)m(e)h(that)f(if)f
Fs(t)2755 3345 y Fr(\003)2873 3381 y Fu(\))2973 3326
y Fp(l)2995 3303 y Fk(\003)2994 3347 y Fl(1)3030 3326
y Fp(\033)3072 3303 y Fk(\003)3070 3347 y Fl(1)2973 3416
y Fr(R)3033 3397 y Fm(d)3033 3433 y(m)3195 3381 y Fw(\026)-53
b Fs(s)3237 3345 y Fr(\003)3338 3381 y Fw(and)257 3543
y Fs(t)292 3507 y Fr(\003)381 3543 y Fu(\))481 3488 y
Fp(l)503 3464 y Fk(\003)502 3508 y Fl(2)538 3488 y Fp(\033)580
3464 y Fk(\003)578 3508 y Fl(2)481 3578 y Fr(R)541 3559
y Fm(d)541 3594 y(n)671 3524 y Fw(\026)670 3543 y Fs(t)705
3507 y Fr(\003)745 3543 y Fw(,)48 b(then)d(\(i\))j(\026)-53
b Fs(s)1248 3507 y Fr(\003)1336 3543 y Fu(\))1436 3488
y Fp(l)1458 3464 y Fk(\003)1457 3508 y Fl(2)1494 3488
y Fp(\033)1536 3464 y Fk(\003)1534 3508 y Fl(2)1436 3578
y Fr(R)1496 3559 y Fm(d)1496 3594 y(n)1629 3543 y Fw(~)g
Fs(s)1671 3507 y Fr(\003)1756 3543 y Fw(or)48 b(\026)-53
b Fs(s)1933 3507 y Fr(\003)2021 3543 y Fw(=)53 b(~)-53
b Fs(s)2192 3507 y Fr(\003)2232 3543 y Fw(,)48 b(\(ii\))2483
3524 y(\026)2482 3543 y Fs(t)2517 3507 y Fr(\003)2605
3543 y Fu(\))2705 3488 y Fp(l)2727 3464 y Fk(\003)2726
3508 y Fl(1)2763 3488 y Fp(\033)2805 3464 y Fk(\003)2803
3508 y Fl(1)2705 3578 y Fr(R)2765 3559 y Fm(d)2765 3594
y(m)2896 3524 y Fw(~)2894 3543 y Fs(t)2929 3507 y Fr(\003)3014
3543 y Fw(or)3147 3524 y(\026)3145 3543 y Fs(t)3180 3507
y Fr(\003)3269 3543 y Fw(=)3395 3524 y(~)3393 3543 y
Fs(t)3428 3507 y Fr(\003)3468 3543 y Fw(,)257 3687 y(and)26
b(\(iii\))h(~)-53 b Fs(s)669 3651 y Fr(\003)736 3660
y Fu(\030)737 3691 y Fw(=)843 3669 y(~)842 3687 y Fs(t)877
3651 y Fr(\003)942 3687 y Fw(for)25 b(some)h(mark)m(ed)g(terms)k(~)-53
b Fs(s)1973 3651 y Fr(\003)2038 3687 y Fw(and)2223 3669
y(~)2221 3687 y Fs(t)2256 3651 y Fr(\003)2296 3687 y
Fw(.)41 b(W)-8 b(e)27 b(distinguish)d(three)j(cases:)257
3928 y(\(a\))33 b Fs(l)446 3892 y Fr(\003)444 3953 y
Fq(1)485 3928 y Fs(\033)544 3892 y Fr(\003)540 3953 y
Fq(1)611 3928 y Fw(=)28 b Fs(l)746 3892 y Fr(\003)744
3953 y Fq(2)786 3928 y Fs(\033)845 3892 y Fr(\003)841
3953 y Fq(2)884 3928 y Fw(,)257 4048 y(\(b\))33 b Fs(l)451
4012 y Fr(\003)449 4073 y Fq(1)491 4048 y Fs(\033)550
4012 y Fr(\003)546 4073 y Fq(1)622 4048 y Fw(is)f(neither)g(a)h
(subterm)f(of)h Fs(l)1654 4012 y Fr(\003)1652 4073 y
Fq(2)1693 4048 y Fs(\033)1752 4012 y Fr(\003)1748 4073
y Fq(2)1824 4048 y Fw(nor)g(con)m(v)m(ersely)-8 b(,)257
4169 y(\(c\))33 b Fs(l)440 4133 y Fr(\003)438 4193 y
Fq(1)480 4169 y Fs(\033)539 4133 y Fr(\003)535 4193 y
Fq(1)611 4169 y Fw(is)f(a)g(prop)s(er)h(subterm)g(of)f
Fs(l)1624 4133 y Fr(\003)1622 4193 y Fq(2)1663 4169 y
Fs(\033)1722 4133 y Fr(\003)1718 4193 y Fq(2)1762 4169
y Fw(.)257 4410 y(\(a\))h(With)f(the)h(aid)e(of)h(\(1\),)g(this)h
(follo)m(ws)e(easily)-8 b(.)257 4530 y(\(b\))49 b(The)h(pro)s(of)e(is)g
(analogous)g(to)g(Prop)s(osition)f(3.19,)52 b(\(1\),)h(case)d(1)e(in)g
([12].)92 b(W)-8 b(e)257 4650 y(ma)m(y)63 b(write)h Fs(t)815
4614 y Fr(\003)934 4650 y Fw(=)80 b Fs(C)1167 4614 y
Fr(\003)1206 4650 y Fw([)p Fs(l)1264 4614 y Fr(\003)1262
4675 y Fp(i)1286 4684 y Fl(1)1325 4650 y Fs(\033)1384
4614 y Fr(\003)1380 4675 y Fp(i)1404 4684 y Fl(1)1444
4650 y Fs(;)17 b(:)g(:)g(:)e(;)i(l)1693 4614 y Fr(\003)1691
4675 y Fp(i)1715 4683 y Fm(p)1756 4650 y Fs(\033)1815
4614 y Fr(\003)1811 4675 y Fp(i)1835 4683 y Fm(p)1875
4650 y Fw(],)71 b(where)65 b Fs(i)2346 4665 y Fp(j)2463
4650 y Fu(2)80 b(f)p Fw(1)p Fs(;)17 b Fw(2)p Fu(g)62
b Fw(for)h(ev)m(ery)i(in-)257 4771 y(dex)59 b Fs(j)77
b Fu(2)72 b(f)p Fw(1)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p
Fu(g)p Fw(.)120 b(Without)57 b(loss)h(of)f(generalit)m(y)-8
b(,)64 b(w)m(e)59 b(ma)m(y)f(assume)g(that)257 4891 y(1)e(=)h
Fs(i)528 4906 y Fq(1)624 4891 y Fu(\024)f Fs(i)790 4906
y Fq(2)886 4891 y Fu(\024)h(\001)17 b(\001)g(\001)55
b(\024)h Fs(i)1359 4906 y Fp(p)1456 4891 y Fw(=)g(2.)94
b(That)49 b(is)g(to)g(sa)m(y)-8 b(,)55 b Fs(t)2513 4855
y Fr(\003)2609 4891 y Fw(=)h Fs(C)2818 4855 y Fr(\003)2857
4891 y Fw([)p Fs(l)2915 4855 y Fr(\003)2913 4916 y Fq(1)2955
4891 y Fs(\033)3014 4855 y Fr(\003)3010 4916 y Fq(1)3053
4891 y Fs(;)17 b(:)g(:)g(:)f(;)h(l)3303 4855 y Fr(\003)3301
4916 y Fq(2)3343 4891 y Fs(\033)3402 4855 y Fr(\003)3398
4916 y Fq(2)3441 4891 y Fw(].)257 5011 y(Then)66 b(\026)-53
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5637 y(19)p eop
%%Page: 20 20
20 19 bop 1292 665 a Fs(t)1327 629 y Fr(\003)1578 665
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701 y(n)2056 665 y Fw(\026)-55 b Fs(u)2106 629 y Fr(\003)2228
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821 y(\177)2466 812 y Fr(R)2526 788 y Fm(d)2526 829 y(m)1291
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1475 y(\()p Fs(mar)s(k)s(s)p Fw(\()p Fs(r)663 1438 y
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Fw(\))f Fu([)h Fs(mar)s(k)s(s)p Fw(\()p Fs(\033)2493
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b Fu(6)p Fw(=)h Fu(;)p Fw(,)d(then)e(w)m(e)f(tak)m(e)257
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b(\026)-54 b Fs(\033)1379 1559 y Fr(\003)1375 1620 y
Fq(2)1460 1595 y Fw(with)42 b(\()p Fs(mar)s(k)s(s)p Fw(\()p
Fs(r)2098 1559 y Fr(\003)2095 1620 y Fq(1)2137 1595 y
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1559 y Fr(\003)2674 1620 y Fq(1)2718 1595 y Fw(\)\))f
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b Fu([)257 1730 y Fs(mar)s(k)s(s)p Fw(\()5 b(\026)-54
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y Fp(l)1840 1651 y Fk(\003)1839 1695 y Fl(2)1879 1675
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1694 y Fr(\003)2993 1730 y Fw(,)42 b(and)e(apply)257
1864 y(Lemma)34 b(4.9;)j(see)g(Figure)d(4.\))52 b(No)m(w)36
b(if)e(w)m(e)j(con)m(tract)f(the)g(redex)g Fs(l)2750
1828 y Fr(\003)2748 1889 y Fq(2)2790 1864 y Fs(\033)2849
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1828 y Fr(\003)3328 1889 y Fq(2)3370 1864 y Fs(\033)3429
1828 y Fr(\003)3425 1889 y Fq(2)3468 1864 y Fw(,)257
1985 y(then)h(w)m(e)g(obtain)i(~)-53 b Fs(s)983 1949
y Fr(\003)1057 1985 y Fw(=)33 b Fs(C)1243 1949 y Fr(\003)1283
1985 y Fw([)p Fs(r)1357 1949 y Fr(\003)1354 2010 y Fq(1)1396
1985 y Fs(\033)1455 1949 y Fr(\003)1451 2010 y Fq(1)1494
1985 y Fs(;)17 b(:)g(:)g(:)f(;)h(r)1760 1949 y Fr(\003)1757
2010 y Fq(2)1799 1985 y Fs(\033)1858 1949 y Fr(\003)1854
2010 y Fq(2)1897 1985 y Fw(].)55 b(Analogously)-8 b(,)35
b(con)m(tracting)h Fs(l)3129 1949 y Fr(\003)3127 2010
y Fq(1)3169 1985 y Fs(\033)3228 1949 y Fr(\003)3224 2010
y Fq(1)3303 1985 y Fw(in)3422 1966 y(\026)3421 1985 y
Fs(t)3456 1949 y Fr(\003)257 2105 y Fw(yields)533 2087
y(~)532 2105 y Fs(t)567 2069 y Fr(\003)634 2105 y Fw(=)27
b Fs(C)814 2069 y Fr(\003)854 2105 y Fw([)p Fs(r)928
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2226 y(\(c\))32 b(W)-8 b(e)32 b(pro)s(ceed)g(in)e(analogy)h(to)g(Prop)s
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2582 y Fq(2)1607 2557 y Fw(])28 b(=)g Fs(C)1843 2516
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y(\026)834 2768 y Fs(C)911 2732 y Fr(\003)950 2768 y
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y Fr(\003)1800 2793 y Fq(1)1842 2768 y Fs(\033)1901 2732
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y(\026)368 2888 y Fs(C)445 2852 y Fr(\003)485 2888 y
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f Fs(r)2888 3696 y Fr(\003)2885 3757 y Fq(1)2927 3732
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3817 y Fr(\003)364 3877 y Fq(2)438 3853 y Fw(\(for)31
b(all)e(extra)j(v)-5 b(ariables)30 b Fs(x)i Fw(and)f
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b Fs(t)3428 3817 y Fr(\003)3468 3853 y Fw(.)257 3973
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y(\026)1621 3973 y Fs(C)1698 3937 y Fr(\003)1737 3973
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Fr(\003)598 4359 y Fq(1)640 4334 y Fs(\033)699 4298 y
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4316 y(\026)1614 4334 y Fs(t)1649 4298 y Fr(\003)1689
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2454 4504 y Fr(\003)2451 4570 y Fq(1)2493 4545 y Fs(\033)2552
4504 y Fr(\003)2548 4570 y Fq(1)2591 4545 y Fw(]])257
4771 y(and)39 b(observ)m(e)h(that)1025 4752 y(\026)1024
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4716 y Fp(l)1258 4692 y Fk(\003)1257 4736 y Fl(1)1294
4716 y Fp(\033)1336 4692 y Fk(\003)1334 4736 y Fl(1)1236
4798 y Fr(R)1416 4752 y Fw(~)1414 4771 y Fs(t)1449 4735
y Fr(\003)1489 4771 y Fw(.)61 b(The)40 b(situation)d(is)h(depicted)h
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Fr(\003)286 4916 y Fq(2)328 4891 y Fs(\033)387 4855 y
Fr(\003)383 4916 y Fq(2)454 4891 y Fw(=)580 4866 y(\026)558
4891 y Fs(C)635 4855 y Fr(\003)674 4891 y Fw([)p Fs(l)732
4855 y Fr(\003)730 4916 y Fq(1)772 4891 y Fs(\033)831
4855 y Fr(\003)827 4916 y Fq(1)870 4891 y Fs(;)17 b(:)g(:)g(:)f(;)h(l)
1120 4855 y Fr(\003)1118 4916 y Fq(1)1160 4891 y Fs(\033)1219
4855 y Fr(\003)1215 4916 y Fq(1)1258 4891 y Fw(])33 b
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4855 y Fr(\003)1631 4891 y Fw([)p Fs(l)1689 4855 y Fr(\003)1687
4916 y Fq(1)1728 4891 y Fs(\033)1787 4855 y Fr(\003)1783
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4855 y Fr(\003)2171 4916 y Fq(1)2214 4891 y Fw(])28 b(=)g
Fs(r)2420 4855 y Fr(\003)2417 4916 y Fq(2)2459 4891 y
Fs(\033)2518 4855 y Fr(\003)2514 4916 y Fq(2)2557 4891
y Fw(.)404 5011 y(Next,)42 b(w)m(e)f(will)d(sho)m(w)j(that)1511
4986 y(\026)1489 5011 y Fs(C)1566 4975 y Fr(\003)1605
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5011 y Fs(\033)1777 4975 y Fr(\003)1773 5036 y Fq(1)1817
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5036 y Fq(1)2121 5011 y Fs(\033)2180 4975 y Fr(\003)2176
5036 y Fq(1)2220 5011 y Fw(])40 b Fu(\))2387 5031 y Fr(R)2447
5012 y Fm(d)2447 5047 y(n)2556 4986 y Fw(~)2534 5011
y Fs(C)2611 4975 y Fr(\003)2650 5011 y Fw([)p Fs(r)2724
4975 y Fr(\003)2721 5036 y Fq(1)2763 5011 y Fs(\033)2822
4975 y Fr(\003)2818 5036 y Fq(1)2862 5011 y Fs(;)17 b(:)g(:)g(:)e(;)i
(r)3127 4975 y Fr(\003)3124 5036 y Fq(1)3166 5011 y Fs(\033)3225
4975 y Fr(\003)3221 5036 y Fq(1)3265 5011 y Fw(].)65
b(T)-8 b(o)257 5132 y(this)45 b(end,)k(recall)44 b(that)h
Fs(l)1215 5096 y Fr(\003)1213 5156 y Fq(2)1255 5132 y
Fs(\033)1314 5096 y Fr(\003)1310 5156 y Fq(2)1402 5132
y Fw(=)1549 5107 y(\026)1527 5132 y Fs(C)1604 5096 y
Fr(\003)1643 5132 y Fw([)p Fs(l)1701 5096 y Fr(\003)1699
5156 y Fq(1)1741 5132 y Fs(\033)1800 5096 y Fr(\003)1796
5156 y Fq(1)1840 5132 y Fs(;)17 b(:)g(:)g(:)f(;)h(l)2090
5096 y Fr(\003)2088 5156 y Fq(1)2129 5132 y Fs(\033)2188
5096 y Fr(\003)2184 5156 y Fq(1)2227 5132 y Fw(])33 b
Fu(\))2387 5151 y Fr(R)2447 5132 y Fm(d)2447 5168 y(n)2526
5132 y Fs(r)2573 5096 y Fr(\003)2570 5156 y Fq(2)2612
5132 y Fs(\033)2671 5096 y Fr(\003)2667 5156 y Fq(2)2711
5132 y Fw(.)81 b(Th)m(us,)50 b(for)44 b(ev)m(ery)257
5252 y Fs(s)303 5216 y Fr(\003)303 5277 y Fp(i)343 5252
y Fw(==)o Fs(t)529 5216 y Fr(\003)529 5277 y Fp(i)607
5252 y Fw(in)37 b Fs(c)768 5216 y Fr(\003)768 5277 y
Fq(2)808 5252 y Fw(,)i(there)g(exist)f(mark)m(ed)h(ground)f
(constructor)h(terms)e Fs(u)2907 5216 y Fr(\003)2907
5277 y Fp(i)2984 5252 y Fw(and)i Fs(v)3231 5216 y Fr(\003)3227
5277 y Fp(i)3308 5252 y Fw(suc)m(h)257 5373 y(that)47
b Fs(s)529 5336 y Fr(\003)529 5397 y Fp(i)569 5373 y
Fs(\033)628 5336 y Fr(\003)624 5397 y Fq(2)699 5373 y
Fu(\))799 5336 y Fr(\003)799 5410 y(R)859 5387 y Fm(d)859
5431 y(n)p Fk(\000)p Fl(1)984 5373 y Fs(u)1040 5336 y
Fr(\003)1040 5397 y Fp(i)1079 5373 y Fw(,)k Fs(t)1192
5336 y Fr(\003)1192 5397 y Fp(i)1232 5373 y Fs(\033)1291
5336 y Fr(\003)1287 5397 y Fq(2)1363 5373 y Fu(\))1463
5336 y Fr(\003)1463 5410 y(R)1523 5387 y Fm(d)1523 5431
y(n)p Fk(\000)p Fl(1)1647 5373 y Fs(v)1698 5336 y Fr(\003)1694
5397 y Fp(i)1738 5373 y Fw(,)f(and)d Fs(u)2075 5336 y
Fr(\003)2075 5397 y Fp(i)2166 5373 y Fu(\030)53 b Fs(v)2347
5336 y Fr(\003)2343 5397 y Fp(i)2386 5373 y Fw(.)87 b(Since)47
b Fu(R)g Fw(is)g(orthogonal,)1828 5637 y(20)p eop
%%Page: 21 21
21 20 bop 1567 665 a Fs(t)1602 629 y Fr(\003)1730 665
y Fw(=)-17 b Fu(\))1889 629 y Fp(l)1911 606 y Fk(\003)1910
650 y Fl(2)1947 629 y Fp(\033)1989 606 y Fk(\003)1987
650 y Fl(2)2116 647 y Fw(\026)2114 665 y Fs(t)2149 629
y Fr(\003)1566 702 y Ft(w)1566 761 y(w)1566 821 y(\177)1643
804 y Fp(l)1665 781 y Fk(\003)1664 825 y Fl(1)1701 804
y Fp(\033)1743 781 y Fk(\003)1741 825 y Fl(1)2113 702
y Ft(w)2113 761 y(w)2113 821 y(\177)2191 804 y Fp(l)2213
781 y Fk(\003)2212 825 y Fl(1)2248 804 y Fp(\033)2290
781 y Fk(\003)2288 825 y Fl(1)1566 965 y Fw(\026)-53
b Fs(s)1608 929 y Fr(\003)2116 947 y Fw(~)2114 965 y
Fs(t)2149 929 y Fr(\003)1663 1195 y Fw(Figure)31 b(5:)257
1475 y(for)52 b(ev)m(ery)j(o)s(ccurrence)f(of)e Fs(l)1371
1439 y Fr(\003)1369 1500 y Fq(1)1410 1475 y Fs(\033)1469
1439 y Fr(\003)1465 1500 y Fq(1)1509 1475 y Fw(,)57 b(there)d(is)e(a)g
(v)-5 b(ariable)50 b Fs(x)63 b Fu(2)f(V)8 b Fs(ar)s Fw(\()p
Fs(l)2947 1490 y Fq(2)2986 1475 y Fw(\))52 b(suc)m(h)i(that)257
1596 y Fs(x\033)371 1560 y Fr(\003)367 1620 y Fq(2)454
1596 y Fw(=)43 b Fs(C)650 1560 y Fr(\003)643 1620 y Fp(x)689
1596 y Fw([)p Fs(l)747 1560 y Fr(\003)745 1620 y Fq(1)787
1596 y Fs(\033)846 1560 y Fr(\003)842 1620 y Fq(1)886
1596 y Fs(;)17 b(:)g(:)g(:)e(;)i(l)1135 1560 y Fr(\003)1133
1620 y Fq(1)1175 1596 y Fs(\033)1234 1560 y Fr(\003)1230
1620 y Fq(1)1273 1596 y Fw(])42 b(con)m(tains)f(this)h(particular)d(o)s
(ccurrence.)72 b(De\014ne)47 b(\026)-54 b Fs(\033)3311
1560 y Fr(\003)3307 1620 y Fq(2)3392 1596 y Fw(b)m(y)257
1716 y Fs(x)5 b Fw(\026)-54 b Fs(\033)371 1680 y Fr(\003)367
1741 y Fq(2)465 1716 y Fw(=)53 b Fs(C)671 1680 y Fr(\003)664
1741 y Fp(x)710 1716 y Fw([)p Fs(r)784 1680 y Fr(\003)781
1741 y Fq(1)823 1716 y Fs(\033)882 1680 y Fr(\003)878
1741 y Fq(1)922 1716 y Fs(;)17 b(:)g(:)g(:)f(;)h(r)1188
1680 y Fr(\003)1185 1741 y Fq(1)1226 1716 y Fs(\033)1285
1680 y Fr(\003)1281 1741 y Fq(1)1325 1716 y Fw(])47 b(for)h(all)d
(those)j(v)-5 b(ariables)47 b Fs(x)h Fw(and)f Fs(y)9
b Fw(\026)-54 b Fs(\033)2822 1680 y Fr(\003)2818 1741
y Fq(2)2914 1716 y Fw(=)53 b Fs(y)t(\033)3154 1680 y
Fr(\003)3150 1741 y Fq(2)3241 1716 y Fw(other-)257 1837
y(wise.)64 b(No)m(w)40 b Fs(l)787 1800 y Fr(\003)785
1861 y Fq(2)832 1837 y Fw(\026)-54 b Fs(\033)886 1800
y Fr(\003)882 1861 y Fq(2)964 1837 y Fu(\))1064 1856
y Fr(R)1124 1837 y Fm(d)1124 1872 y(n)1210 1837 y Fs(r)1257
1800 y Fr(\003)1254 1861 y Fq(2)1301 1837 y Fw(\026)g
Fs(\033)1355 1800 y Fr(\003)1351 1861 y Fq(2)1433 1837
y Fw(=)1570 1811 y(~)1548 1837 y Fs(C)1625 1800 y Fr(\003)1665
1837 y Fw([)p Fs(r)1739 1800 y Fr(\003)1736 1861 y Fq(1)1778
1837 y Fs(\033)1837 1800 y Fr(\003)1833 1861 y Fq(1)1876
1837 y Fs(;)17 b(:)g(:)g(:)f(;)h(r)2142 1800 y Fr(\003)2139
1861 y Fq(1)2181 1837 y Fs(\033)2240 1800 y Fr(\003)2236
1861 y Fq(1)2279 1837 y Fw(].)64 b(In)40 b(order)f(to)g(see)h(this,)h
(infer)257 1957 y(from)e(the)i(inductiv)m(e)f(h)m(yp)s(othesis)i(on)e
Fs(`)g Fw(in)g(conjunction)g(with)g Fs(s)2748 1921 y
Fr(\003)2748 1982 y Fp(i)2787 1957 y Fs(\033)2846 1921
y Fr(\003)2842 1982 y Fq(2)2918 1957 y Fu(\))3018 1921
y Fr(\003)3018 1994 y(R)3078 1971 y Fm(d)3078 2015 y(n)p
Fk(\000)p Fl(1)3203 1957 y Fs(u)3259 1921 y Fr(\003)3259
1982 y Fp(i)3338 1957 y Fw(and)257 2100 y Fs(s)303 2064
y Fr(\003)303 2124 y Fp(i)343 2100 y Fs(\033)402 2064
y Fr(\003)398 2124 y Fq(2)474 2100 y Fu(\))574 2119 y
Fr(R)634 2100 y Fm(d)634 2136 y(m)729 2100 y Fs(s)775
2064 y Fr(\003)775 2124 y Fp(i)819 2100 y Fw(\026)-54
b Fs(\033)873 2064 y Fr(\003)869 2124 y Fq(2)941 2100
y Fw(that)29 b(there)g(is)g(a)f(mark)m(ed)h(ground)g(constructor)g
(term)35 b(\026)-55 b Fs(u)3033 2064 y Fr(\003)3033 2124
y Fp(i)3100 2100 y Fw(suc)m(h)30 b(that)257 2220 y Fs(s)303
2184 y Fr(\003)303 2245 y Fp(i)348 2220 y Fw(\026)-54
b Fs(\033)402 2184 y Fr(\003)398 2245 y Fq(2)474 2220
y Fu(\))574 2184 y Fr(\003)574 2257 y(R)634 2235 y Fm(d)634
2279 y(n)p Fk(\000)p Fl(1)765 2220 y Fw(\026)f Fs(u)815
2184 y Fr(\003)815 2245 y Fp(i)881 2220 y Fw(and)27 b
Fs(u)1121 2184 y Fr(\003)1121 2245 y Fp(i)1188 2192 y
Fu(\030)1189 2224 y Fw(=)1299 2220 y(\026)-55 b Fs(u)1349
2184 y Fr(\003)1349 2245 y Fp(i)1388 2220 y Fw(.)42 b(Analogously)-8
b(,)27 b(there)h(is)f(a)g(mark)m(ed)g(ground)g(construc-)257
2363 y(tor)37 b(term)j(\026)-53 b Fs(v)707 2327 y Fr(\003)703
2388 y Fp(i)784 2363 y Fw(suc)m(h)38 b(that)f Fs(t)1259
2327 y Fr(\003)1259 2388 y Fp(i)1304 2363 y Fw(\026)-54
b Fs(\033)1358 2327 y Fr(\003)1354 2388 y Fq(2)1430 2363
y Fu(\))1530 2327 y Fr(\003)1530 2400 y(R)1590 2377 y
Fm(d)1590 2421 y(n)p Fk(\000)p Fl(1)1718 2363 y Fw(\026)h
Fs(v)1765 2327 y Fr(\003)1761 2388 y Fp(i)1805 2363 y
Fw(,)38 b(and)f Fs(v)2115 2327 y Fr(\003)2111 2388 y
Fp(i)2189 2335 y Fu(\030)2190 2367 y Fw(=)2306 2363 y(\026)-53
b Fs(v)2353 2327 y Fr(\003)2349 2388 y Fp(i)2392 2363
y Fw(.)57 b(Hence)38 b(the)g(claim)c(follo)m(ws)257 2506
y(from)k(\026)-55 b Fs(u)544 2470 y Fr(\003)544 2530
y Fp(i)610 2506 y Fu(\030)28 b Fs(u)771 2470 y Fr(\003)771
2530 y Fp(i)838 2506 y Fu(\030)g Fs(v)994 2470 y Fr(\003)990
2530 y Fp(i)1061 2506 y Fu(\030)k Fw(\026)-53 b Fs(v)1217
2470 y Fr(\003)1213 2530 y Fp(i)1256 2506 y Fw(.)2135
b Fj(\003)257 2725 y Fo(Corollary)36 b(4.11)49 b Fv(F)-7
b(or)34 b(every)h Fs(n)28 b Fu(2)g Fi(N)9 b Fv(,)76 b
Fu(\))1895 2745 y Fr(R)1955 2726 y Fm(d)1955 2761 y(n)2071
2725 y Fv(is)34 b(c)-5 b(on\015uent)35 b(mo)-5 b(dulo)2933
2698 y Fu(\030)2933 2730 y Fw(=)3010 2725 y Fv(.)257
2945 y(Pr)g(o)g(of)67 b Fw(Immediate)32 b(consequence)j(of)d(Lemma)f
(4.10.)1171 b Fj(\003)257 3164 y Fo(Theorem)37 b(4.12)49
b Fv(F)-7 b(or)34 b(every)h Fs(n)28 b Fu(2)g Fi(N)9 b
Fv(,)76 b Fu(\))1872 3179 y Fr(R)1932 3187 y Fm(n)2048
3164 y Fv(is)35 b(c)-5 b(on\015uent)35 b(mo)-5 b(dulo)2910
3137 y Fu(\030)2911 3169 y Fw(=)2987 3164 y Fv(.)257
3384 y(Pr)g(o)g(of)86 b Fw(Because)43 b(of)74 b Fu(\))1205
3403 y Fr(R)1265 3384 y Fm(d)1265 3420 y(n)1388 3384
y Fu(\022)j(\))1642 3399 y Fr(R)1702 3407 y Fm(n)1781
3384 y Fw(,)44 b(Prop)s(osition)c(4.13,)k(and)e(Lemma)f(4.9,)j(w)m(e)
257 3504 y(conclude)36 b(b)m(y)f(the)h(generalization)c(of)i(Staples')h
(result)g(\(see)h(Section)f(2\))f(that)67 b Fu(\))3356
3519 y Fr(R)3416 3527 y Fm(n)257 3625 y Fw(is)32 b(con\015uen)m(t)i(mo)
s(dulo)1125 3597 y Fu(\030)1126 3629 y Fw(=)1235 3625
y(if)d(and)i(only)f(if)64 b Fu(\))1950 3644 y Fr(R)2010
3625 y Fm(d)2010 3661 y(n)2122 3625 y Fw(is)32 b(con\015uen)m(t)i(mo)s
(dulo)2989 3597 y Fu(\030)2990 3629 y Fw(=)3067 3625
y(.)324 b Fj(\003)257 3844 y Fo(Prop)s(osition)36 b(4.13)49
b Fv(If)25 b Fs(s)1250 3808 y Fr(\003)1325 3844 y Fu(\))1425
3808 y Fr(\003)1425 3870 y(R)1485 3878 y Fm(n)1566 3844
y Fs(t)1601 3808 y Fr(\003)1641 3844 y Fv(,)i(then)f(ther)-5
b(e)26 b(ar)-5 b(e)25 b(marke)-5 b(d)25 b(terms)h Fs(u)2947
3808 y Fr(\003)3012 3844 y Fv(and)f Fs(v)3243 3808 y
Fr(\003)3308 3844 y Fv(such)257 3965 y(that)36 b Fs(s)503
3929 y Fr(\003)577 3965 y Fu(\))677 3929 y Fr(\003)677
3998 y(R)737 3979 y Fm(d)737 4015 y(n)818 3965 y Fs(u)874
3929 y Fr(\003)913 3965 y Fv(,)f Fs(t)1013 3929 y Fr(\003)1087
3965 y Fu(\))1187 3929 y Fr(\003)1187 3998 y(R)1247 3979
y Fm(d)1247 4015 y(n)1329 3965 y Fs(v)1380 3929 y Fr(\003)1419
3965 y Fv(,)g(and)f Fs(u)1729 3929 y Fr(\003)1796 3937
y Fu(\030)1797 3969 y Fw(=)1901 3965 y Fs(v)1952 3929
y Fr(\003)1991 3965 y Fv(.)257 4184 y(Pr)-5 b(o)g(of)49
b Fw(W)-8 b(e)23 b(pro)s(ceed)h(b)m(y)g(induction)e(on)h(the)g(depth)h
Fs(n)f Fw(of)f Fs(s)2371 4148 y Fr(\003)2443 4184 y Fu(\))2543
4148 y Fr(\003)2543 4209 y(R)2603 4217 y Fm(n)2682 4184
y Fs(t)2717 4148 y Fr(\003)2757 4184 y Fw(.)40 b(The)24
b(prop)s(osition)257 4305 y(holds)29 b(v)-5 b(acuously)30
b(for)f Fs(n)f Fw(=)g(0.)42 b(So)29 b(let)g Fs(n)f(>)f
Fw(0.)43 b(W)-8 b(e)29 b(pro)s(ceed)i(further)e(b)m(y)i(induction)d(on)
257 4425 y(the)e(length)f Fs(`)g Fw(of)f(the)i(reduction)f(sequence)j
Fs(s)1908 4389 y Fr(\003)1980 4425 y Fu(\))2080 4389
y Fr(\003)2080 4450 y(R)2140 4458 y Fm(n)2219 4425 y
Fs(t)2254 4389 y Fr(\003)2293 4425 y Fw(.)41 b(Again,)26
b(the)f(case)h Fs(`)i Fw(=)g(0)c(holds)257 4545 y(v)-5
b(acuously)d(.)47 b(Supp)s(ose)35 b(the)f(claim)e(is)h(true)h(for)f
Fs(`)p Fw(.)47 b(In)34 b(order)g(to)f(sho)m(w)i(it)e(for)g
Fs(`)23 b Fw(+)f(1,)34 b(w)m(e)257 4666 y(consider)g
Fs(s)684 4630 y Fr(\003)752 4666 y Fw(=)28 b Fs(C)933
4630 y Fr(\003)973 4666 y Fw([)p Fs(l)1031 4630 y Fr(\003)1070
4666 y Fs(\033)1129 4630 y Fr(\003)1169 4666 y Fs(;)17
b(:)g(:)g(:)f(;)h(l)1419 4630 y Fr(\003)1458 4666 y Fs(\033)1517
4630 y Fr(\003)1557 4666 y Fw(])28 b Fu(\))1712 4630
y Fp(l)1734 4606 y Fk(\003)1770 4630 y Fp(\033)1812 4606
y Fk(\003)1712 4691 y Fr(R)1772 4699 y Fm(n)1881 4666
y Fs(C)1958 4630 y Fr(\003)1998 4666 y Fw([)p Fs(r)2072
4630 y Fr(\003)2111 4666 y Fs(\033)2170 4630 y Fr(\003)2209
4666 y Fs(;)17 b(:)g(:)g(:)f(;)h(r)2475 4630 y Fr(\003)2514
4666 y Fs(\033)2573 4630 y Fr(\003)2612 4666 y Fw(])29
b(=)2774 4647 y(\026)2773 4666 y Fs(t)2808 4630 y Fr(\003)2876
4666 y Fu(\))2976 4630 y Fp(`)2976 4691 y Fr(R)3036 4699
y Fm(n)3111 4666 y Fs(t)3146 4630 y Fr(\003)3186 4666
y Fw(,)k(where)257 4786 y Fs(s)303 4750 y Fr(\003)375
4786 y Fu(\))475 4801 y Fr(R)535 4809 y Fm(n)616 4768
y Fw(\026)614 4786 y Fs(t)649 4750 y Fr(\003)720 4786
y Fw(b)m(y)g(a)e(mark)m(ed)h(v)m(ersion)g(of)f(the)h(rule)f
Fs(\032)d Fw(:)g Fs(l)i Fu(!)d Fs(r)k Fu(\()c Fs(s)2649
4801 y Fq(1)2688 4786 y Fw(==)p Fs(t)2875 4801 y Fq(1)2915
4786 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(s)3196 4801 y
Fp(k)3238 4786 y Fw(==)p Fs(t)3425 4801 y Fp(k)3468 4786
y Fw(.)257 4907 y(W)-8 b(e)53 b(sho)m(w)g(that)f(there)h(are)g(mark)m
(ed)f(terms)59 b(\026)-55 b Fs(u)2103 4870 y Fr(\003)2194
4907 y Fw(and)56 b(\026)-53 b Fs(v)2454 4870 y Fr(\003)2545
4907 y Fw(suc)m(h)54 b(that)e Fs(s)3062 4870 y Fr(\003)3134
4907 y Fu(\))3234 4926 y Fr(R)3294 4907 y Fm(d)3294 4942
y(n)3379 4907 y Fw(\026)-55 b Fs(u)3429 4870 y Fr(\003)3468
4907 y Fw(,)259 5009 y(\026)257 5027 y Fs(t)292 4991
y Fr(\003)365 5027 y Fu(\))465 5046 y Fr(R)525 5027 y
Fm(d)525 5063 y(n)607 5027 y Fw(\026)j Fs(v)655 4991
y Fr(\003)694 5027 y Fw(,)40 b(and)45 b(\026)-55 b Fs(u)1013
4991 y Fr(\003)1090 4999 y Fu(\030)1091 5031 y Fw(=)1209
5027 y(\026)i Fs(v)1256 4991 y Fr(\003)1296 5027 y Fw(.)61
b(The)40 b(whole)f(claim)d(then)j(follo)m(ws)e(from)h(the)h(inductiv)m
(e)257 5147 y(h)m(yp)s(othesis)g(on)f Fs(`)f Fw(in)g(com)m(bination)e
(with)i(Corollary)f(4.11)h(and)h(Lemma)e(4.9.)58 b(Since)257
5268 y Fs(s)303 5232 y Fr(\003)375 5268 y Fu(\))475 5283
y Fr(R)535 5291 y Fm(n)616 5249 y Fw(\026)614 5268 y
Fs(t)649 5232 y Fr(\003)689 5268 y Fw(,)36 b(there)g(are)g(mark)m(ed)f
(ground)h(constructor)g(terms)f Fs(u)2701 5232 y Fr(\003)2701
5292 y Fp(i)2776 5268 y Fw(and)g Fs(v)3019 5232 y Fr(\003)3015
5292 y Fp(i)3094 5268 y Fw(suc)m(h)h(that)257 5388 y
Fs(s)303 5352 y Fr(\003)303 5413 y Fp(i)343 5388 y Fs(\033)402
5352 y Fr(\003)474 5388 y Fu(\))574 5352 y Fr(\003)574
5413 y(R)634 5422 y Fm(n)p Fk(\000)p Fl(1)759 5388 y
Fs(u)815 5352 y Fr(\003)815 5413 y Fp(i)854 5388 y Fw(,)f
Fs(t)951 5352 y Fr(\003)951 5413 y Fp(i)991 5388 y Fs(\033)1050
5352 y Fr(\003)1122 5388 y Fu(\))1222 5352 y Fr(\003)1222
5413 y(R)1282 5422 y Fm(n)p Fk(\000)p Fl(1)1407 5388
y Fs(v)1458 5352 y Fr(\003)1454 5413 y Fp(i)1497 5388
y Fw(,)h(and)f Fs(u)1808 5352 y Fr(\003)1808 5413 y Fp(i)1879
5388 y Fu(\030)e Fs(v)2040 5352 y Fr(\003)2036 5413 y
Fp(i)2079 5388 y Fw(.)51 b(By)36 b(the)g(inductiv)m(e)f(h)m(yp)s
(othesis)h(on)1828 5637 y(21)p eop
%%Page: 22 22
22 21 bop 257 573 a Fs(n)34 b Fw(and)f(the)g(fact)g(that)g
Fs(u)1168 537 y Fr(\003)1168 597 y Fp(i)1240 573 y Fw(is)g(a)g(normal)e
(form,)h(w)m(e)i(conclude)g(that)f(there)h(are)f(mark)m(ed)257
693 y(terms)54 b(\026)-55 b Fs(u)600 657 y Fr(\003)600
718 y Fp(i)686 693 y Fw(and)51 b(\026)-53 b Fs(v)941
657 y Fr(\003)937 718 y Fp(i)1028 693 y Fw(suc)m(h)48
b(that)f Fs(s)1534 657 y Fr(\003)1534 718 y Fp(i)1574
693 y Fs(\033)1633 657 y Fr(\003)1705 693 y Fu(\))1805
657 y Fr(\003)1805 730 y(R)1865 708 y Fm(d)1865 752 y(n)p
Fk(\000)p Fl(1)1996 693 y Fw(\026)-55 b Fs(u)2046 657
y Fr(\003)2046 718 y Fp(i)2137 666 y Fu(\030)2138 697
y Fw(=)2268 693 y Fs(u)2324 657 y Fr(\003)2324 718 y
Fp(i)2410 693 y Fw(and)47 b Fs(t)2649 657 y Fr(\003)2649
718 y Fp(i)2689 693 y Fs(\033)2748 657 y Fr(\003)2820
693 y Fu(\))2920 657 y Fr(\003)2920 730 y(R)2980 708
y Fm(d)2980 752 y(n)p Fk(\000)p Fl(1)3109 693 y Fw(\026)-53
b Fs(v)3156 657 y Fr(\003)3152 718 y Fp(i)3248 666 y
Fu(\030)3249 697 y Fw(=)3378 693 y Fs(v)3429 657 y Fr(\003)3425
718 y Fp(i)3468 693 y Fw(.)257 842 y(Note)47 b(that)52
b(\026)-55 b Fs(u)788 805 y Fr(\003)788 866 y Fp(i)878
842 y Fu(\030)52 b Fs(u)1063 805 y Fr(\003)1063 866 y
Fp(i)1153 842 y Fu(\030)g Fs(v)1333 805 y Fr(\003)1329
866 y Fp(i)1423 842 y Fu(\030)k Fw(\026)-53 b Fs(v)1603
805 y Fr(\003)1599 866 y Fp(i)1642 842 y Fw(.)85 b(So)47
b(if)e Fu(V)8 b Fs(ar)s Fw(\()p Fs(r)s Fw(\))51 b Fu(\022)g(V)8
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805 y Fr(\003)3182 842 y Fu(\))3282 861 y Fr(R)3342 842
y Fm(d)3342 877 y(n)3422 823 y Fw(\026)3421 842 y Fs(t)3456
805 y Fr(\003)257 962 y Fw(and)g(the)h(claim)c(follo)m(ws.)85
b(Supp)s(ose)48 b(otherwise)g(that)e Fu(V)8 b Fs(ar)s
Fw(\()p Fs(r)s Fw(\))52 b Fu(6\022)h(V)8 b Fs(ar)s Fw(\()p
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Fs(x)c Fu(2)h(V)8 b Fs(ar)s Fw(\()p Fs(t)1531 1097 y
Fp(j)1567 1082 y Fw(\))32 b(for)f(some)h Fs(s)2075 1046
y Fr(\003)2075 1107 y Fp(j)2114 1082 y Fw(==)p Fs(t)2301
1046 y Fr(\003)2301 1107 y Fp(j)2340 1082 y Fw(.)44 b(Since)31
b Fs(t)2699 1046 y Fr(\003)2699 1107 y Fp(j)2739 1082
y Fs(\033)2798 1046 y Fr(\003)2870 1082 y Fu(\))2970
1046 y Fr(\003)2970 1120 y(R)3030 1097 y Fm(d)3030 1141
y(n)p Fk(\000)p Fl(1)3159 1082 y Fw(\026)-53 b Fs(v)3206
1046 y Fr(\003)3202 1107 y Fp(j)3273 1055 y Fu(\030)3274
1087 y Fw(=)3378 1082 y Fs(v)3429 1046 y Fr(\003)3425
1107 y Fp(j)3468 1082 y Fw(,)257 1225 y Fs(t)292 1189
y Fr(\003)292 1250 y Fp(j)379 1225 y Fw(is)47 b(a)f(mark)m(ed)i
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y Fr(\003)1510 1383 y(R)1570 1360 y Fm(d)1570 1404 y(n)p
Fk(\000)p Fl(1)1698 1346 y Fw(\026)-52 b Fs(v)1746 1309
y Fr(\003)1742 1370 y Fp(x)1836 1346 y Fw(for)50 b(some)h(ground)f
(constructor)i(subterm)261 1488 y(\026)-53 b Fs(v)308
1452 y Fr(\003)304 1513 y Fp(x)394 1488 y Fw(of)50 b(\026)-53
b Fs(v)570 1452 y Fr(\003)566 1513 y Fp(j)609 1488 y
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y Fp(j)2885 1488 y Fw(,)k(ev)m(ery)f(o)s(ccur-)257 1609
y(rence)41 b(of)d Fs(x\033)750 1573 y Fr(\003)829 1609
y Fw(in)g Fs(t)984 1573 y Fr(\003)984 1633 y Fp(j)1024
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1609 y Fw(b)s(ecause)40 b(redexes)h(with)e(iden)m(tical)e(marks)257
1729 y(are)42 b(shared)g(and)f(are)h(th)m(us)g(reduced)h(sim)m
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1729 y Fm(d)2720 1774 y(n)p Fk(\000)p Fl(1)2845 1729
y Fw(.)70 b(De\014ne)47 b(\026)-54 b Fs(\033)3312 1693
y Fr(\003)3392 1729 y Fw(b)m(y)257 1863 y Fs(x)5 b Fw(\026)-54
b Fs(\033)371 1827 y Fr(\003)465 1863 y Fw(=)56 b(\026)-52
b Fs(v)645 1827 y Fr(\003)641 1888 y Fp(x)732 1863 y
Fw(for)47 b(ev)m(ery)j Fs(x)j Fu(2)h(E)9 b(V)f Fs(ar)s
Fw(\()p Fs(\032)p Fw(\))47 b(and)h Fs(y)9 b Fw(\026)-54
b Fs(\033)2115 1827 y Fr(\003)2207 1863 y Fw(=)53 b Fs(y)t(\033)2447
1827 y Fr(\003)2533 1863 y Fw(for)47 b(ev)m(ery)j Fs(y)56
b Fu(2)e(V)8 b Fs(ar)s Fw(\()p Fs(l)r Fw(\).)257 1984
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y Fw(=)50 b Fs(C)789 1948 y Fr(\003)829 1984 y Fw([)p
Fs(r)903 1948 y Fr(\003)947 1984 y Fw(\026)-54 b Fs(\033)1001
1948 y Fr(\003)1040 1984 y Fs(;)17 b(:)g(:)g(:)f(;)h(r)1306
1948 y Fr(\003)1350 1984 y Fw(\026)-54 b Fs(\033)1404
1948 y Fr(\003)1443 1984 y Fw(].)84 b(Since)46 b Fs(z)t(\033)1957
1948 y Fr(\003)2048 1984 y Fw(=)k Fs(z)10 b Fw(\026)-55
b Fs(\033)2282 1948 y Fr(\003)2368 1984 y Fw(for)45 b(ev)m(ery)j
Fs(z)55 b Fu(2)c(V)8 b Fs(ar)s Fw(\()p Fs(l)r Fw(\))46
b(and)257 2104 y Fs(z)t(\033)365 2068 y Fr(\003)438 2104
y Fu(\))538 2068 y Fr(\003)538 2141 y(R)598 2119 y Fm(d)598
2163 y(n)p Fk(\000)p Fl(1)723 2104 y Fs(z)10 b Fw(\026)-55
b Fs(\033)831 2068 y Fr(\003)905 2104 y Fw(for)34 b(ev)m(ery)i
Fs(z)f Fu(2)c(E)9 b(V)e Fs(ar)s Fw(\()p Fs(\032)p Fw(\))35
b(w)m(e)g(deriv)m(e)2316 2086 y(\026)2315 2104 y Fs(t)2350
2068 y Fr(\003)2422 2104 y Fu(\))2522 2068 y Fr(\003)2522
2141 y(R)2582 2119 y Fm(d)2582 2163 y(n)p Fk(\000)p Fl(1)2710
2104 y Fw(\026)-52 b Fs(v)2758 2068 y Fr(\003)2831 2104
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b Fs(v)870 2337 y Fr(\003)910 2373 y Fw(.)71 b(This)42
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2493 y Fs(s)303 2457 y Fr(\003)303 2518 y Fp(i)348 2493
y Fw(\026)-54 b Fs(\033)402 2457 y Fr(\003)474 2493 y
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2552 y(n)p Fk(\000)p Fl(1)765 2493 y Fw(~)f Fs(u)815
2457 y Fr(\003)815 2518 y Fp(i)881 2466 y Fu(\030)882
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y Fw(,)32 b Fs(s)2640 2457 y Fr(\003)2640 2518 y Fp(i)2679
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b(con\015uen)m(t)h(mo)s(dulo)2070 2609 y Fu(\030)2070
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b Fs(t)2839 2600 y Fr(\003)2839 2661 y Fp(i)2884 2636
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y Fu(\))3109 2600 y Fr(\003)3109 2673 y(R)3169 2651 y
Fm(d)3169 2695 y(n)p Fk(\000)p Fl(1)3298 2636 y Fw(~)i
Fs(v)3345 2600 y Fr(\003)3341 2661 y Fp(i)3418 2609 y
Fu(\030)3419 2640 y Fw(=)261 2779 y(\026)g Fs(v)308 2743
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b Fs(u)1688 2743 y Fr(\003)1688 2804 y Fp(i)1772 2779
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2743 y Fr(\003)2200 2804 y Fp(i)2286 2779 y Fw(that)42
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4143 y Fw(\))33 b Fu(\))2219 4158 y Fr(R)2315 4143 y
Fs(g)2366 4107 y Fq(2)2405 4143 y Fw(\()p Fs(a)2494 4107
y Fq(3)2533 4143 y Fs(;)17 b(a)2628 4107 y Fq(4)2668
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y Fw(\()p Fs(a)436 4228 y Fq(3)476 4264 y Fs(;)17 b(a)571
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4743 y Fm(n)2693 4720 y Fw(\))46 b Fv(and)g Fs(N)10 b(F)k
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23 22 bop 257 573 a Fx(5)161 b(Concluding)53 b(Remarks)257
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b(ev)m(ery)h(almost)d(functional)f(CTRS)k(is)257 1394
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b(gonal)27 b Fw(systems)k(except)f(for)e(the)h(last)f(one.)404
1635 y(W)-8 b(e)34 b(stress)i(that)e(the)g(results)h(remain)d(v)-5
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257 1755 y(conditional)d(critical)h(pairs.)42 b(A)33
b(conditional)d(critical)g(pair)1373 1972 y Fu(h)p Fs(C)7
b Fw([)p Fs(r)1560 1987 y Fq(2)1600 1972 y Fw(])p Fs(\033)n(;)17
b(r)1768 1987 y Fq(1)1807 1972 y Fs(\033)t Fu(i)28 b(\()f
Fs(c)2102 1987 y Fq(1)2141 1972 y Fs(\033)n(;)17 b(c)2280
1987 y Fq(2)2320 1972 y Fs(\033)257 2189 y Fw(induced)34
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(rewrite)i(rules)h Fs(l)2791 2204 y Fq(1)2858 2189 y
Fu(!)28 b Fs(r)3030 2204 y Fq(1)3097 2189 y Fu(\()f Fs(c)3266
2204 y Fq(1)3338 2189 y Fw(and)257 2309 y Fs(l)286 2324
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2309 y Fu(\()f Fs(c)789 2324 y Fq(2)865 2309 y Fw(is)i(called)f
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2324 y Fq(1)2452 2309 y Fs(\033)n(;)17 b(c)2591 2324
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userdict /end-hook known{end-hook}if
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