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268.2262 391.5571 Tm (i)Tj /F10 1 Tf 11.9552 0 0 11.9552 275.5268 393.3504 Tm ()Tj /F3 1 Tf 1.0556 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 294.9778 391.5571 Tm (i)Tj /F2 1 Tf 0.4042 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 311.0625 393.3504 Tm 0 Tc ()Tj /F4 1 Tf 8.3686 0 0 8.3686 317.8939 391.5571 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 321.874 393.3504 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 328.7054 391.5571 Tm (i)Tj /F2 1 Tf 0.4042 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 227.6673 379.0042 Tm 0 Tc ()Tj /F4 1 Tf 8.3686 0 0 8.3686 234.4986 377.2109 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 238.4787 379.0042 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 245.3101 377.2109 Tm (j)Tj /F10 1 Tf 11.9552 0 0 11.9552 253.6678 379.0042 Tm ()Tj /F3 1 Tf 1.0556 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 273.1188 377.2109 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 278.1559 379.0042 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 284.9873 377.2109 Tm (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 312.8777 379.0042 Tm (for)Tj /F10 1 Tf 1.5306 0 TD (|)Tj /F3 1 Tf 0.2778 0 TD (i)Tj /F10 1 Tf 0.5667 0 TD ()Tj /F3 1 Tf 1 0 TD (j)Tj /F10 1 Tf 0.4691 0 TD 0.2777 Tc (|)Tj /F1 1 Tf 1.6111 0 TD 0 Tc (2)Tj 10.752 0.4388 TD (\(1)Tj /F3 1 Tf 0.8889 0 TD (.)Tj /F1 1 Tf 0.2777 0 TD (1\))Tj -37.5223 -2.1667 TD (and)Tj /F3 1 Tf 1.8562 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 101.0222 356.5534 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 105.0022 358.3467 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 112.262 363.4605 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.0093 TD (i)Tj /F10 1 Tf 11.9552 0 0 11.9552 128.4184 358.3467 Tm ()Tj /F3 1 Tf 1.0556 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 148.2979 363.4605 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.0093 TD (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 161.1337 358.3467 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 167.9641 356.5534 Tm (i)Tj /F10 1 Tf 11.9552 0 0 11.9552 175.2657 358.3467 Tm ()Tj /F3 1 Tf 1.0555 0 TD ()Tj /F1 1 Tf 0.4663 0 TD [(,)-262.7(where)]TJ /F3 1 Tf 3.3438 0 TD ()Tj /F1 1 Tf 0.7114 0 TD [(is)-245(the)-245(n)27.8(ullstring.)-415.1(In)-245.1(other)-245.1(w)27.8(ords,)-262.8(if)]TJ /F3 1 Tf 14.9834 0 TD (B)Tj /F4 1 Tf 8.3686 0 0 8.3686 430.138 356.5534 Tm (n)Tj /F1 1 Tf 11.9552 0 0 11.9552 439.5756 358.3467 Tm [(denotes)-245.1(the)-245(group)]TJ -30.7462 -1.2 TD [(generated)-412.7(b)27.7(y)-412.7(a)-412.7(sequence)-412.8(of)-412.7(generators)]TJ /F3 1 Tf 17.3846 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 286.667 342.2072 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 292.0299 344.0005 Tm (,)Tj /F3 1 Tf 0.7103 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 307.3534 342.2072 Tm (2)Tj /F1 1 Tf 11.9552 0 0 11.9552 312.7163 344.0005 Tm (,)Tj /F3 1 Tf 0.7103 0 TD 0.1667 Tc (...)Tj /F1 1 Tf 1.1667 0 TD 0 Tc (,)Tj /F3 1 Tf 0.7103 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 350.4797 342.2072 Tm (n)Tj /F11 1 Tf 0.7062 0 TD ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 374.1579 344.0005 Tm [(sub)-55.6(ject)-412.7(to)-412.7(the)-412.7(relations)-412.8(\(1.1\),)]TJ -25.2743 -1.2 TD [(then)-352.8(the)-352.8(isotop)27.7(y)-352.8(problem)-352.9(for)]TJ /F3 1 Tf 12.9696 0 TD (n)Tj /F1 1 Tf 0.953 0 TD [(strand)-352.8(braids)-352.9(b)-27.8(ecomes)-352.9(the)]TJ /F5 1 Tf 11.9057 0 TD [(wor)51(d)-375.6(p)0(r)51.1(oblem)]TJ /F1 1 Tf 6.1718 0 TD [(for)-352.8(the)-352.8(presen-)]TJ -32.0001 -1.2 TD [(tation)-333.3(\(1.1\))-333.4(of)-333.4(the)-333.4(group)]TJ /F3 1 Tf 11.0306 0 TD (B)Tj /F4 1 Tf 8.3686 0 0 8.3686 212.9405 313.5148 Tm (n)Tj /F1 1 Tf 11.9552 0 0 11.9552 219.4481 315.308 Tm (.)Tj -12.3334 -2.0724 TD [(In)-305.8(this)-305.8(text)-305.8(it)-305.8(will)-305.7(b)-27.8(e)-305.8(con)27.8(v)27.8(enien)27.7(t)-305.8(t)0(o)-305.8(w)27.8(ork)-305.7(with)-305.8(braids)-305.8(that)-305.8(in)27.8(v)27.8(olv)27.7(e)-305.7(a)0(n)-305.8(u)0(n)27.8(b)-27.8(ounded)-305.8(n)27.8(u)0(m)27.8(b)-27.8(er)]TJ 0 -1.2 TD [(of)-302.7(strands.)-434.3(This)-302.7(amoun)27.7(ts)-302.7(to)-302.7(consider)-302.8(the)-302.7(direct)-302.8(limit)]TJ /F3 1 Tf 23.1698 0 TD (B)Tj /F11 1 Tf 8.3686 0 0 8.3686 358.0663 274.3925 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 371.8137 276.1858 Tm [(of)-302.7(the)-302.8(groups)]TJ /F3 1 Tf 5.9998 0 TD (B)Tj /F4 1 Tf 8.3686 0 0 8.3686 452.6107 274.3925 Tm (n)Tj /F1 1 Tf 11.9552 0 0 11.9552 462.7368 276.1858 Tm [(\(with)-302.8(resp)-27.7(ect)]TJ -32.6835 -1.2 TD [(to)-251.8(inclusion\),)]TJ /F5 1 Tf 5.9143 0 TD (i.e.)Tj /F1 1 Tf 1.38 0 TD [(,)-268.1(the)-251.8(group)-251.8(generated)-251.8(b)27.7(y)-251.7(an)-251.8(innite)-251.7(sequence)]TJ /F3 1 Tf 19.4026 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 397.9965 260.0463 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 403.3594 261.8395 Tm (,)Tj /F3 1 Tf 0.5458 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 416.7163 260.0463 Tm (2)Tj /F1 1 Tf 11.9552 0 0 11.9552 422.0792 261.8395 Tm (,)Tj /F3 1 Tf 0.5458 0 TD 0.1667 Tc (...)Tj /F1 1 Tf 1.1667 0 TD 0 Tc [(,)-268.1(sub)-55.6(ject)-251.8(to)-251.7(\(1.1\).)]TJ -30.9952 -1.2 TD [(So)-264(braid)-264.1(w)27.8(ords)-264.1(will)-264.1(b)-27.8(e)-264.1(arbitrary)-264.1(nite)-264.1(sequences)-264.1(of)-264.1(letters)]TJ /F3 1 Tf 24.9851 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 377.9616 252.6072 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.042 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 390.7965 247.4934 Tm [(.)-421.4(The)]TJ /F5 1 Tf 2.6854 0 TD [(br)51.1(aid)]TJ /F1 1 Tf 2.4235 0 TD [(asso)-27.8(ciated)-264.1(with)]TJ -31.7749 -1.2 TD [(a)-383.6(braid)-383.6(w)27.7(ord)]TJ /F3 1 Tf 6.073 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (,)Tj /F5 1 Tf 0.674 0 TD (i.e.)Tj /F1 1 Tf 1.38 0 TD [(,)-396.1(its)-383.6(class)-383.6(in)-383.6(the)-383.6(group)]TJ /F3 1 Tf 10.3892 0 TD (B)Tj /F11 1 Tf 8.3686 0 0 8.3686 311.3126 231.3538 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 321.4416 233.147 Tm [(,)-396.1(i)0(s)-383.6(denoted)-383.7(b)27.8(y)]TJ ET 405.09 240.148 8.881 -0.478 re f BT /F3 1 Tf 11.9552 0 0 11.9552 405.0919 233.147 Tm (w)Tj /F1 1 Tf 0.7428 0 TD [(.)-595.3(I)0(f)-383.6(the)-383.6(braid)]TJ /F3 1 Tf 6.36 0 TD ()Tj /F1 1 Tf 1.002 0 TD [(is)-383.6(the)]TJ -35.9665 -1.2 TD [(class)-281.2(of)-281(the)-281.1(w)27.7(ord)]TJ /F3 1 Tf 7.4716 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(,)-291.6(w)27.7(e)-281.1(s)0(a)27.8(y)-281.1(also)-281(that)]TJ /F3 1 Tf 7.7327 0 TD (w)Tj /F1 1 Tf 1.0239 0 TD [(is)-281(a)-281.2(decomp)-27.8(osition)-281.2(for)]TJ /F3 1 Tf 9.7494 0 TD ()Tj /F1 1 Tf 0.6183 0 TD 0.4271 Tc [(.I)427.1(f)]TJ /F3 1 Tf 1.6526 0 TD 0 Tc (w)Tj /F1 1 Tf 1.0239 0 TD [(is)-281.1(a)-281.1(braid)-281.1(w)27.7(ord,)-291.5(the)]TJ /F5 1 Tf -30.0153 -1.2 TD (length)Tj /F1 1 Tf 2.8928 0 TD (of)Tj /F3 1 Tf 1.1173 0 TD (w)Tj /F1 1 Tf 1.0546 0 TD [(is)-311.8(the)-311.8(n)27.8(u)0(m)27.7(b)-27.8(er)-311.7(of)-311.8(letters)-311.8(o)-27.7(ccurring)-311.9(in)]TJ /F3 1 Tf 16.0102 0 TD (w)Tj /F1 1 Tf 1.0546 0 TD [(\(hence)-311.8(the)-311.8(n)27.8(u)0(m)27.7(b)-27.8(er)-311.7(of)-311.8(crossings)-311.9(in)-311.7(the)]TJ -22.1295 -1.2 TD [(asso)-27.8(ciated)-351.8(braid)-351.8(diagram\),)-356.6(and)-351.8(the)]TJ /F5 1 Tf 15.697 0 TD (width)Tj /F1 1 Tf 2.6773 0 TD (of)Tj /F3 1 Tf 1.1574 0 TD (w)Tj /F1 1 Tf 1.0946 0 TD [(is)-351.8(the)-351.9(width)-351.8(of)-351.9(the)-351.8(smallest)-351.9(domain)-351.9(that)]TJ -20.6263 -1.2 TD [(includes)-353.7(the)-353.6(non-trivial)-353.6(part)-353.7(of)]TJ /F3 1 Tf 13.8903 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(:)-484.9(formally)-353.7(the)-353.6(width)-353.6(of)]TJ /F3 1 Tf 10.4854 0 TD (w)Tj /F1 1 Tf 1.0964 0 TD (is)Tj /F3 1 Tf 1.0258 0 TD (n)Tj /F1 1 Tf 0.9538 0 TD (if)Tj /F3 1 Tf 0.9369 0 TD (n)Tj /F10 1 Tf 0.6693 0 TD ()Tj /F1 1 Tf 0.8468 0 TD [(2)-353.6(i)0(s)-353.6(the)-353.6(dierence)]TJ -30.6477 -1.2 TD [(b)-27.8(e)0(t)27.7(w)27.8(een)-429.9(the)-429.9(lo)27.8(w)27.7(est)-429.8(and)-429.9(the)-430(highest)-429.9(indices)-430(of)-429.9(letters)-430(o)-27.8(ccurring)-429.9(in)]TJ /F3 1 Tf 29.8427 0 TD (w)Tj /F1 1 Tf 1.1727 0 TD [(\(for)-429.9(instance)-429.9(the)]TJ -31.0153 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[(\).)-435.2(Finally)-305.5(w)27.8(e)-305.4(sa)27.8(y)-305.5(that)]TJ -28.957 -1.2 TD [(a)-337.7(braid)-337.8(w)27.8(ord)-337.8(is)]TJ /F5 1 Tf 6.9454 0 TD [(fr)51.1(e)51.2(ely)-361.9(r)51.2(e)51.1(duc)51.1(e)51.2(d)]TJ /F1 1 Tf 6.1429 0 TD [(if)-337.7(it)-337.8(includes)-337.8(no)-337.8(sub)27.8(w)27.8(ord)-337.8(of)-337.8(the)-337.7(form)]TJ /F3 1 Tf 16.3575 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 430.8613 116.5841 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 434.8414 118.3774 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 442.1021 123.4913 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 458.9749 118.3774 Tm (or)Tj /F3 1 Tf 1.2294 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 480.9325 123.4913 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.0093 TD (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 493.7684 118.3774 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 500.5998 116.5841 Tm (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 504.5798 118.3774 Tm [(,)-338.8(and)]TJ -36.1835 -1.2 TD 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0 8.3686 226.4448 691.1586 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 230.4259 692.9519 Tm 0.1666 Tc (,)Tj /F4 1 Tf 8.3686 0 0 8.3686 242.5704 691.1586 Tm 0 Tc (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 247.6075 692.9519 Tm 0.2778 Tc (\)=)Tj /F12 1 Tf 1.7222 2.0501 TD 0 Tc ()Tj 0 -0.9001 TD ()Tj 0 -1.8 TD ()Tj /F3 1 Tf 1.0555 1.8223 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 287.6473 705.1731 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 291.6274 706.9663 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 298.4588 705.1731 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 315.4521 706.9663 Tm (for)Tj /F10 1 Tf 1.5305 0 TD (|)Tj /F3 1 Tf 0.2778 0 TD (i)Tj /F10 1 Tf 0.5667 0 TD ()Tj /F3 1 Tf 1 0 TD (j)Tj /F10 1 Tf 0.4691 0 TD (|)Tj /F1 1 Tf 0.5556 0 TD 0.2777 Tc [(=1)277.7(,)]TJ /F3 1 Tf -7.2968 -1.2 TD 0 Tc ()Tj /F4 1 Tf 8.3686 0 0 8.3686 287.6473 690.8269 Tm (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 315.4521 692.6202 Tm (for)Tj /F10 1 Tf 1.5305 0 TD (|)Tj /F3 1 Tf 0.2778 0 TD (i)Tj /F10 1 Tf 0.5667 0 TD ()Tj /F3 1 Tf 1 0 TD (j)Tj /F10 1 Tf 0.4691 0 TD 0.2778 Tc (|)Tj /F1 1 Tf 1.6111 0 TD 0 Tc (2,)Tj /F3 1 Tf -8.3523 -1.2 TD ()Tj /F1 1 Tf 2.8972 0 TD (for)Tj /F3 1 Tf 1.5305 0 TD (i)Tj /F1 1 Tf 0.6223 0 TD (=)Tj /F3 1 Tf 1.0555 0 TD (j)Tj /F1 1 Tf 0.4691 0 TD (.)Tj -24.0412 -2.1938 TD [(In)-340.5(this)-340.5(denition)-340.5(the)-340.5(letter)-340.5(R)-340.5(stands)-340.6(for)-340.5(righ)27.7(t:)-458.7(righ)27.7(t)-340.4(reduction)-340.6(corresp)-27.8(onds)-340.5(to)-340.5(mo)27.7(ving)]TJ /F5 1 Tf 0 -1.2 TD (right)Tj /F1 1 Tf 2.505 0 TD [(the)-473.4(negativ)27.7(e)-473.3(generators)-473.5(in)-473.3(order)-473.4(to)-473.3(gather)-473.4(them)-473.4(at)-473.4(the)-473.3(end)-473.4(of)-473.4(the)-473.4(w)27.8(ord,)-508.4(while)]TJ -2.505 -1.2 TD [(p)-27.8(ositiv)27.7(e)-363.6(generators)-363.6(are)-363.6(gathered)-363.6(at)-363.5(the)-363.6(b)-27.8(eginning.)-535.1(Of)-363.6(course,)-371.1(lik)27.7(e)-363.6(for)-363.5(handle)-363.7(reduction,)]TJ T* [(w)27.8(e)-271.8(shall)-271.9(iterate)]TJ /F3 1 Tf 6.796 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-reduction)-271.8(and)-271.8(consider)-271.9(w)27.8(ords)-271.9(that)-271.8(can)-271.9(b)-27.8(e)-271.8(deduced)-271.8(from)-271.9(an)-271.8(initial)-271.9(w)27.8(ord)]TJ -7.563 -1.2 TD [(in)-333.3(an)27.8(y)-333.4(n)27.8(um)27.8(b)-27.8(e)0(r)-333.4(o)0(f)]TJ /F3 1 Tf 7.8362 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-reduction)-333.3(steps.)]TJ /F6 1 Tf -8.6032 -2.1372 TD [(Example)-369(0.2.)]TJ /F1 1 Tf 7.0293 0 TD [(In)-320.8(this)-320.8(section)-320.8(w)27.8(e)-320.8(shall)-320.8(illustrate)-320.8(our)-320.8(constructions)-320.8(using)-320.9(the)-320.7(w)27.7(ord)]TJ /F3 1 Tf 29.1618 0 TD (w)Tj /F2 1 Tf 8.3686 0 0 8.3686 513.2304 567.318 Tm (0)Tj /F1 1 Tf 11.9552 0 0 11.9552 521.9138 569.1113 Tm (=)Tj /F3 1 Tf -37.6334 -1.2 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 78.8314 552.9718 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 84.1943 554.7651 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 91.455 559.8789 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 104.2899 554.7651 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 111.5507 559.8789 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 124.3855 554.7651 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 131.2169 552.9718 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 136.5798 554.7651 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 143.4112 552.9718 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 148.7751 554.7651 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 155.6065 552.9718 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 160.9694 554.7651 Tm [(.)-426.1(Only)-278.6(one)-278.5(w)27.7(ord)-278.5(can)-278.6(b)-27.8(e)-278.5(deduced)-278.6(from)]TJ /F3 1 Tf 16.548 0 TD (w)Tj /F2 1 Tf 8.3686 0 0 8.3686 367.3624 552.9718 Tm (0)Tj /F1 1 Tf 11.9552 0 0 11.9552 376.0548 554.7651 Tm [(using)-278.5(one)-278.5(step)-278.6(of)]TJ /F3 1 Tf 7.4862 0 TD (R)Tj /F1 1 Tf 0.7671 0 TD (-reduction,)Tj -33.6863 -1.2 TD (namely)Tj /F3 1 Tf 3.571 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 121.5233 538.6256 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 126.8872 540.4189 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 134.1469 545.5327 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 146.9818 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 153.8132 538.6256 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 159.1771 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 166.0085 538.6256 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 171.3713 540.4189 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 178.6311 545.5327 Tm ()Tj /F2 1 Tf 0.893 0 TD (1)Tj -0.9442 -0.992 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 191.467 540.4189 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 198.7268 545.5327 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.944 -0.992 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 211.5626 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 218.394 538.6256 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 223.7569 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 230.5883 538.6256 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 235.9512 540.4189 Tm [(.)-740.8(Similarly)]TJ /F3 1 Tf 5.3702 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 307.4132 544.7576 Tm (2)Tj -0.0513 -0.8731 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 312.7761 540.4189 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 320.0369 545.5327 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 332.8717 540.4189 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 340.1325 545.5327 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 352.9674 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 359.7988 538.6256 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 365.1616 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 371.993 538.6256 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 382.5226 540.4189 Tm (and)Tj /F3 1 Tf 2.0433 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 413.7813 538.6256 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 419.1442 540.4189 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 426.405 545.5327 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 439.2399 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 446.0712 538.6256 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 451.4341 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 458.2655 538.6256 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 463.6294 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 470.4608 538.6256 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 475.8237 540.4189 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 483.0834 545.5327 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.944 -0.992 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 495.9193 540.4189 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 502.7507 538.6256 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 513.2802 540.4189 Tm (can)Tj -36.9113 -1.2 TD [(b)-27.8(e)-382.1(deduced)-382.2(in)-382.1(t)27.7(w)27.8(o)-382.2(steps,)]TJ /F5 1 Tf 11.3506 0 TD (etc.)Tj /F1 1 Tf 1.5589 0 TD [(W)83.4(e)-382.1(can)-382.1(c)27.8(hec)27.8(k)-382.1(that)]TJ /F5 1 Tf 8.6118 0 TD (any)Tj /F1 1 Tf 1.941 0 TD [(sequence)-382.2(of)]TJ /F3 1 Tf 5.3809 0 TD (R)Tj /F1 1 Tf 0.7671 0 TD [(-reductions)-382.2(from)]TJ /F3 1 Tf 7.6364 0 TD (w)Tj /F2 1 Tf 8.3686 0 0 8.3686 525.8491 524.2794 Tm (0)Tj /F1 1 Tf 11.9552 0 0 11.9552 72 511.7265 Tm [(ends)-303.7(after)-303.6(v)27.8(e)-303.6(steps)-303.7(in)-303.7(the)-303.6(w)27.8(ord)]TJ /F3 1 Tf 14.1476 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 248.3975 516.0652 Tm (2)Tj -0.0513 -0.8731 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 253.7604 511.7265 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 260.5918 509.9332 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 265.9547 511.7265 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 272.786 509.9332 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 278.1499 511.7265 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 285.4097 516.8403 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.944 -0.992 TD (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 298.2456 511.7265 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 305.5053 516.8403 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.944 -0.992 TD (2)Tj /F1 1 Tf 11.9552 0 0 11.9552 318.3412 511.7265 Tm [(,)-309.6(whic)27.8(h)-303.7(i)0(s)]TJ /F3 1 Tf 4.3946 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-irreducible)-303.7(since)-303.7(no)-303.6(negativ)27.7(e)]TJ -25.767 -1.2 TD [(letter)-333.9(precedes)-334(an)27.7(y)-333.9(p)-27.8(ositiv)27.7(e)-333.9(one.)-446.2(Figure)-334(2.1)-333.9(illustrates)]TJ /F3 1 Tf 23.8863 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-reduction)-334(in)-333.9(the)-333.9(Ca)27.7(yley)-333.9(graph.)]TJ -24.6533 -1.2 TD [(\(W)83.3(e)-355.1(recall)-355.3(that)-355.2(the)-355.2(Ca)27.8(yley)-355.2(graph)-355.2(of)]TJ /F3 1 Tf 16.1029 0 TD (B)Tj /F11 1 Tf 8.3686 0 0 8.3686 273.5811 481.2408 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 287.9562 483.0341 Tm [(is)-355.2(an)-355.2(orien)27.8(ted)-355.1(graph)-355.3(with)-355.2(lab)-27.8(elled)-355.2(arro)27.7(ws)-355.2(suc)27.8(h)]TJ -18.0638 -1.2 TD [(that)-291.7(the)-291.6(set)-291.7(of)-291.6(v)27.8(ertices)-291.7(is)]TJ /F3 1 Tf 10.9637 0 TD (B)Tj /F11 1 Tf 8.3686 0 0 8.3686 212.1405 466.8946 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 222.2695 468.6879 Tm [(,)-300(and)-291.7(there)-291.6(is)-291.7(a)]TJ /F3 1 Tf 6.7526 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 309.8291 466.8946 Tm (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 313.8092 468.6879 Tm [(-lab)-27.9(elled)-291.6(arro)27.7(w)-291.6(b)-27.8(et)27.7(w)27.8(een)-291.7(from)-291.6(the)-291.6(v)27.8(ertex)]TJ /F3 1 Tf 17.5665 0 TD ()Tj /F1 1 Tf -37.7928 -1.2 TD [(to)-420(the)-420(v)27.8(ertex)]TJ /F3 1 Tf 6.2348 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 153.9317 458.6804 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 162.3063 454.3417 Tm [(if)-419.9(and)-420(only)-420(if)]TJ /F3 1 Tf 6.3187 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 245.2403 458.6804 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 253.615 454.3417 Tm [(is)-419.9(equal)-420(to)]TJ /F3 1 Tf 5.1265 0 TD 0.0528 Tc ()Tj /F4 1 Tf 8.3686 0 0 8.3686 329.1278 452.5484 Tm 0 Tc (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 333.1079 454.3417 Tm [(.\))-704.3(Observ)27.7(e)-420(that,)-441.6(if)-420(the)-420(arro)27.7(ws)-419.9(of)-419.9(the)]TJ -21.8406 -1.2 TD [(Ca)27.8(yley)-312.7(graph)-312.7(are)-312.6(alw)27.7(a)27.8(y)0(s)-312.6(giv)27.8(en)-312.7(a)-312.6(left-to-righ)27.7(t)-312.6(orien)27.7(tation,)]TJ /F3 1 Tf 25.0495 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-reduction)-312.7(alw)27.7(a)27.8(y)0(s)-312.6(transforms)]TJ -25.8165 -1.2 TD [(a)-306.5(w)27.7(ord)-306.5(in)27.7(to)-306.5(another)-306.6(one)-306.5(suc)27.8(h)-306.5(that)-306.6(the)-306.5(path)-306.6(asso)-27.8(ciated)-306.5(with)-306.6(the)-306.6(latter)-306.5(lies)-306.6(on)-306.5(the)-306.5(righ)27.7(t)-306.5(o)0(f)]TJ T* [(the)-333.4(path)-333.4(asso)-27.8(ciated)-333.3(with)-333.4(the)-333.3(initial)-333.4(w)27.8(ord.)]TJ /F3 1 Tf 18.2098 -10.7829 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 296.532 280.5982 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 303.3066 313.0057 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 310.138 311.2124 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 320.3148 330.0139 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 327.1462 328.2206 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 279.4959 313.0057 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 286.3273 311.2124 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 259.0864 323.2104 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 265.9178 321.4171 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 238.677 343.6199 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 245.5083 341.8266 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 276.0947 358.9275 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 282.926 357.1342 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 253.9846 369.1322 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 260.8159 367.3389 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 305.0082 367.4306 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 311.8396 365.6373 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 352.6297 367.4306 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 359.461 365.6373 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 305.0082 340.2186 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 311.8396 338.4254 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 352.6297 340.2186 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 359.461 338.4254 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 286.2994 387.8401 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 293.1308 386.0468 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 320.3148 387.8401 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 327.1462 386.0468 Tm (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 174.0473 255.1785 Tm [(Figure)-383.3(2.1)]TJ /F1 1 Tf 5.1104 0 TD (:)Tj /F3 1 Tf 0.7222 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-reduction)-333.4(of)]TJ /F3 1 Tf 5.9195 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 330.5464 253.3853 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 335.9093 255.1785 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 343.1701 260.2933 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 356.005 255.1785 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 363.2657 260.2933 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 376.1006 255.1785 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 382.932 253.3853 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 388.2949 255.1785 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 395.1263 253.3853 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 400.4902 255.1785 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 407.3216 253.3853 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 72 230.0049 Tm [(It)-298.3(is)-298.2(clear)-298.3(that,)-305.3(if)]TJ /F3 1 Tf 7.6732 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 172.6147 234.3426 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 179.5347 230.0049 Tm [(is)-298.2(deduced)-298.3(from)]TJ /F3 1 Tf 7.153 0 TD (w)Tj /F1 1 Tf 1.0411 0 TD (using)Tj /F3 1 Tf 2.5815 0 TD (R)Tj /F1 1 Tf 0.7671 0 TD [(-reduction,)-305.3(then)]TJ /F3 1 Tf 7.2729 0 TD (w)Tj /F1 1 Tf 1.0411 0 TD (and)Tj /F3 1 Tf 1.9093 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 448.6317 234.3426 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 455.5507 230.0049 Tm [(are)-298.2(equiv)55.5(alen)27.8(t,)]TJ -32.0824 -1.2 TD [(and)-264.1(it)-264(is)-264(not)-264.1(hard)-264(to)-264(see)-264.1(that)-264(at)-264(most)-264(one)]TJ /F3 1 Tf 17.8127 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-irreducible)-264.1(w)27.7(ord)-264(can)-264(b)-27.8(e)-264.1(deduced)-264.1(from)-264.1(a)-264(giv)27.7(en)]TJ -18.5797 -1.2 TD [(w)27.8(ord)-362.6(using)]TJ /F3 1 Tf 5.1501 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-reduction.)-532.2(It)-362.6(is)-362.5(less)-362.6(ob)27.8(vious)-362.6(that)]TJ /F3 1 Tf 15.1198 0 TD (R)Tj /F1 1 Tf 0.767 0 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696.1509 Tm (0)Tj /F1 1 Tf 11.9552 0 0 11.9552 189.5591 697.9442 Tm (,)Tj /F3 1 Tf 0.6111 0 TD 0.1667 Tc (...)Tj /F1 1 Tf 1.1667 0 TD 0 Tc (,)Tj /F3 1 Tf 0.6112 0 TD (v)Tj /F4 1 Tf 8.3686 0 0 8.3686 223.9133 696.1509 Tm (m)Tj /F1 1 Tf 11.9552 0 0 11.9552 236.9803 697.9442 Tm (are)Tj /F3 1 Tf 1.6694 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 264.1992 703.058 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.042 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 281.0192 697.9442 Tm (with)Tj /F10 1 Tf 2.2778 0 TD (|)Tj /F3 1 Tf 0.2778 0 TD (k)Tj /F10 1 Tf 0.7743 0 TD ()Tj /F3 1 Tf 1 0 TD (j)Tj /F10 1 Tf 0.4691 0 TD 0.2777 Tc (|)Tj /F1 1 Tf 1.6111 0 TD 0 Tc [(2\))-333.4(in)27.8(to)]TJ /F3 1 Tf -14.0172 -2.5174 TD (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 195.8695 666.0547 Tm (0)Tj /F3 1 Tf 11.9552 0 0 11.9552 205.2184 667.848 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 212.4782 672.9629 Tm ()Tj /F4 1 Tf 0.8929 0 TD (e)Tj -0.9442 -1.0094 TD (j)Tj /F2 1 Tf 0.5306 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 229.1915 667.848 Tm 0 Tc ()Tj /F4 1 Tf 8.3686 0 0 8.3686 236.4513 672.7845 Tm (d)Tj -0.0512 -0.943 TD (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 242.0264 667.848 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 249.2862 672.7845 Tm (e)Tj -0.0512 -0.943 TD (j)Tj /F2 1 Tf 0.5305 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 269.9846 667.848 Tm 0 Tc (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 275.7798 666.0547 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 281.1427 667.848 Tm 0.1667 Tc [(...)-166.6(v)]TJ /F4 1 Tf 8.3686 0 0 8.3686 304.8707 666.0547 Tm 0 Tc (m)Tj /F11 1 Tf 1.0137 0 TD ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 330.1739 667.848 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 337.4346 672.9629 Tm ()Tj /F4 1 Tf 0.8929 0 TD (e)Tj -0.9442 -1.0094 TD (j)Tj /F2 1 Tf 0.5306 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 354.1479 667.848 Tm 0 Tc ()Tj /F4 1 Tf 8.3686 0 0 8.3686 361.4077 672.7845 Tm (d)Tj -0.0512 -0.943 TD (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 366.9828 667.848 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 374.2426 672.7845 Tm (e)Tj -0.0513 -0.943 TD (j)Tj /F2 1 Tf 0.5306 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 394.941 667.848 Tm 0 Tc (v)Tj /F4 1 Tf 8.3686 0 0 8.3686 400.7352 666.0547 Tm (m)Tj /F3 1 Tf 11.9552 0 0 11.9552 409.8172 667.848 Tm (.)Tj /F1 1 Tf 8.0987 0 TD (\(2)Tj /F3 1 Tf 0.8889 0 TD (.)Tj /F1 1 Tf 0.2777 0 TD (2\))Tj -37.5223 -2.4167 TD [(can)-557.3(b)-27.8(e)-557.2(decomp)-27.8(osed)-557.2(in)27.7(to)-557.2(a)-557.2(series)-557.2(of)-557.3(microsteps)-557.3(o)0(f)-557.2(the)-557.3(ab)-27.8(o)27.8(v)27.8(e)-557.2(t)27.7(y)0(p)-27.8(es.)-1116.1(Assume)-557.3(for)]TJ 0 -1.2 TD [(instance)-528.6(that)]TJ /F3 1 Tf 6.4516 0 TD (e)Tj /F1 1 Tf 0.9942 0 TD [(is)-528.5(+1)-528.6(and)]TJ /F3 1 Tf 5.1467 0 TD (d)Tj /F1 1 Tf 1.0491 0 TD (is)Tj /F10 1 Tf 1.2008 0 TD ()Tj /F1 1 Tf 0.7778 0 TD [(1.)-1030.1(Then)-528.6(reduction)-528.6(can)-528.6(b)-27.8(e)-528.6(done)-528.6(b)27.7(y)-528.6(mo)27.8(ving)-528.6(righ)27.7(t)]TJ -15.6201 -1.2 TD [(the)-497.5(initial)]TJ /F3 1 Tf 4.9395 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 137.8839 608.4707 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 142.921 610.264 Tm [(.)-937.1(First)-497.6(transforming)]TJ /F3 1 Tf 9.9098 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 268.2262 608.4707 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 273.2643 610.264 Tm (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 279.0585 608.4707 Tm (0)Tj /F1 1 Tf 11.9552 0 0 11.9552 290.3701 610.264 Tm [(in)27.7(to)]TJ /F3 1 Tf 2.1919 0 TD (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 322.3701 608.4707 Tm (0)Tj /F3 1 Tf 11.9552 0 0 11.9552 327.734 610.264 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 334.5644 608.4707 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 345.5502 610.264 Tm [(can)-497.5(b)-27.8(e)-497.6(made)-497.5(b)27.7(y)-497.5(a)-497.5(sequence)-497.6(of)]TJ /F3 1 Tf 14.516 0 TD (L)Tj /F1 1 Tf 0.6806 0 TD (-)Tj -38.0779 -1.2 TD [(reductions)-414.6(and)]TJ /F3 1 Tf 6.9485 0 TD (P)Tj /F1 1 Tf 0.7809 0 TD [(-equiv)55.5(alences.)-688.2(Indeed)-414.5(b)27.8(y)-414.6(h)27.8(y)0(p)-27.8(othesis)]TJ /F3 1 Tf 16.4483 0 TD (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 366.8444 594.1245 Tm (0)Tj /F1 1 Tf 11.9552 0 0 11.9552 377.1627 595.9178 Tm [(con)27.7(tains)-414.5(only)-414.6(generators)]TJ /F3 1 Tf 11.2047 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 518.3771 601.0316 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.042 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 72 581.5716 Tm (with)Tj /F10 1 Tf 2.2966 0 TD (|)Tj /F3 1 Tf 0.2778 0 TD (k)Tj /F10 1 Tf 0.7868 0 TD ()Tj /F3 1 Tf 1.0126 0 TD (j)Tj /F10 1 Tf 0.469 0 TD 0.3091 Tc (|)Tj /F1 1 Tf 1.6738 0 TD 0 Tc [(2:)-482.1(for)-352.2(suc)27.8(h)]TJ /F3 1 Tf 5.0836 0 TD (k)Tj /F1 1 Tf 0.9042 0 TD (transforming)Tj /F3 1 Tf 5.9466 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 299.4162 579.7783 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 304.4533 581.5716 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 311.2847 579.7783 Tm (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 321.3768 581.5716 Tm [(in)27.7(to)]TJ /F3 1 Tf 2.0466 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 352.6754 579.7783 Tm (k)Tj /F3 1 Tf 11.9552 0 0 11.9552 358.5574 581.5716 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 365.3888 579.7783 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 374.6361 581.5716 Tm [(is)-352.1(a)]TJ /F3 1 Tf 1.8765 0 TD (P)Tj /F1 1 Tf 0.7809 0 TD [(-equiv)55.5(alence,)-356.9(and)-352.2(trans-)]TJ -27.9717 -1.2 TD (forming)Tj /F3 1 Tf 3.758 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 123.7589 565.4321 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 128.796 567.2253 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 136.0568 572.3392 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.042 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 153.604 567.2253 Tm [(in)27.8(to)]TJ /F3 1 Tf 2.0885 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 185.8331 572.3392 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.042 TD (k)Tj /F3 1 Tf 11.9552 0 0 11.9552 198.668 567.2253 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 205.4994 565.4321 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 215.2488 567.2253 Tm [(is)-394.1(an)]TJ /F3 1 Tf 2.516 0 TD (L)Tj /F1 1 Tf 0.6806 0 TD [(-reduction.)-626.8(Then)-394.2(w)27.8(e)-394.1(nd)-394.1(the)-394.1(pattern)]TJ /F3 1 Tf 17.0196 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 463.7679 565.4321 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 468.805 567.2253 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 476.0648 572.3392 Tm ()Tj /F2 1 Tf 0.893 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F2 1 Tf 0.5305 0 TD -0.0001 Tc (+1)Tj /F1 1 Tf 11.9552 0 0 11.9552 492.7781 567.2253 Tm 0 Tc [(,)-409.3(whic)27.7(h)]TJ -35.1963 -1.3937 TD [(b)-27.8(ecomes)]TJ /F3 1 Tf 3.9984 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 127.0615 555.6767 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.0093 TD (j)Tj /F2 1 Tf 0.5305 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 143.7748 550.5629 Tm 0 Tc ()Tj /F11 1 Tf 8.3686 0 0 8.3686 151.0356 555.6767 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 163.8705 550.5629 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 170.7018 548.7696 Tm (j)Tj /F2 1 Tf 0.5305 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 187.8436 550.5629 Tm 0 Tc ()Tj /F4 1 Tf 8.3686 0 0 8.3686 194.675 548.7696 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 203.9442 550.5629 Tm -0.0277 Tc [(by)-381.7(a)-27.7(n)]TJ /F3 1 Tf 2.8191 0 TD 0 Tc (L)Tj /F1 1 Tf 0.6806 0 TD [(-reduction.)-506.4(So,)-359.1(at)-354(this)-353.9(p)-27.8(oin)27.7(t,)-359.1(w)27.7(e)-354(ha)27.8(v)27.8(e)-354(transformed)-354.1(the)]TJ -14.5363 -1.2 TD [(initial)-333.4(w)27.8(ord)-333.4(in)27.8(to)]TJ /F3 1 Tf 7.8061 -2.4168 TD (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 171.1183 505.5307 Tm (0)Tj /F3 1 Tf 11.9552 0 0 11.9552 180.4662 507.324 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 187.726 512.4378 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.0093 TD (j)Tj /F2 1 Tf 0.5305 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 204.4394 507.324 Tm 0 Tc ()Tj /F11 1 Tf 8.3686 0 0 8.3686 211.7001 512.4378 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 224.535 507.324 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 231.3664 505.5307 Tm (j)Tj /F2 1 Tf 0.5305 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 248.5081 507.324 Tm 0 Tc ()Tj /F4 1 Tf 8.3686 0 0 8.3686 255.3395 505.5307 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 264.3626 507.324 Tm (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 270.1569 505.5307 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 279.5059 507.324 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 286.7656 512.4378 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.0093 TD (j)Tj /F2 1 Tf 0.5305 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 307.464 507.324 Tm 0 Tc (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 313.2593 505.5307 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 318.6222 507.324 Tm 0.1667 Tc [(...)-166.6(v)]TJ /F4 1 Tf 8.3686 0 0 8.3686 342.3502 505.5307 Tm 0 Tc (m)Tj /F11 1 Tf 1.0137 0 TD ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 367.6533 507.324 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 374.9141 512.4378 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F2 1 Tf 0.5305 0 TD -0.0001 Tc (+1)Tj /F3 1 Tf 11.9552 0 0 11.9552 395.6125 507.324 Tm 0 Tc (v)Tj /F4 1 Tf 8.3686 0 0 8.3686 401.4067 505.5307 Tm (m)Tj /F3 1 Tf 11.9552 0 0 11.9552 414.4737 507.324 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 421.7335 512.4378 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 434.5684 507.324 Tm (.)Tj /F1 1 Tf 6.0283 0 TD (\(2)Tj /F3 1 Tf 0.8889 0 TD (.)Tj /F1 1 Tf 0.2777 0 TD (3\))Tj -37.5223 -2.4168 TD (After)Tj /F3 1 Tf 2.6222 0 TD (m)Tj /F1 1 Tf 1.2198 0 TD [(suc)27.8(h)-341.7(sequences)-341.8(of)-341.7(reductions,)-343.9(and)-341.7(a)-341.7(last)]TJ /F3 1 Tf 17.7857 0 TD (L)Tj /F1 1 Tf 0.6806 0 TD [(-reduction)-341.8(to)-341.7(delete)-341.7(the)-341.7(nal)-341.8(pattern)]TJ /F3 1 Tf -22.3083 -1.2 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 78.8314 462.2919 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 83.8685 464.0851 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 91.1293 469.2 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 103.9641 464.0851 Tm [(,)-504.3(w)27.8(e)-470.1(reac)27.7(h)-470(the)-470.1(form)-470.1(\(2.2\),)-504.3(as)-470.1(w)27.8(e)-470.1(wished.)-854.7(The)-470.1(argumen)27.8(t)-470.1(i)0(s)-470.1(similar)-470.1(in)-470.1(the)-470.1(case)]TJ /F3 1 Tf -2.6737 -1.2 TD (e)Tj /F1 1 Tf 0.8847 0 TD (=)Tj /F10 1 Tf 1.1968 0 TD ()Tj /F1 1 Tf 0.7778 0 TD (1,)Tj /F3 1 Tf 1.2171 0 TD (d)Tj /F1 1 Tf 0.9396 0 TD [(=)-419.1(1)0(,)-439.2(with)]TJ /F3 1 Tf 4.7764 0 TD (N)Tj /F1 1 Tf 0.9125 0 TD [(-equiv)55.5(alences)-418.2(instead)-418.1(of)]TJ /F3 1 Tf 10.8488 0 TD (P)Tj /F1 1 Tf 0.7809 0 TD [(-equiv)55.5(alences.)-698.8(In)-418.1(the)-418.1(case)-418.1(when)-418.2(the)]TJ -22.3345 -1.2 TD [(exp)-27.8(onen)27.8(ts)]TJ /F3 1 Tf 4.6728 0 TD (e)Tj /F1 1 Tf 0.7718 0 TD (and)Tj /F3 1 Tf 1.9173 0 TD (d)Tj /F1 1 Tf 0.8267 0 TD [(ha)27.7(v)27.8(e)-306.1(the)-306.3(same)-306.2(sign,)-311.6(w)27.8(e)-306.2(use)]TJ /F3 1 Tf 11.9149 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-reduction)-306.3(in)-306.2(order)-306.3(to)-306.2(mo)27.7(v)27.8(e)-306.1(left)-306.3(the)-306.2(nal)]TJ -20.8706 -1.2 TD (generator)Tj /F3 1 Tf 4.45 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 132.4613 426.1613 Tm ()Tj /F4 1 Tf 0.8929 0 TD (e)Tj -0.9442 -1.0093 TD (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 149.052 421.0465 Tm [(in)-333.3(a)-333.3(symmetric)-333.4(w)27.8(a)27.7(y)83.4(.)]TJ /F16 1 Tf 31.1883 0 TD ()Tj /F1 1 Tf -37.6334 -2.4278 TD [(It)-273.3(follo)27.8(ws)-273.4(from)-273.4(this)-273.4(result)-273.3(that)-273.4(the)-273.4(set)-273.4(of)-273.4(all)-273.4(w)27.8(ords)-273.4(that)-273.4(can)-273.4(b)-27.8(e)-273.3(deduced)-273.4(b)27.7(y)-273.4(reduction)-273.4(from)]TJ 0 -1.2 TD [(a)-361.6(giv)27.7(en)-361.6(w)27.8(ord)]TJ /F3 1 Tf 6.0043 0 TD (w)Tj /F1 1 Tf 1.1045 0 TD [(is)-361.6(included)-361.6(in)-361.6(the)-361.6(closure)-361.7(of)-361.6(the)-361.6(w)27.8(ord)]TJ /F3 1 Tf 16.7984 0 TD (w)Tj /F1 1 Tf 1.1044 0 TD (under)Tj /F3 1 Tf 2.8644 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-)-361.7(and)]TJ /F3 1 Tf 2.6677 0 TD (L)Tj /F1 1 Tf 0.6806 0 TD [(-reduction)-361.7(and)]TJ /F3 1 Tf -31.9912 -1.2 TD (P)Tj /F1 1 Tf 0.7809 0 TD [(-)-291.9(and)]TJ /F3 1 Tf 2.5283 0 TD (N)Tj /F1 1 Tf 0.9125 0 TD [(-equiv)55.5(alence.)-430.8(W)83.4(e)-292(ha)27.8(v)27.8(e)-292(men)27.7(tioned)-292(that)-291.9(the)-292(closure)-292(of)-292(a)-291.9(w)27.7(ord)-292(under)]TJ /F3 1 Tf 28.9753 0 TD (R)Tj /F1 1 Tf 0.767 0 TD (-reduction)Tj -33.964 -1.2 TD [(alone,)-287(or)-275.3(under)]TJ /F3 1 Tf 6.7877 0 TD (L)Tj /F1 1 Tf 0.6805 0 TD [(-reduction)-275.4(alone,)-287(is)-275.4(certainly)-275.5(nite.)-425.1(Things)-275.4(are)-275.3(not)-275.4(so)-275.3(simple)-275.5(when)-275.3(b)-27.8(oth)]TJ -7.4682 -1.2 TD [(op)-27.8(erations)-333.4(are)-333.4(allo)27.8(w)27.7(e)0(d)-333.3(sim)27.8(ultaneously)83.3(,)-333.4(a)0(s)-333.3(sho)27.7(ws)-333.3(the)-333.3(sequence)]TJ /F3 1 Tf 2.7288 -2.4168 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 111.454 303.9506 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 116.8169 305.7439 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 124.0777 310.8587 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.9921 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 136.9126 305.7439 Tm 1.1666 Tc (,)Tj /F11 1 Tf 8.3686 0 0 8.3686 161.4416 310.8587 Tm 0 Tc ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.9921 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 174.2765 305.7439 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 181.5372 310.8587 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.9921 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 194.3721 305.7439 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 201.2035 303.9506 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 206.5664 305.7439 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 213.3977 303.9506 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 218.7616 305.7439 Tm 1.1666 Tc (,)Tj /F11 1 Tf 8.3686 0 0 8.3686 243.2907 310.8587 Tm 0 Tc ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.9921 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 256.1255 305.7439 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 262.9569 303.9506 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 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214.8662 642.2062 Tm [(Figure)-383.3(2.3)]TJ /F1 1 Tf 5.1104 0 TD [(:)-444.4(W)83.3(ord)-333.3(traced)-333.4(in)]TJ /F3 1 Tf 7.6722 0 TD (S)Tj -24.7328 -1.4795 TD 0.0359 Tc (v)Tj /F4 1 Tf 8.3686 0 0 8.3686 85.055 622.7253 Tm 0 Tc (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 93.9337 624.5186 Tm (\()Tj /F5 1 Tf 0.3888 0 TD [(r)51.1(esp.)]TJ /F3 1 Tf 2.5173 0 TD 0.0359 Tc (v)Tj /F11 1 Tf 8.3686 0 0 8.3686 142.1599 629.6323 Tm 0 Tc ()Tj /F2 1 Tf 0.893 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 154.9958 624.5186 Tm [(\))-321.3(there)-321.3(exist)-321.3(p)-27.8(ositiv)27.7(e)-321.3(w)27.8(ords)]TJ /F3 1 Tf 12.1563 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 307.1711 628.8572 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 310.5255 624.5186 Tm (,)Tj /F3 1 Tf 0.6014 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 324.5599 628.8572 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 334.5106 624.5186 Tm [(suc)27.8(h)-321.3(that)]TJ /F3 1 Tf 4.3982 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 393.9348 628.8572 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 397.2892 624.5186 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 404.1206 622.7253 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 409.1577 624.5186 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 416.001 628.8572 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 425.9527 624.5186 Tm [(b)-27.8(elongs)-321.3(to)]TJ /F3 1 Tf 4.787 0 TD (S)Tj /F1 1 Tf 0.9921 0 TD (and)Tj /F3 1 Tf 1.9324 0 TD (uv)Tj /F1 1 Tf -37.3182 -1.2 TD [(is)-333.4(equiv)55.6(alen)27.8(t)-333.4(t)0(o)]TJ /F3 1 Tf 6.9778 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 162.2645 614.511 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 169.604 610.1723 Tm (\()Tj /F5 1 Tf 0.3888 0 TD [(r)51.1(esp.)]TJ /F1 1 Tf 2.5173 0 TD (to)Tj /F3 1 Tf 1.2222 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 225.8022 614.511 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 229.1567 610.1723 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 235.988 608.379 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 241.0251 610.1723 Tm (\).)Tj -14.1383 -2.1795 TD [(Observ)27.7(e)-387.2(that)-387.2(there)-387.3(are)-387.2(alw)27.7(a)27.7(y)0(s)-387.2(innitely)-387.3(man)27.8(y)-387.2(w)27.8(ords)-387.3(traced)-387.2(in)-387.2(an)27.8(y)-387.2(set)]TJ /F3 1 Tf 31.5438 0 TD (S)Tj /F1 1 Tf 1.058 0 TD [(that)-387.3(con)27.7(tains)]TJ -32.6018 -1.2 TD [(at)-396.7(least)-396.8(one)-396.8(nonempt)27.8(y)-396.7(w)27.7(ord:)-571.2(if)-396.8(the)-396.8(w)27.8(ord)]TJ /F3 1 Tf 18.6094 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 301.3101 567.9766 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 311.0904 569.7698 Tm [(is)-396.8(traced)-396.7(in)]TJ /F3 1 Tf 5.4207 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD [(,)-412.6(s)0(o)-396.7(are)-396.7(all)-396.8(w)27.8(ords)-396.8(\()]TJ /F3 1 Tf 8.4883 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 492.2272 567.9766 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 497.2653 569.7698 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 504.5251 574.8846 Tm ()Tj /F2 1 Tf 0.8928 0 TD (1)Tj /F4 1 Tf -0.944 -1.0094 TD (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 517.3599 569.7698 Tm (\))Tj /F4 1 Tf 8.3686 0 0 8.3686 522.0095 574.1085 Tm (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 527.8914 569.7698 Tm (.)Tj -38.1334 -1.2 TD [(Nev)27.8(ertheless)-301.1(w)27.8(e)-301(shall)-301.1(see)-300.9(in)-301(Section)-301(3)-301(h)0(o)27.8(w)-301(the)-301(fact)-301(that)-300.9(some)-301.1(w)27.8(ords)-301(are)-301(traced)-301(in)-301(a)-301(nite)]TJ T* [(set)-282.7(can)-282.8(b)-27.8(e)-282.7(used)-282.8(as)-282.8(a)-282.7(strong)-282.8(b)-27.8(oundedness)-282.9(h)27.8(y)0(p)-27.8(othesis.)-427.6(Presen)27.7(tly)-282.7(w)27.8(e)-282.7(observ)27.7(e)-282.7(that,)-292.9(pro)27.8(vided)]TJ T* [(that)-379.6(the)-379.5(set)]TJ /F3 1 Tf 5.5887 0 TD (S)Tj /F1 1 Tf 1.0504 0 TD [(is)-379.5(regular)-379.6(enough,)-391.2(the)-379.5(set)-379.6(of)-379.6(the)-379.5(w)27.8(ords)-379.6(traced)-379.6(in)]TJ /F3 1 Tf 21.8348 0 TD (S)Tj /F1 1 Tf 1.0504 0 TD [(v)27.8(eries)-379.6(go)-27.8(o)-27.7(d)-379.6(closure)]TJ -29.5243 -1.2 TD [(prop)-27.8(erties.)]TJ /F6 1 Tf 0 -2.1795 TD [(Lemma)-428.1(2.3.)]TJ /F13 1 Tf 6.5568 0 TD [(Assume)-372.2(that)]TJ /F3 1 Tf 5.95 0 TD (S)Tj /F13 1 Tf 1.0431 0 TD [(is)-372.3(the)-372.3(set)-372.2(of)-372.3(all)-372.2(p)-27.8(ositiv)27.7(e)-372.2(decomp)-27.9(ositions)-372.2(of)-372.3(some)-372.3(p)-27.8(ositiv)27.8(e)]TJ -13.5498 -1.2 TD [(braid.)-534.5(Then)-363.4(the)-363.4(set)-363.4(of)-363.4(the)-363.4(w)27.8(ords)-363.4(traced)-363.4(in)]TJ /F3 1 Tf 19.1831 0 TD (S)Tj /F13 1 Tf 1.0343 0 TD [(from)-363.4(some)-363.4(giv)27.8(en)-363.3(p)-27.9(oin)27.8(t)-363.4(i)0(s)-363.4(closed)-363.5(under)]TJ /F3 1 Tf 17.0935 0 TD (R)Tj /F13 1 Tf 0.767 0 TD (-)Tj -38.0778 -1.2 TD (and)Tj /F3 1 Tf 1.9444 0 TD (L)Tj /F13 1 Tf 0.6806 0 TD [(-reductions,)-333.5(and)-333.3(under)]TJ /F3 1 Tf 10.2333 0 TD (P)Tj /F13 1 Tf 0.7809 0 TD [(-)-333.3(and)]TJ /F3 1 Tf 2.6112 0 TD (N)Tj /F13 1 Tf 0.9124 0 TD [(-equiv)55.5(alence.)]TJ /F5 1 Tf -17.1628 -2.1795 TD [(Pr)51.1(o)51.1(of.)]TJ /F1 1 Tf 3.4613 0 TD [(Consider)-466.6(rst)-466.7(the)-466.6(case)-466.7(of)]TJ /F3 1 Tf 11.8832 0 TD (R)Tj /F1 1 Tf 0.767 0 TD [(-reduction)-466.8(\(Figure)-466.7(2.4\).)-844.3(Assume)-466.7(that)-466.6(some)-466.7(w)27.8(ord)]TJ /F3 1 Tf -16.1114 -1.2 TD 0.0359 Tc (v)Tj /F11 1 Tf 8.3686 0 0 8.3686 85.4844 422.3487 Tm 0 Tc ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0094 TD (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 98.3193 417.2338 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 105.1507 415.4406 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 110.1878 417.2338 Tm (v)Tj /F11 1 Tf 8.3686 0 0 8.3686 116.4124 421.5726 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 123.5078 417.2338 Tm [(is)-312.9(traced)-313(in)]TJ /F3 1 Tf 5.1694 0 TD (S)Tj /F1 1 Tf 0.9838 0 TD (from)Tj /F3 1 Tf 2.3435 0 TD (u)Tj /F1 1 Tf 0.5725 0 TD [(.)-437.6(W)83.4(e)-313(h)0(a)27.8(v)27.8(e)-313(to)-313(sho)27.7(w)-312.9(that)-312.9(the)-313(w)27.8(ord)]TJ /F3 1 Tf 14.6644 0 TD 0.0359 Tc (vf)Tj /F1 1 Tf 1.1178 0 TD 0 Tc (\()Tj /F3 1 Tf 0.3889 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 432.0916 415.4406 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 437.1288 417.2338 Tm 0.1666 Tc (,)Tj /F4 1 Tf 8.3686 0 0 8.3686 449.2733 415.4406 Tm 0 Tc (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 453.2533 417.2338 Tm (\))Tj /F3 1 Tf 0.3889 0 TD (f)Tj /F1 1 Tf 0.5972 0 TD (\()Tj /F3 1 Tf 0.3888 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 476.5231 415.4406 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 481.5612 417.2338 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 488.3926 415.4406 Tm (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 492.3726 417.2338 Tm (\))Tj /F11 1 Tf 8.3686 0 0 8.3686 497.0212 421.5726 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 509.8571 417.2338 Tm (v)Tj /F11 1 Tf 8.3686 0 0 8.3686 516.0807 421.5726 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 523.1761 417.2338 Tm (is)Tj -37.739 -1.2 TD [(also)-304.5(traced)-304.6(in)]TJ /F3 1 Tf 6.1442 0 TD (S)Tj /F1 1 Tf 0.9754 0 TD (from)Tj /F3 1 Tf 2.3351 0 TD (u)Tj /F1 1 Tf 0.5724 0 TD [(.)-434.9(N)0(o)27.7(w)-304.5(the)-304.5(h)27.8(y)0(p)-27.8(othesis)-304.6(means)-304.6(that)-304.5(there)-304.6(exist)-304.5(p)-27.9(ositiv)27.8(e)-304.6(braid)-304.6(w)27.7(ords)]TJ /F3 1 Tf -10.0271 -1.2 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 78.8433 392.8802 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 82.1978 388.5414 Tm (,)Tj /F3 1 Tf 0.5621 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 95.7619 392.8802 Tm ()Tj /F2 1 Tf 0 -0.873 TD (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 105.1248 388.5414 Tm (and)Tj /F3 1 Tf 1.8832 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 134.4827 392.8802 Tm ()Tj /F2 1 Tf 0 -0.873 TD (2)Tj /F1 1 Tf 11.9552 0 0 11.9552 143.8456 388.5414 Tm [(suc)27.8(h)-272(that)-272.1(b)-27.8(oth)]TJ /F3 1 Tf 6.5995 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 229.588 392.8802 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 232.9425 388.5414 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 239.7738 386.7481 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 243.7539 388.5414 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 250.5973 392.8802 Tm ()Tj /F2 1 Tf 0 -0.873 TD (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 259.9601 388.5414 Tm (and)Tj /F3 1 Tf 1.8833 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 289.3181 392.8802 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 292.6725 388.5414 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 299.5038 386.7481 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 304.541 388.5414 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 311.3843 392.8802 Tm ()Tj /F2 1 Tf 0 -0.873 TD (2)Tj /F1 1 Tf 11.9552 0 0 11.9552 320.7482 388.5414 Tm [(b)-27.8(elong)-272.1(to)]TJ /F3 1 Tf 4.2941 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD [(,)-284.3(and)]TJ /F3 1 Tf 2.4453 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 416.1823 392.8802 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 419.5367 388.5414 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 426.3681 386.7481 Tm (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 433.601 388.5414 Tm [(is)-272(equiv)55.5(alen)27.8(t)-272(t)0(o)]TJ /F3 1 Tf 6.794 0 TD (uv)Tj /F1 1 Tf 1.093 0 TD (.)Tj -38.1334 -1.2 TD [(By)-293.1(h)27.8(y)0(p)-27.8(othesis)-293.1(the)-293(p)-27.8(ositiv)27.7(e)-293.1(w)27.8(ords)]TJ /F3 1 Tf 14.5873 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 253.2384 378.5339 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 256.5918 374.1952 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 263.4232 372.402 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 267.4032 374.1952 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 274.2476 378.5339 Tm ()Tj /F2 1 Tf 0 -0.873 TD (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 283.8605 374.1952 Tm (and)Tj /F3 1 Tf 1.9042 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 313.4695 378.5339 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 316.8229 374.1952 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 323.6543 372.402 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 328.6914 374.1952 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 335.5357 378.5339 Tm ()Tj /F2 1 Tf 0 -0.873 TD (2)Tj /F1 1 Tf 11.9552 0 0 11.9552 345.1497 374.1952 Tm [(are)-293(equiv)55.5(alen)27.8(t,)-301.1(and)-293(so)-293.1(are)-293(the)-293(w)27.7(ords)]TJ /F3 1 Tf -22.8478 -1.2 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 78.8314 358.0557 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 82.8114 359.849 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 89.6558 364.1877 Tm ()Tj /F2 1 Tf 0 -0.873 TD (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 100.804 359.849 Tm (and)Tj /F3 1 Tf 2.0326 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 131.9352 358.0557 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 136.9724 359.849 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 143.8157 364.1877 Tm ()Tj /F2 1 Tf 0 -0.873 TD (2)Tj /F1 1 Tf 11.9552 0 0 11.9552 149.9268 359.849 Tm [(.)-708.7(B)0(y)-421.5([10])-421.5(\(or)-421.4([6]\))-421.4(w)27.8(e)-421.5(kno)27.7(w)-421.4(that)-421.5(this)-421.4(implies)-421.5(the)-421.4(existence)-421.5(of)-421.5(a)-421.4(p)-27.8(ositiv)27.7(e)]TJ -6.5182 -1.2 TD [(w)27.8(ord)]TJ /F3 1 Tf 2.475 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 108.4334 349.8415 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 118.5285 345.5028 Tm (satisfying)Tj /F3 1 Tf 4.9856 -1.9038 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 184.9753 327.6786 Tm ()Tj /F2 1 Tf 0 -0.9431 TD (1)Tj /F10 1 Tf 11.9552 0 0 11.9552 194.407 322.7422 Tm ()Tj /F3 1 Tf 1.0555 0 TD (f)Tj /F1 1 Tf 0.5973 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 225.6468 320.9489 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 230.6839 322.7422 Tm 0.1666 Tc (,)Tj /F4 1 Tf 8.3686 0 0 8.3686 242.8284 320.9489 Tm 0 Tc (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 246.8085 322.7422 Tm (\))Tj /F3 1 Tf 0.3889 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 258.3024 327.6786 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 288.3228 322.7422 Tm (and)Tj /F3 1 Tf 3.9444 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 342.3233 327.6786 Tm ()Tj /F2 1 Tf 0 -0.9431 TD (2)Tj /F10 1 Tf 11.9552 0 0 11.9552 351.7539 322.7422 Tm ()Tj /F3 1 Tf 1.0556 0 TD (f)Tj /F1 1 Tf 0.5972 0 TD (\()Tj /F3 1 Tf 0.3888 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 382.9938 320.9489 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 388.0319 322.7422 Tm 0.1666 Tc (,)Tj /F4 1 Tf 8.3686 0 0 8.3686 400.1764 320.9489 Tm 0 Tc (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 404.1564 322.7422 Tm (\))Tj /F3 1 Tf 0.3889 0 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 415.6493 327.6786 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 421.7604 322.7422 Tm (.)Tj /F1 1 Tf -29.256 -1.9038 TD [(This)-387.4(sho)27.7(ws)-387.4(that)-387.5(the)-387.5(w)27.8(ord)]TJ /F3 1 Tf 11.79 0 TD (f)Tj /F1 1 Tf 0.5972 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 231.5726 298.1883 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 236.6097 299.9815 Tm 0.1666 Tc (,)Tj /F4 1 Tf 8.3686 0 0 8.3686 248.7542 298.1883 Tm 0 Tc (i)Tj /F1 1 Tf 11.9552 0 0 11.9552 252.7343 299.9815 Tm (\))Tj /F3 1 Tf 0.3889 0 TD (f)Tj /F1 1 Tf 0.5972 0 TD (\()Tj /F3 1 Tf 0.3888 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 276.004 298.1883 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 279.985 299.9815 Tm 0.1666 Tc (,)Tj /F4 1 Tf 8.3686 0 0 8.3686 292.1295 298.1883 Tm 0 Tc (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 297.1666 299.9815 Tm (\))Tj /F11 1 Tf 8.3686 0 0 8.3686 301.8162 304.3192 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 319.2827 299.9815 Tm [(is)-387.4(traced)-387.5(in)]TJ /F3 1 Tf 5.3929 0 TD (S)Tj /F1 1 Tf 1.0583 0 TD (from)Tj /F3 1 Tf 2.418 0 TD (uv)Tj /F1 1 Tf 1.093 0 TD [(,)-401(whic)27.7(h)-387.4(i)0(s)-387.5(exactly)]TJ -30.6464 -1.2 TD [(what)-333.4(w)27.8(e)-333.4(need.)]TJ /F3 1 Tf 14.0841 -7.3588 TD (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 247.2219 201.9989 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 277.7953 214.6684 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 284.6266 212.8752 Tm (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 277.7953 173.8495 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 284.6266 172.0562 Tm (i)Tj /F3 1 Tf 11.9552 0 0 11.9552 311.8107 141.5337 Tm (v)Tj 0 8.2513 TD (v)Tj /F11 1 Tf 8.3686 0 0 8.3686 318.0354 244.5185 Tm ()Tj /F3 1 Tf 11.9552 0 0 11.9552 342.4249 233.3763 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 349.2692 237.715 Tm ()Tj /F2 1 Tf 0 -0.873 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 342.4249 151.7384 Tm (u)Tj /F11 1 Tf 8.3686 0 0 8.3686 349.2692 156.0772 Tm ()Tj /F2 1 Tf T* (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 366.2366 197.6602 Tm (u)Tj 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[(is)-242.2(the)-242.1(n)27.8(ullstring.)-414.2(So)]TJ -29.9712 -1.2 TD [(assume)-281.1(that)-281.1(the)-281(result)-281(is)-281(pro)27.7(v)27.8(e)0(d)-281(for)]TJ /F3 1 Tf 15.5533 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(,)-291.5(and)-281.1(consider)-281.1(the)-281(case)-281.1(of)-281(the)-281(w)27.8(ords)]TJ /F3 1 Tf 15.6143 0 TD 0.0269 Tc (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 469.6359 748.8124 Tm 0 Tc ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9441 -1.042 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 482.4717 743.6986 Tm [(.)-427(Assume)]TJ -34.3343 -1.2 TD [(rst)-339.6(that)]TJ /F3 1 Tf 4.2429 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 131.6055 733.6912 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 139.0177 729.3524 Tm (is)Tj /F3 1 Tf 1.0118 0 TD 0.0269 Tc (w)Tj /F4 1 Tf 8.3686 0 0 8.3686 166.8254 727.5591 Tm 0 Tc (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 172.7073 729.3524 Tm [(.)-463(B)0(y)-339.5(construction)]TJ /F3 1 Tf 8.0531 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 278.8812 727.5591 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 285.1906 729.3524 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 298.7208 733.6912 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 302.0742 729.3524 Tm 0.3395 Tc [(\)i)339.5(s)]TJ /F3 1 Tf 1.7402 0 TD 0 Tc (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 332.7761 727.5591 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 339.0854 729.3524 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\),)-341.1(and)]TJ /F3 1 Tf 2.9583 0 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 397.588 727.5591 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 404.3527 729.3524 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 417.8829 733.6912 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 421.2364 729.3524 Tm 0.3395 Tc [(\)i)339.5(s)]TJ /F3 1 Tf 1.7402 0 TD 0 Tc (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 451.6453 727.5591 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 458.41 729.3524 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7427 0 TD (\))Tj /F3 1 Tf 0.3889 0 TD (v)Tj /F1 1 Tf 0.8601 0 TD [(for)-339.5(some)]TJ -34.7023 -1.2 TD [(p)-27.8(ositiv)27.7(e)-330.6(w)27.7(ord)]TJ /F3 1 Tf 6.1696 0 TD (v)Tj /F1 1 Tf 0.5206 0 TD [(,)-331.1(namely)-330.7(the)-330.6(righ)27.7(t)-330.6(n)27.8(umerator)-330.7(of)]TJ /F3 1 Tf 14.2148 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 331.8207 713.213 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 338.5844 715.0062 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (\))Tj /F11 1 Tf 8.3686 0 0 8.3686 356.7631 719.3449 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 369.599 715.0062 Tm ()Tj /F4 1 Tf 8.3686 0 0 8.3686 376.4304 713.213 Tm (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 386.2645 715.0062 Tm [(\(Figure)-330.7(2.6\).)-443.6(It)-330.6(follo)27.8(ws)-330.7(that)]TJ /F3 1 Tf -26.2869 -1.2 TD (uv)Tj /F1 1 Tf 1.4012 0 TD [(b)-27.8(elongs)-308.2(to)]TJ /F3 1 Tf 4.7608 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 167.2189 704.9988 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 170.5734 700.66 Tm [(\))-308.2(for)-308.2(ev)27.8(ery)-308.2(w)27.7(ord)]TJ /F3 1 Tf 7.2689 0 TD (u)Tj /F1 1 Tf 0.8807 0 TD (in)Tj /F3 1 Tf 1.1415 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\),)-313.2(and,)-313.2(therefore,)-313.3(ev)27.7(ery)-308.2(w)27.8(ord)-308.2(traced)-308.3(in)]TJ /F3 1 Tf 16.8808 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (\))Tj -38.0223 -1.2 TD (from)Tj /F3 1 Tf 2.4087 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 110.6949 684.5205 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 117.0052 686.3138 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\))-378.3(i)0(s)-378.2(also)-378.3(traced)-378.2(in)]TJ /F3 1 Tf 8.1828 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 249.9118 690.6525 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 253.2653 686.3138 Tm [(\))-378.3(from)]TJ /F3 1 Tf 3.1759 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 301.1317 684.5205 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 307.4421 686.3138 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\).)-579.1(This)-378.2(applies)-378.3(in)-378.3(particular)-378.2(to)]TJ /F3 1 Tf 14.0981 0 TD (w)Tj /F1 1 Tf 1.121 0 TD (itself.)Tj -36.0445 -1.2 TD [(So)-434.5(it)-434.4(only)-434.6(remains)-434.5(to)-434.5(consider)-434.5(the)-434.5(case)-434.5(of)-434.5(the)-434.5(nal)-434.5(letter)]TJ /F3 1 Tf 26.2388 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 392.5211 670.1743 Tm (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 403.5975 671.9676 Tm (of)Tj /F3 1 Tf 1.2401 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 427.3036 676.3063 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 430.657 671.9676 Tm 0.748 Tc [(.N)748(o)775.8(w)]TJ /F3 1 Tf 3.4047 0 TD 0 Tc (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 481.2583 670.1743 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 487.5677 671.9676 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 501.0979 676.3063 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 504.4513 671.9676 Tm (\))Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 1.1773 0 TD (is)Tj -37.739 -1.2 TD [(equiv)55.6(alen)27.8(t)-333.4(b)27.8(y)-333.4(construction)-333.4(to)]TJ /F3 1 Tf 13.0917 0 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 238.1191 655.8281 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 244.4284 657.6214 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\),)-333.3(and)-333.3(the)-333.4(equiv)55.6(alence)]TJ /F3 1 Tf -4.4275 -1.7954 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 214.6331 634.3636 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 220.9425 636.1569 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (\))Tj /F3 1 Tf 0.7222 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 249.9377 634.3636 Tm (k)Tj /F3 1 Tf 11.9552 0 0 11.9552 259.8047 636.1569 Tm (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 269.7026 634.3636 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 276.4672 636.1569 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (\))Tj /F10 1 Tf 1 0 TD ()Tj /F3 1 Tf 1.3888 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 328.4543 634.3636 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 334.7646 636.1569 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 348.2939 641.0934 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 351.6483 636.1569 Tm (\))Tj /F3 1 Tf 0.7222 0 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 369.8879 634.3636 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 376.6526 636.1569 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 390.1818 641.0934 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 393.5363 636.1569 Tm (\))Tj -26.8952 -1.7953 TD [(sho)27.7(ws)-475.6(that)-475.7(this)-475.7(nal)-475.6(letter)]TJ /F3 1 Tf 12.5921 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 229.3719 612.9001 Tm (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 240.9405 614.6934 Tm [(also)-475.6(satises)-475.6(the)-475.7(dening)-475.7(condition)-475.7(for)-475.6(a)-475.7(w)27.8(ord)-475.7(traced)]TJ -14.1312 -1.2 TD (in)Tj /F3 1 Tf 1.1667 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 107.4979 604.6859 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 110.8513 600.3472 Tm (\).)Tj -1.2497 -1.6657 TD [(Assume)-485.4(no)27.8(w)-485.4(that)]TJ /F3 1 Tf 8.4117 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 205.3539 584.7726 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 214.5096 580.4338 Tm (is)Tj /F3 1 Tf 1.1576 0 TD 0.0269 Tc (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 244.4901 585.5477 Tm 0 Tc ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.042 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 257.325 580.4338 Tm [(.)-900.6(There)-485.4(exists)-485.4(a)-485.3(p)-27.8(ositiv)27.7(e)-485.3(w)27.8(ord)]TJ /F3 1 Tf 14.5995 0 TD (v)Tj /F1 1 Tf 1.006 0 TD [(\(namely)-485.4(the)-485.4(left)]TJ -31.1072 -1.2 TD [(denominator)-436.4(of)]TJ /F3 1 Tf 7.1809 0 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 167.455 564.2944 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 173.7644 566.0876 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (\))Tj /F3 1 Tf 0.3888 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 199.204 571.2015 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 212.0388 566.0876 Tm [(\))-436.3(suc)27.8(h)-436.4(that)]TJ /F3 1 Tf 5.4533 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 287.1323 564.2944 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 293.4426 566.0876 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 306.9719 570.4264 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 310.3263 566.0876 Tm 0.4363 Tc [(\)i)436.3(s)]TJ /F3 1 Tf 1.9337 0 TD 0.0359 Tc (vD)Tj /F15 1 Tf 5.9776 0 0 5.9776 349.5651 564.2944 Tm 0 Tc (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 355.8755 566.0876 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\),)-462.1(while)]TJ /F3 1 Tf 3.8428 0 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 424.9514 564.2944 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 431.7161 566.0876 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 445.2453 570.4264 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 448.5998 566.0876 Tm 0.4363 Tc [(\)i)436.3(s)]TJ /F3 1 Tf 1.9337 0 TD 0 Tc (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 481.3231 564.2944 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 488.0877 566.0876 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\).)-753.3(So)]TJ -35.9357 -1.2 TD [(no)27.8(w)-340.4(the)-340.5(w)27.7(ord)]TJ /F3 1 Tf 6.3019 0 TD 0.0359 Tc (vu)Tj /F1 1 Tf 1.4335 0 TD 0 Tc [(b)-27.8(elongs)-340.5(to)]TJ /F3 1 Tf 4.8253 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 243.7151 556.0802 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 247.0695 551.7415 Tm [(\))-340.4(for)-340.5(ev)27.8(ery)-340.5(w)27.7(ord)]TJ /F3 1 Tf 7.3979 0 TD (u)Tj /F1 1 Tf 0.9128 0 TD (of)Tj /F3 1 Tf 1.146 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\).)-465.7(It)-340.5(follo)27.8(ws)-340.4(that)]TJ /F3 1 Tf 7.6871 0 TD (w)Tj /F1 1 Tf 1.0833 0 TD [(is)-340.5(traced)]TJ -34.6736 -1.2 TD (in)Tj /F3 1 Tf 1.1143 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 106.8712 541.7339 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 110.2256 537.3952 Tm [(\))-280.9(from)]TJ /F3 1 Tf 2.9814 0 TD 0.0359 Tc (vD)Tj /F15 1 Tf 5.9776 0 0 5.9776 161.9905 535.6019 Tm 0 Tc (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 168.3009 537.3952 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(\),)-291.5(whic)27.8(h)-281(i)0(s)]TJ /F3 1 Tf 4.7201 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 248.1574 535.6019 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 254.4668 537.3952 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 267.997 541.7339 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 271.3514 537.3952 Tm [(\).)-427.1(Again)-281(it)-281(remains)-281.1(to)-281(consider)-281.1(the)-281(nal)-281.1(letter)]TJ /F3 1 Tf 20.0554 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 518.3771 542.509 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.042 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 72 523.049 Tm (of)Tj /F3 1 Tf 1.1389 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 94.4966 527.3878 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 97.8501 523.049 Tm 0.4445 Tc [(.N)444.5(o)472.2(w)]TJ /F3 1 Tf 3 0 TD 0 Tc (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 143.6134 521.2557 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 149.9238 523.049 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 163.453 527.3878 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 166.8075 523.049 Tm (\))Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 180.3377 527.3878 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 187.6762 523.049 Tm [(is)-333.4(equiv)55.6(alen)27.8(t)-333.4(t)0(o)]TJ /F3 1 Tf 6.9778 0 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 280.7024 521.2557 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 287.0127 523.049 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 300.542 527.3878 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 303.8964 523.049 Tm [(\),)-333.3(and)-333.3(the)-333.4(equiv)55.5(alence)]TJ /F3 1 Tf -8.4103 -1.7953 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 212.9554 499.7922 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 219.2657 501.5855 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 232.795 506.521 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 236.1494 501.5855 Tm (\))Tj /F3 1 Tf 0.7223 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 251.6154 499.7922 Tm (k)Tj /F3 1 Tf 11.9552 0 0 11.9552 261.4825 501.5855 Tm (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 271.3803 499.7922 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 278.144 501.5855 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (\))Tj /F10 1 Tf 1 0 TD ()Tj /F3 1 Tf 1.3888 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 330.131 499.7922 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 336.4413 501.5855 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 349.9706 506.521 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 353.325 501.5855 Tm (\))Tj /F3 1 Tf 0.7223 0 TD (N)Tj /F15 1 Tf 5.9776 0 0 5.9776 371.5646 499.7922 Tm (R)Tj /F1 1 Tf 11.9552 0 0 11.9552 378.3293 501.5855 Tm (\()Tj /F3 1 Tf 0.3889 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 391.8595 506.521 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 395.213 501.5855 Tm (\))Tj -27.0354 -1.7954 TD [(giv)27.8(es)-333.4(the)-333.4(conclusion.)]TJ /F16 1 Tf 37.6334 0 TD ()Tj /F3 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416.527 307.3668 Tm (\))Tj -28.8183 -2.4831 TD [(Gathering)-333.4(the)-333.4(previous)-333.4(lemmas)-333.4(giv)27.8(es)-333.4(the)-333.4(main)-333.4(result)-333.3(of)-333.4(this)-333.4(section:)]TJ /F6 1 Tf 0 -2.1312 TD [(Prop)-32(osition)-451.1(2.6.)]TJ /F13 1 Tf 8.7886 0 TD (Let)Tj /F3 1 Tf 1.8506 0 TD (w)Tj /F13 1 Tf 1.1352 0 TD [(b)-27.8(e)-392.2(an)27.7(y)-392.2(braid)-392.3(w)27.7(ord.)-621.3(Then)-392.3(all)-392.3(w)27.8(ords)]TJ /F3 1 Tf 16.1278 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 414.4538 256.5404 Tm ()Tj /F13 1 Tf 11.9552 0 0 11.9552 422.4977 252.2017 Tm [(that)-392.4(can)-392.3(b)-27.8(e)-392.3(deduced)]TJ -29.3177 -1.2 TD (from)Tj /F3 1 Tf 2.3639 0 TD (w)Tj /F13 1 Tf 1.0762 0 TD [(using)-333.4(handle)-333.3(reduction)-333.4(are)-333.4(traced)-333.4(in)]TJ /F3 1 Tf 16.1806 0 TD (S)Tj /F1 1 Tf 0.6708 0 TD (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (\))Tj /F13 1 Tf 0.7222 0 TD (\(from)Tj /F3 1 Tf 2.7528 0 TD (D)Tj /F15 1 Tf 5.9776 0 0 5.9776 379.5597 236.0621 Tm (L)Tj /F1 1 Tf 11.9552 0 0 11.9552 385.87 237.8555 Tm (\()Tj /F3 1 Tf 0.3888 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD (\))Tj /F13 1 Tf 0.3889 0 TD (\).)Tj /F1 1 Tf -27.7745 -2.1313 TD [(This)-322.7(result)-322.7(giv)27.8(es)-322.6(the)-322.7(b)-27.8(oundedness)-322.7(prop)-27.8(ert)27.7(y)-322.6(w)27.8(e)-322.6(w)27.8(ere)-322.7(lo)-27.7(oking)-322.7(for:)-439.1(it)-322.7(means)-322.7(that)-322.7(all)-322.6(w)27.7(ords)]TJ 0 -1.2 TD [(that)-301.4(are)-301.4(deduced)-301.5(from)-301.5(a)-301.4(giv)27.8(en)-301.4(w)27.8(ord)-301.5(using)-301.4(reduction)-301.5(ha)27.8(v)27.8(e)-301.4(to)-301.5(remain)-301.4(traced)-301.5(in)-301.4(some)]TJ /F5 1 Tf 36.1879 0 TD (nite)Tj /F1 1 Tf -36.1879 -1.2 TD [(region)-346.2(of)-346.2(the)-346.1(Ca)27.8(yley)-346.2(graph)-346.3(of)]TJ /F3 1 Tf 13.2213 0 TD (B)Tj /F11 1 Tf 8.3686 0 0 8.3686 239.1313 181.8903 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 249.2603 183.6836 Tm 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[(ular)-274(w)27.7(e)-274(still)-274(ha)27.7(v)27.8(e)-274(no)-274(argumen)27.7(t)-274(t)0(o)-274(pro)27.8(v)27.8(e)-274.1(that)-274(cycles)-274(are)-274.1(imp)-27.8(ossible)-274.1(in)-274(reduction)-274.1(sequences.)]TJ T* [(The)-381.6(latter)-381.5(prop)-27.8(ert)27.8(y)-381.6(will)-381.5(follo)27.8(w)-381.5(from)-381.5(the)-381.5(existence)-381.5(of)-381.5(the)]TJ /F5 1 Tf 25.3317 0 TD [(or)51.1(der)51.1(e)51.2(d)]TJ /F1 1 Tf 3.5248 0 TD [(structure)-381.6(pro)27.7(vided)-381.5(b)27.8(y)]TJ -28.8565 -1.2 TD [(Corollary)-333.4(1.3.)]TJ 0 -2.5317 TD [(The)-340.1(main)-340(idea)-340.1(is)-340(as)-340.1(follo)27.8(ws.)-464.6(Let)-340(us)-340(in)27.7(tro)-27.8(duce,)-341.8(for)-340(an)27.8(y)-340.1(braid)-340.1(w)27.8(ord)]TJ /F3 1 Tf 28.9827 0 TD (w)Tj /F1 1 Tf 1.0829 0 TD [(of)-340(length)]TJ /F3 1 Tf 4.2078 0 TD ()Tj /F1 1 Tf 0.4167 0 TD 0.3416 Tc (,a)Tj /F5 1 Tf 1.4595 0 TD 0 Tc (char-)Tj -36.1496 -1.2 TD [(acteristic)-328.2(function)]TJ /F12 1 Tf 8.3835 0 TD ()Tj /F3 1 Tf -0.177 0 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()Tj /F14 1 Tf 5.9776 0 0 5.9776 206.7995 705.6404 Tm (1)Tj /F4 1 Tf 8.3686 0 0 8.3686 180.8508 693.2299 Tm ()Tj /F14 1 Tf 5.9776 0 0 5.9776 186.39 692.0343 Tm (1)Tj /F4 1 Tf 8.3686 0 0 8.3686 191.0565 693.2299 Tm ()Tj /F17 1 Tf 5.9776 0 0 5.9776 196.8966 697.1083 Tm ()Tj /F14 1 Tf 1.0833 0 TD (1)Tj -1.1337 -1.131 TD (2)Tj /F4 1 Tf 8.3686 0 0 8.3686 197.858 737.4501 Tm ()Tj /F14 1 Tf 5.9776 0 0 5.9776 203.6981 741.0585 Tm (2)Tj -0.0502 -1.0037 TD (1)Tj /F4 1 Tf 8.3686 0 0 8.3686 194.4568 652.4109 Tm ()Tj /F17 1 Tf 5.9776 0 0 5.9776 200.2969 656.2894 Tm ()Tj /F14 1 Tf 1.0833 0 TD (1)Tj -1.1335 -1.131 TD (1)Tj /F4 1 Tf 8.3686 0 0 8.3686 163.8426 635.4027 Tm ()Tj /F14 1 Tf 5.9776 0 0 5.9776 169.3828 634.2072 Tm (1)Tj /F4 1 Tf 8.3686 0 0 8.3686 174.0483 635.4027 Tm ()Tj /F17 1 Tf 5.9776 0 0 5.9776 179.8884 639.2822 Tm ()Tj /F14 1 Tf 1.0833 0 TD (1)Tj -1.1335 -1.1312 TD (2)Tj /F4 1 Tf 8.3686 0 0 8.3686 191.0306 635.4027 Tm ()Tj /F17 1 Tf 5.9776 0 0 5.9776 196.8707 639.2822 Tm ()Tj /F14 1 Tf 1.0833 0 TD (1)Tj -1.1337 -1.1312 TD (1)Tj /F4 1 Tf 8.3686 0 0 8.3686 194.4568 621.7967 Tm ()Tj /F17 1 Tf 5.9776 0 0 5.9776 200.2969 625.6752 Tm ()Tj /F14 1 Tf 1.0833 0 TD (2)Tj -1.1335 -1.131 TD (1)Tj ET 194.49 725.908 11.582 -0.478 re f BT /F4 1 Tf 8.3686 0 0 8.3686 194.4568 720.4428 Tm (w)Tj /F14 1 Tf 5.9776 0 0 5.9776 201.3729 719.2473 Tm (0)Tj /F6 1 Tf 11.9552 0 0 11.9552 140.0319 594.5838 Tm [(Figure)-383.3(3.1)]TJ /F1 1 Tf 5.1103 0 TD [(:)-444.5(The)-333.4(c)27.8(haracteristic)-333.4(function)-333.4(of)]TJ /F3 1 Tf 13.9834 0 TD (w)Tj /F2 1 Tf 8.3686 0 0 8.3686 376.8598 592.7905 Tm (0)Tj /F1 1 Tf 11.9552 0 0 11.9552 385.5432 594.5838 Tm (=)Tj /F3 1 Tf 1.0556 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 404.9943 592.7905 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 410.3572 594.5838 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 417.6179 599.6986 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 430.4528 594.5838 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 437.7126 599.6986 Tm ()Tj /F2 1 Tf 0.893 0 TD (1)Tj -0.9442 -0.992 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 450.5485 594.5838 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 457.3799 592.7905 Tm (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 462.7427 594.5838 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 469.5741 592.7905 Tm (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 474.937 594.5838 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 481.7684 592.7905 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 72 559.4724 Tm [(in)-303.7(braid)-303.7(w)27.8(ords:)-429.7(w)27.8(e)-303.6(sa)27.8(y)-303.7(that)]TJ /F3 1 Tf 12.2424 0 TD (x)Tj /F1 1 Tf 0.8752 0 TD [(is)-303.7(a)-303.7(p)-27.8(osition)-303.7(of)-303.7(the)-303.6(letter)]TJ /F3 1 Tf 11.0026 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 367.6204 564.5863 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.042 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 384.0867 559.4724 Tm (in)Tj /F3 1 Tf 1.137 0 TD (w)Tj /F1 1 Tf 1.0464 0 TD [(just)-303.7(to)-303.6(express)-303.8(that)-303.7(the)]TJ /F3 1 Tf -28.2882 -1.2 TD (x)Tj /F1 1 Tf 0.5715 0 TD [(-th)-333.4(letter)-333.3(of)]TJ /F3 1 Tf 5.4195 0 TD (w)Tj /F1 1 Tf 1.0762 0 TD [(\(starting)-333.4(from)-333.4(the)-333.4(left\))-333.3(is)-333.4(that)-333.4(letter.)]TJ /F6 1 Tf -7.0672 -2.2143 TD (Denition.)Tj /F1 1 Tf 5.9681 0 TD [(The)-377.1(in)27.8(teger)]TJ /F3 1 Tf 5.4513 0 TD (x)Tj /F1 1 Tf 0.9486 0 TD [(is)-377(a)]TJ /F5 1 Tf 1.9263 0 TD [(critic)51.1(al)]TJ /F1 1 Tf 3.3797 0 TD [(p)-27.8(osition)-377.1(of)-377(the)-377(letter)]TJ /F3 1 Tf 9.5164 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 404.3258 522.9922 Tm (e)Tj -0.0513 -0.923 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 414.2864 518.6535 Tm (in)Tj /F3 1 Tf 1.2103 0 TD (w)Tj /F1 1 Tf 1.1198 0 TD (\()Tj /F3 1 Tf 0.3889 0 TD (e)Tj /F1 1 Tf 0.8162 0 TD (=)Tj /F10 1 Tf 1.1284 0 TD ()Tj /F1 1 Tf 0.7778 0 TD [(1\))-377(if)]TJ /F3 1 Tf 2.2262 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 513.2155 522.9922 Tm (e)Tj -0.0513 -0.923 TD (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 523.1761 518.6535 Tm (is)Tj -37.739 -1.2 TD (the)Tj /F3 1 Tf 1.7031 0 TD (x)Tj /F1 1 Tf 0.5715 0 TD [(-th)-314.2(letter)-314.3(of)]TJ /F3 1 Tf 5.3621 0 TD (w)Tj /F1 1 Tf 1.057 0 TD [(and,)-318.1(in)-314.2(addition,)-318.1(some)]TJ /F3 1 Tf 10.0478 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 302.8882 502.514 Tm (k)Tj /F1 1 Tf 11.9552 0 0 11.9552 308.7701 504.3073 Tm [(-handle)-314.2(of)]TJ /F3 1 Tf 4.6562 0 TD (w)Tj /F1 1 Tf 1.057 0 TD [(b)-27.8(egins)-314.2(at)-314.2(this)-314.3(p)-27.8(osition)-314.2(\(w)27.8(e)-314.2(d)0(o)]TJ /F5 1 Tf -25.518 -1.2 TD (not)Tj /F1 1 Tf 1.7389 0 TD [(require)-333.4(that)-333.4(this)-333.4(handle)-333.3(b)-27.8(e)-333.4(p)-27.8(ermitted\).)]TJ /F6 1 Tf -1.7389 -2.2143 TD [(Example)-492.8(0.4.)]TJ /F1 1 Tf 7.4863 0 TD [(In)-428.5(the)-428.5(w)27.8(ord)]TJ /F3 1 Tf 5.7327 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 236.8677 461.6951 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 242.2307 463.4883 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 249.4904 467.8271 Tm (2)Tj -0.0512 -0.873 TD (2)Tj /F3 1 Tf 11.9552 0 0 11.9552 254.8543 463.4883 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 262.1141 468.6032 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.944 -0.992 TD (3)Tj /F3 1 Tf 11.9552 0 0 11.9552 274.95 463.4883 Tm ()Tj /F11 1 Tf 8.3686 0 0 8.3686 282.2097 468.6032 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.944 -0.992 TD (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 295.0446 463.4883 Tm ()Tj /F2 1 Tf 8.3686 0 0 8.3686 301.876 461.6951 Tm (3)Tj /F1 1 Tf 11.9552 0 0 11.9552 312.3626 463.4883 Tm [(there)-428.5(are)-428.5(t)27.8(w)27.7(o)-428.5(handles,)-452.4(and)-428.5(therefore)-428.6(t)27.8(w)27.7(o)]TJ -20.1053 -1.2 TD [(critical)-333.4(p)-27.8(ositions:)-444.6(1)-333.3(i)0(s)-333.4(a)-333.3(critical)-333.4(p)-27.8(osition)-333.3(of)]TJ /F3 1 Tf 18.8889 0 TD ()Tj /F2 1 Tf 8.3686 0 0 8.3686 304.6516 447.3489 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 310.0144 449.1422 Tm [(,)-333.4(and)-333.3(4)-333.3(i)0(s)-333.4(a)-333.3(critical)-333.4(p)-27.8(osition)-333.4(of)]TJ /F3 1 Tf 13.5139 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 478.8364 454.257 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj -0.9442 -0.992 TD (3)Tj /F1 1 Tf 11.9552 0 0 11.9552 491.6712 449.1422 Tm (.)Tj -35.1038 -2.2142 TD [(W)83.4(e)-477.9(x)-477.8(no)27.8(w)-477.9(for)-477.8(a)-477.8(while)-477.9(a)-477.8(pair)-477.9(of)-477.9(w)27.8(ords)-477.9(\()]TJ /F3 1 Tf 18.9312 0 TD 0.0268 Tc [(w,)-139.9(w)]TJ /F11 1 Tf 8.3686 0 0 8.3686 321.3997 427.0081 Tm 0 Tc ()Tj /F1 1 Tf 11.9552 0 0 11.9552 324.7532 422.6705 Tm [(\))-477.9(suc)27.8(h)-477.9(that)]TJ /F3 1 Tf 5.578 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 400.3198 427.0081 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 409.3858 422.6705 Tm [(is)-477.9(obtained)-477.9(from)]TJ /F3 1 Tf 7.9141 0 TD (w)Tj /F1 1 Tf 1.2207 0 TD -0.0278 Tc (by)Tj -37.3557 -1.2 TD 0 Tc [(reducing)-384.7(exactly)-384.7(one)]TJ /F3 1 Tf 9.4901 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 192.2869 406.5309 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 197.324 408.3242 Tm [(-handle.)-598.5(Our)-384.7(aim)-384.7(is)-384.6(to)-384.7(establish)-384.7(a)-384.7(corresp)-27.8(ondence)-384.7(b)-27.8(e)0(t)27.7(w)27.8(een)-384.7(the)]TJ -10.4828 -1.2 TD [(critical)-409.9(p)-27.8(ositions)-409.9(in)]TJ /F3 1 Tf 8.9379 0 TD (w)Tj /F1 1 Tf 1.1527 0 TD [(and)-409.9(the)-409.9(critical)-409.9(p)-27.8(ositions)-409.9(in)]TJ /F3 1 Tf 12.7577 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 354.0354 398.3158 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 357.3898 393.978 Tm [(.)-674(W)83.3(e)-409.8(assume)-409.9(that)-409.9(the)]TJ /F3 1 Tf 10.3246 0 TD ()Tj /F4 1 Tf 8.3686 0 0 8.3686 487.6533 392.1848 Tm (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 492.6904 393.978 Tm (-handle)Tj -35.189 -1.2 TD [(that)-297.9(is)-297.8(reduced)-297.8(from)]TJ /F3 1 Tf 9.119 0 TD (w)Tj /F1 1 Tf 1.0406 0 TD (to)Tj /F3 1 Tf 1.1867 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 216.527 383.9696 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 223.4411 379.6318 Tm [(corresp)-27.8(onds)-297.9(to)-297.8(the)-297.8(p)-27.8(ositions)]TJ /F3 1 Tf 12.4967 0 TD (p)Tj /F1 1 Tf 0.8009 0 TD (and)Tj /F3 1 Tf 1.9089 0 TD (q)Tj /F1 1 Tf 0.7801 0 TD (in)Tj /F3 1 Tf 1.1312 0 TD (w)Tj /F1 1 Tf 0.7428 0 TD [(,)-304.9(and,)-304.9(in)-297.8(this)-297.8(case,)]TJ -30.5281 -1.2 TD [(w)27.8(e)-364.8(sa)27.8(y)-364.8(that)]TJ /F3 1 Tf 5.4611 0 TD (p)Tj /F1 1 Tf 0.8679 0 TD [(is)-364.8(the)]TJ /F5 1 Tf 2.7907 0 TD (active)Tj /F1 1 Tf 2.8948 0 TD [(critical)-364.8(p)-27.8(osition)-364.8(in)-364.8(the)-364.8(reduction)-364.9(of)]TJ /F3 1 Tf 15.811 0 TD (w)Tj /F1 1 Tf 1.1076 0 TD (to)Tj /F3 1 Tf 1.2537 0 TD (w)Tj /F11 1 Tf 8.3686 0 0 8.3686 441.7684 369.6234 Tm ()Tj /F1 1 Tf 11.9552 0 0 11.9552 445.1228 365.2856 Tm [(.)-538.8(W)83.3(e)-364.8(write)]TJ /F3 1 Tf 5.1601 0 TD (r)Tj /F1 1 Tf 0.8438 0 TD (for)Tj -37.214 -1.2 TD [(the)-293.7(p)-27.8(osition)-293.8(of)]TJ /F3 1 Tf 6.5534 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 157.607 356.0533 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.944 -1.0093 TD (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 173.9537 350.9394 Tm (in)Tj /F3 1 Tf 1.1271 0 TD (w)Tj /F1 1 Tf 1.0366 0 TD [(that)-293.8(immediately)-293.8(precedes)]TJ /F3 1 Tf 11.7506 0 TD (p)Tj /F1 1 Tf 0.7969 0 TD [(\(if)-293.7(it)-293.8(exists\),)-301.7(and)]TJ /F3 1 Tf 7.5272 0 TD (s)Tj /F2 1 Tf 8.3686 0 0 8.3686 445.4217 349.1461 Tm (1)Tj /F1 1 Tf 11.9552 0 0 11.9552 450.7845 350.9394 Tm (,)Tj /F3 1 Tf 0.5794 0 TD 0.1667 Tc (...)Tj /F1 1 Tf 1.1667 0 TD 0 Tc (,)Tj /F3 1 Tf 0.5794 0 TD (s)Tj /F4 1 Tf 8.3686 0 0 8.3686 484.1903 349.1461 Tm (m)Tj /F1 1 Tf 11.9552 0 0 11.9552 496.7841 350.9394 Tm [(for)-293.7(the)]TJ -35.5314 -1.3937 TD [(p)-27.8(ositions)-333.4(of)]TJ /F3 1 Tf 5.3444 0 TD ()Tj /F11 1 Tf 8.3686 0 0 8.3686 143.1542 339.3907 Tm ()Tj /F2 1 Tf 0.8929 0 TD (1)Tj /F4 1 Tf -0.9442 -1.0093 TD (j)Tj /F2 1 Tf 0.5306 0 TD -0.0001 Tc (+1)Tj /F1 1 Tf 11.9552 0 0 11.9552 163.8525 334.2769 Tm 0 Tc [(b)-27.8(e)0(t)27.8(w)27.7(een)]TJ /F3 1 Tf 3.8611 0 TD (p)Tj /F1 1 Tf 0.8365 0 TD (and)Tj /F3 1 Tf 1.9444 0 TD (q)Tj /F1 1 Tf 0.8156 0 TD (in)Tj /F3 1 Tf 1.1667 0 TD (w)Tj /F1 1 Tf 1.0761 0 TD [(\(if)-333.4(they)-333.4(exists\).)-444.5(So)]TJ /F3 1 Tf 8.4834 0 TD (w)Tj /F1 1 Tf 1.0762 0 TD [(can)-333.3(b)-27.8(e)-333.3(written)-333.4(as)]TJ /F3 1 Tf -21.4578 -4.1277 TD (w)Tj /F1 1 Tf 1.354 0 TD 0.6111 Tc (=\()Tj /F3 1 Tf 1.7778 0 TD 0 Tc (w)Tj /F2 1 Tf 8.3686 0 0 8.3686 183.5766 283.1367 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 192.9245 313.6224 Tm (r)Tj /F10 1 Tf 0 -1.2 TD ()Tj /F3 1 Tf T* ()Tj /F4 1 Tf 8.3686 0 0 8.3686 200.1853 289.2687 Tm (b)Tj -0.0513 -0.8902 TD (j)Tj /F1 1 Tf 11.9552 0 0 11.9552 206.5315 284.93 Tm (\))Tj /F3 1 Tf 0.7222 0 TD (u)Tj /F2 1 Tf 8.3686 0 0 8.3686 222.0095 283.1367 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 231.3574 313.6224 Tm (p)Tj /F10 1 Tf 0 -1.2 TD ()Tj /F3 1 Tf T* ()Tj /F4 1 Tf 8.3686 0 0 8.3686 238.6182 289.2687 Tm (e)Tj -0.0513 -0.8902 TD (j)Tj /F3 1 Tf 11.9552 0 0 11.9552 248.9485 284.93 Tm (v)Tj /F2 1 Tf 8.3686 0 0 8.3686 254.7437 283.1367 Tm (1)Tj /F3 1 Tf 11.9552 0 0 11.9552 264.0916 313.6224 Tm 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[(is)-320(critical)-320.1(means)-320.1(that)-320.1(there)-320(exists)]TJ
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[(y>)35.8(x)]TJ
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[(o)-27.8(ccurs)-374(in)]TJ
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[(b)-27.8(e)0(t)27.7(w)27.8(een)]TJ
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[(.)-566.3(I)0(t)-373.9(follo)27.8(ws)-374(that)]TJ
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[(o)-27.8(ccurs)-262.9(in)]TJ
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[(b)-27.8(e)0(t)27.8(w)27.7(een)]TJ
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[(+)-135.8(2)-290.1(since,)-298.8(in)-290.1(this)-290.2(case,)]TJ
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[(inducess)-290.3(an)-290.1(order-preserving)-290.2(bijection)-290.2(b)-27.8(e)0(t)27.8(w)27.7(een)-290.1(the)-290.1(p)-27.8(ositions)-290.2(of)]TJ
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[(the)-324.9(letters)]TJ
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0.4416 Tc
[(.I)441.6(f)]TJ
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[(+)-205.2(1)0(,)-326.6(and)-324.8(some)-324.9(letters)]TJ
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(d)Tj
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()Tj
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-13.928 -1.3938 TD
[(some)-302(letter)]TJ
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[(b)-27.8(e)0(t)27.7(w)27.8(een)-301.9(t)27.8(w)27.7(o)-301.9(letters)]TJ
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[(.)-434(Hence)]TJ
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[(\))-301.9(i)0(s)-302(a)-301.9(critical)-301.9(p)-27.8(osition)-301.9(of)]TJ
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()Tj
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[(is)-342(a)-342.1(critical)-342.1(p)-27.8(osition)-342.2(of)]TJ
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(k)Tj
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[(,)-344.2(then)]TJ
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[(necessarily)-299.3(there)-299.3(exists)]TJ
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(is)Tj
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[(is)-299.3(a)-299.3(critical)-299.3(p)-27.8(osition)-299.3(of)]TJ
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[(.)-433.1(So,)]TJ
-36.367 -1.2 TD
[(if)-296.7(w)27.8(e)-296.7(dene,)-304(for)-296.6(suc)27.8(h)-296.7(p)-27.8(ositions)]TJ
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[(\)t)296.7(ob)268.9(e)]TJ
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[(\),)-304(then)-296.7(the)-296.7(critical)-296.7(p)-27.8(ositions)-296.7(of)]TJ
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()Tj
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(in)Tj
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()Tj
/F1 1 Tf
11.9552 0 0 11.9552 100.6326 435.3987 Tm
[(are)-269.2(exactly)-269.2(the)-269.2(heirs)-269.1(\()]TJ
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(i.e.)Tj
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[(,)-282(the)-269.2(images)-269.2(under)]TJ
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[(\))-269.2(o)0(f)-269.1(the)-269.2(similar)-269.2(critical)-269.3(p)-27.8(ositions)-269.2(in)]TJ
/F3 1 Tf
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-38.1334 -2.0968 TD
[(W)83.4(e)-314.6(consider)-314.8(no)27.8(w)-314.7(the)-314.6(critical)-314.7(p)-27.8(ositions)-314.7(of)]TJ
/F3 1 Tf
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8.3686 0 0 8.3686 294.1509 415.4455 Tm
()Tj
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(j)Tj
/F1 1 Tf
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0.4382 Tc
[(.I)438.2(f)]TJ
/F3 1 Tf
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[(is)-314.7(suc)27.8(h)-314.6(a)-314.7(p)-27.8(osition)-314.7(in)]TJ
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[(,)-318.3(and)]TJ
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[(is)-314.6(none)]TJ
-35.3688 -1.2 TD
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(q)Tj
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(r)Tj
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[(,)-352.1(then)]TJ
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(h)Tj
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(x)Tj
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[(\))-348.4(i)0(s)-348.4(again)-348.4(a)-348.3(critical)-348.4(p)-27.8(osition)-348.4(of)-348.4(the)-348.4(same)-348.4(letter)-348.4(in)]TJ
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(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 440.4991 400.3233 Tm
()Tj
/F1 1 Tf
11.9552 0 0 11.9552 443.8535 395.9845 Tm
[(,)-352.1(for)-348.4(nothing)-348.5(has)]TJ
-31.104 -1.2 TD
[(b)-27.8(een)-415.1(c)27.8(hanged)-415.1(b)-27.8(e)0(t)27.8(w)27.7(een)]TJ
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(x)Tj
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[(and)-415.1(the)-415(next)-415.1(p)-27.8(osition)-415.1(of)]TJ
/F3 1 Tf
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/F11 1 Tf
8.3686 0 0 8.3686 349.331 386.7531 Tm
()Tj
/F2 1 Tf
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(1)Tj
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-0.9442 -1.0094 TD
(j)Tj
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11.9552 0 0 11.9552 362.1659 381.6383 Tm
[(.)-689.6(F)83.4(or)-415.1(suc)27.8(h)-415(p)-27.8(ositions)]TJ
/F3 1 Tf
9.468 0 TD
(x)Tj
/F1 1 Tf
0.5716 0 TD
[(,)-435.4(w)27.7(e)-415.1(tak)27.8(e)]TJ
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(H)Tj
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(x)Tj
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(\)=)Tj
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(h)Tj
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(x)Tj
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0 Tc
(v)Tj
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(2)Tj
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(u)Tj
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(2)Tj
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11.9552 0 0 11.9552 210.412 679.5781 Tm
[(.)-551.6(The)-369.1(example)-369.1(of)-369.1(the)-369(pair)-369.1(\()]TJ
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()Tj
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(1)Tj
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(2)Tj
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()Tj
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8.3686 0 0 8.3686 388.9743 684.6918 Tm
()Tj
/F2 1 Tf
0.8929 0 TD
(1)Tj
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(1)Tj
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(1)Tj
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(2)Tj
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0.1666 Tc
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()Tj
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(1)Tj
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()Tj
/F2 1 Tf
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(1)Tj
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(2)Tj
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()Tj
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(1)Tj
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(2)Tj
/F1 1 Tf
11.9552 0 0 11.9552 491.7988 679.5781 Tm
[(\))-369(sho)27.7(ws)]TJ
-35.1144 -1.2 TD
[(that)-361.3(there)-361.3(is)-361.2(no)-361.3(hop)-27.7(e)-361.2(t)0(o)-361.3(obtain)-361.3(a)-361.2(surjectiv)27.7(e)-361.3(mapping)-361.3(of)-361.2(the)]TJ
/F3 1 Tf
26.4904 0 TD
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 395.5288 663.4386 Tm
(j)Tj
/F2 1 Tf
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-0.0001 Tc
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11.9552 0 0 11.9552 412.6705 665.2319 Tm
0 Tc
[(-handles)-361.3(of)]TJ
/F3 1 Tf
5.1447 0 TD
(w)Tj
/F1 1 Tf
1.1041 0 TD
[(on)27.8(to)-361.3(the)]TJ
/F3 1 Tf
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()Tj
/F4 1 Tf
8.3686 0 0 8.3686 78.8314 649.0923 Tm
(j)Tj
/F2 1 Tf
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(+1)Tj
/F1 1 Tf
11.9552 0 0 11.9552 95.9731 650.8856 Tm
0 Tc
[(-handles)-422.2(of)]TJ
/F3 1 Tf
5.2666 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 167.8167 655.2234 Tm
()Tj
/F1 1 Tf
11.9552 0 0 11.9552 171.1711 650.8856 Tm
[(,)-444.4(s)0(o)-422.1(w)27.8(e)-422.2(shall)-422.1(not)-422.2(try)-422.2(to)-422.1(extend)-422.3(further)-422.2(the)-422.1(denition)-422.2(of)-422.2(the)-422.1(heiring)]TJ
-8.2952 -1.2 TD
(function)Tj
/F3 1 Tf
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(H)Tj
/F1 1 Tf
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[(.)-444.5(H)0(o)27.8(w)27.7(ev)27.8(er)-333.3(w)27.7(e)-333.3(can)-333.3(summarize)-333.4(the)-333.4(previous)-333.4(easy)-333.4(analysis)-333.4(as)]TJ
/F6 1 Tf
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[(Lemma)-497.9(3.1.)]TJ
/F13 1 Tf
6.8357 0 TD
[(Assume)-432.9(that)]TJ
/F3 1 Tf
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(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 235.187 611.5571 Tm
()Tj
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11.9552 0 0 11.9552 243.7161 607.2184 Tm
[(is)-432.9(obtained)-433(from)]TJ
/F3 1 Tf
7.7792 0 TD
(w)Tj
/F13 1 Tf
1.1758 0 TD
[(b)27.8(y)-432.9(reducing)-433(one)]TJ
/F3 1 Tf
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()Tj
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8.3686 0 0 8.3686 448.2162 605.4251 Tm
(j)Tj
/F13 1 Tf
11.9552 0 0 11.9552 453.2543 607.2184 Tm
[(-handle.)-743.1(Then)]TJ
-31.8903 -1.2 TD
[(the)-424.1(critical)-424.1(p)-27.8(ositions)-424.2(in)]TJ
/F3 1 Tf
10.7936 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 209.9198 597.2109 Tm
()Tj
/F13 1 Tf
11.9552 0 0 11.9552 218.3432 592.8722 Tm
[(are)-424.1(the)-424.1(heirs)-424.1(of)-424(critical)-424.2(p)-27.8(ositions)-424.1(in)]TJ
/F3 1 Tf
16.2713 0 TD
(w)Tj
/F13 1 Tf
0.7428 0 TD
[(,)-446.8(augmen)27.8(ted)-424.1(with)-424.1(at)]TJ
-29.2552 -1.2 TD
(most)Tj
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(m)Tj
/F1 1 Tf
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0.2262 Tc
(+1)Tj
/F13 1 Tf
1.8433 0 TD
0 Tc
[(new)-339.4(critical)-339.4(p)-27.8(ositions)-339.4(of)]TJ
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10.7602 0 TD
()Tj
/F11 1 Tf
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()Tj
/F2 1 Tf
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(1)Tj
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(j)Tj
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(+1)Tj
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0 Tc
[(,)-340.8(where)]TJ
/F3 1 Tf
3.5163 0 TD
(m)Tj
/F13 1 Tf
1.2174 0 TD
[(is)-339.4(the)-339.3(n)27.7(u)0(m)27.8(b)-27.8(er)-339.4(of)-339.3(letters)]TJ
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()Tj
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(1)Tj
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(j)Tj
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0 Tc
[(in)-339.3(the)]TJ
/F3 1 Tf
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()Tj
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8.3686 0 0 8.3686 78.8314 562.3865 Tm
(j)Tj
/F13 1 Tf
11.9552 0 0 11.9552 83.8685 564.1798 Tm
[(-handle)-333.4(that)-333.4(is)-333.4(reduced)-333.4(from)]TJ
/F3 1 Tf
12.8167 0 TD
(w)Tj
/F13 1 Tf
1.0762 0 TD
(to)Tj
/F3 1 Tf
1.2222 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 273.4525 568.5186 Tm
()Tj
/F13 1 Tf
11.9552 0 0 11.9552 276.807 564.1798 Tm
(.)Tj
/F1 1 Tf
-17.1313 -2.4526 TD
[(W)83.4(e)-297.8(arriv)27.8(e)-297.8(t)0(o)-297.7(the)-297.8(core)-297.8(of)-297.8(the)-297.8(argumen)27.7(t.)-432.6(The)-297.8(fact)-297.8(that)-297.8(a)-297.8(p)-27.8(osition)-297.8(is)-297.7(activ)27.7(e)-297.7(i)0(n)-297.8(a)-297.7(sequence)-297.9(of)]TJ
0 -1.2 TD
[(handle)-363.6(reductions)-363.7(do)-27.8(es)-363.5(not)-363.6(forbid)-363.6(its)-363.6(heirs)-363.6(to)-363.5(b)-27.8(e)-363.6(still)-363.6(activ)27.8(e)-363.6(subsequen)27.7(tly)83.4(.)-535.1(But)-363.6(w)27.8(e)-363.6(shall)]TJ
T*
[(asso)-27.8(ciate)-414.1(with)-414(ev)27.8(ery)-414(critical)-414.1(p)-27.8(osition)-414(of)-414(the)-414(initial)-414.1(w)27.7(ord)]TJ
/F3 1 Tf
25.2233 0 TD
(w)Tj
/F1 1 Tf
1.1568 0 TD
[(and)-414.1(its)-414(heirs)-414.1(a)-414(path)-414(traced)]TJ
-26.3801 -1.2 TD
[(in)-357.2(the)-357.1(Ca)27.8(yley)-357.2(graph)-357.2(of)]TJ
/F3 1 Tf
10.2885 0 TD
(S)Tj
/F1 1 Tf
0.6708 0 TD
(\()Tj
/F3 1 Tf
0.3888 0 TD
(w)Tj
/F1 1 Tf
0.7428 0 TD
[(\))-357.2(i)0(n)-357.1(suc)27.8(h)-357.2(a)-357.1(w)27.7(a)27.8(y)-357.1(that)-357.2(some)-357.1(c)27.8(haracteristic)-357.3(trace)-357.2(is)-357.1(left)-357.2(b)-27.8(ehind)]TJ
-12.091 -1.2 TD
[(on)-399.7(the)-399.7(path)-399.7(at)-399.7(eac)27.8(h)-399.7(time)-399.7(some)-399.7(hair)-399.7(of)-399.7(the)-399.7(considered)-399.8(p)-27.8(osition)-399.7(is)-399.7(activ)27.8(e.)-643.5(The)-399.6(ordering)]TJ
T*
[(phenomenon)-317.2(of)-317.1(Prop)-27.8(osition)-317.1(1.1)-317.2(will)-317.1(then)-317.1(imply)-317.1(that)-317.2(suc)27.8(h)-317(traces)-317.2(can)-317.1(app)-27.8(ear)-317.2(only)-317.1(nitely)]TJ
T*
[(man)27.8(y)-338.6(times)-338.6()-338.5(and)-338.6(henceforth)-338.6(force)-338.6(the)-338.5(termination)-338.6(of)-338.6(handle)-338.6(reduction.)-460.2(T)83.3(o)-338.5(this)-338.6(end)-338.5(w)27.8(e)]TJ
T*
[(shall)-379.6(study)-379.5(the)]TJ
/F5 1 Tf
6.9552 0 TD
[(pr)51.1(exes)]TJ
/F1 1 Tf
3.6162 0 TD
[(asso)-27.8(ciated)-379.5(with)-379.6(the)-379.5(critical)-379.6(p)-27.8(ositions)-379.6(and)-379.5(their)-379.5(transformations)]TJ
-10.5713 -1.2 TD
[(under)-333.3(heiring.)]TJ
/F6 1 Tf
0 -2.4526 TD
(Denition.)Tj
/F1 1 Tf
5.9563 0 TD
-0.0833 Tc
[(Fo)-83.3(r)]TJ
/F3 1 Tf
1.8348 0 TD
0 Tc
(w)Tj
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1.1164 0 TD
[(a)-373.6(braid)-373.6(w)27.7(ord)-373.5(and)]TJ
/F3 1 Tf
8.0278 0 TD
(x)Tj
/F1 1 Tf
0.9452 0 TD
[(a)-373.6(p)-27.8(osition)-373.7(in)]TJ
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(w)Tj
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(,)Tj
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()Tj
/F1 1 Tf
0.6059 0 TD
(\()Tj
/F3 1 Tf
0.3889 0 TD
0.0269 Tc
[(w,)-139.7(x)]TJ
/F1 1 Tf
1.7587 0 TD
0.3736 Tc
[(\)i)373.6(st)373.6(h)373.6(e)]TJ
/F5 1 Tf
3.5708 0 TD
0 Tc
[(pr)51.1(ex)]TJ
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2.7413 0 TD
(of)Tj
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1.1792 0 TD
(w)Tj
/F1 1 Tf
1.1165 0 TD
(that)Tj
-36.5779 -1.2 TD
[(ends)-333.4(at)-333.4(p)-27.8(osition)]TJ
/F3 1 Tf
7.3167 0 TD
(x)Tj
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0.5715 0 TD
(,)Tj
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(i.e.)Tj
/F1 1 Tf
1.38 0 TD
[(,)-333.3(the)-333.4(w)27.8(ord)-333.4(made)-333.4(of)-333.4(the)]TJ
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10.3361 0 TD
(x)Tj
/F1 1 Tf
0.9049 0 TD
[(rst)-333.3(letters)-333.4(of)]TJ
/F3 1 Tf
6.2667 0 TD
(w)Tj
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0.7428 0 TD
(.)Tj
/F6 1 Tf
-28.1298 -2.4526 TD
[(Lemma)-513(3.2.)]TJ
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6.8962 0 TD
[(Assume)-446.1(that)]TJ
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6.0977 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 236.2252 351.4395 Tm
()Tj
/F13 1 Tf
11.9552 0 0 11.9552 244.9116 347.1008 Tm
[(is)-446.1(obtained)-446.1(from)]TJ
/F3 1 Tf
7.8188 0 TD
(w)Tj
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1.1889 0 TD
[(b)27.8(y)-446.1(reducing)-446.1(one)-446.1(handle.)-782.6(Assume)]TJ
-23.471 -1.2 TD
(that)Tj
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2.2302 0 TD
(x)Tj
/F13 1 Tf
0.9684 0 TD
[(is)-396.9(a)-396.9(critical)-396.9(p)-27.8(osition)-396.9(of)]TJ
/F3 1 Tf
10.4429 0 TD
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 242.3472 337.0933 Tm
(e)Tj
-0.0512 -0.923 TD
(k)Tj
/F13 1 Tf
11.9552 0 0 11.9552 252.546 332.7546 Tm
(in)Tj
/F3 1 Tf
1.2302 0 TD
(w)Tj
/F13 1 Tf
0.7428 0 TD
[(,)-412.7(and)-397(that)]TJ
/F3 1 Tf
4.9288 0 TD
(x)Tj
/F11 1 Tf
8.3686 0 0 8.3686 341.8919 337.0933 Tm
()Tj
/F13 1 Tf
11.9552 0 0 11.9552 349.9915 332.7546 Tm
[(is)-396.9(the)-397(heir)-396.9(of)]TJ
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6.1237 0 TD
(x)Tj
/F13 1 Tf
0.9685 0 TD
(in)Tj
/F3 1 Tf
1.2303 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 458.3672 337.0933 Tm
()Tj
/F13 1 Tf
11.9552 0 0 11.9552 461.7216 332.7546 Tm
[(.)-635.1(Then)-397(there)]TJ
-32.5986 -1.2 TD
[(exists)-393.8(a)-393.7(w)27.8(ord)]TJ
/F3 1 Tf
6.2506 0 TD
(u)Tj
/F13 1 Tf
0.9663 0 TD
[(suc)27.8(h)-393.7(that)]TJ
/F3 1 Tf
4.543 0 TD
()Tj
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0.6058 0 TD
(\()Tj
/F3 1 Tf
0.3889 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 233.3639 322.7472 Tm
()Tj
/F3 1 Tf
11.9552 0 0 11.9552 236.7183 318.4084 Tm
0.1666 Tc
(,x)Tj
/F11 1 Tf
8.3686 0 0 8.3686 248.8638 322.7472 Tm
0 Tc
()Tj
/F1 1 Tf
11.9552 0 0 11.9552 252.2182 318.4084 Tm
(\))Tj
/F13 1 Tf
0.7826 0 TD
[(is)-393.8(equiv)55.6(alen)27.8(t)-393.7(t)0(o)]TJ
/F3 1 Tf
7.159 0 TD
()Tj
/F1 1 Tf
0.6059 0 TD
(\()Tj
/F3 1 Tf
0.3888 0 TD
0.0269 Tc
[(w,)-139.8(x)]TJ
/F1 1 Tf
1.7588 0 TD
0 Tc
(\))Tj
/F3 1 Tf
0.3889 0 TD
(u)Tj
/F13 1 Tf
0.5724 0 TD
(,)Tj
/F3 1 Tf
0.6866 0 TD
(u)Tj
/F13 1 Tf
0.9662 0 TD
[(is)-393.8(traced)-393.7(in)]TJ
/F3 1 Tf
5.4117 0 TD
(S)Tj
/F1 1 Tf
0.6708 0 TD
(\()Tj
/F3 1 Tf
0.3889 0 TD
(w)Tj
/F1 1 Tf
0.7428 0 TD
(\))Tj
/F13 1 Tf
0.7826 0 TD
(from)Tj
/F3 1 Tf
-36.3807 -1.2 TD
(D)Tj
/F15 1 Tf
5.9776 0 0 5.9776 81.8979 302.269 Tm
(L)Tj
/F1 1 Tf
11.9552 0 0 11.9552 88.2072 304.0622 Tm
(\()Tj
/F3 1 Tf
0.3889 0 TD
(w)Tj
/F1 1 Tf
0.7428 0 TD
(\))Tj
/F3 1 Tf
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()Tj
/F1 1 Tf
0.6059 0 TD
(\()Tj
/F3 1 Tf
0.3888 0 TD
0.0269 Tc
[(w,)-139.7(x)]TJ
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1.7588 0 TD
0 Tc
(\))Tj
/F13 1 Tf
0.3888 0 TD
(,)Tj
/F3 1 Tf
0.5847 0 TD
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 158.2057 308.4009 Tm
(e)Tj
-0.0513 -0.923 TD
(k)Tj
/F13 1 Tf
11.9552 0 0 11.9552 167.2478 304.0622 Tm
[(do)-27.8(es)-300.3(not)-300.2(o)-27.8(ccur)-300.2(in)]TJ
/F3 1 Tf
7.7648 0 TD
(u)Tj
/F13 1 Tf
0.5725 0 TD
[(,)-306.8(and)]TJ
/F3 1 Tf
2.4959 0 TD
()Tj
/F11 1 Tf
8.3686 0 0 8.3686 304.0219 309.176 Tm
()Tj
/F4 1 Tf
0.8929 0 TD
(e)Tj
-0.9442 -1.042 TD
(k)Tj
/F13 1 Tf
11.9552 0 0 11.9552 320.2172 304.0622 Tm
[(o)-27.8(ccurs)-300.3(\(once\))-300.2(in)]TJ
/F3 1 Tf
7.2146 0 TD
(u)Tj
/F13 1 Tf
0.8727 0 TD
[(just)-300.2(in)-300.3(case)]TJ
/F3 1 Tf
5.1618 0 TD
(x)Tj
/F13 1 Tf
0.8717 0 TD
[(is)-300.3(activ)27.8(e)]TJ
-34.8832 -1.2 TD
(from)Tj
/F3 1 Tf
2.3639 0 TD
(w)Tj
/F13 1 Tf
1.0762 0 TD
(to)Tj
/F3 1 Tf
1.2222 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 136.6187 294.0547 Tm
()Tj
/F13 1 Tf
11.9552 0 0 11.9552 139.9731 289.716 Tm
(.)Tj
/F5 1 Tf
-5.6857 -2.4526 TD
[(Pr)51.1(o)51.1(of.)]TJ
/F1 1 Tf
3.3132 0 TD
[(W)83.4(e)-413(k)27.8(eep)-413.1(the)-413(previous)-413.1(notations)-413(for)]TJ
/F3 1 Tf
16.1058 0 TD
(w)Tj
/F1 1 Tf
0.7428 0 TD
(,)Tj
/F3 1 Tf
0.7107 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 330.414 264.7337 Tm
()Tj
/F1 1 Tf
11.9552 0 0 11.9552 338.7049 260.395 Tm
[(and)-413(their)-413(decomp)-27.9(ositions,)-433(and)-413(come)]TJ
-22.3088 -1.2 TD
[(bac)27.7(k)-284.9(t)0(o)-285(the)-284.9(study)-285(that)-284.9(precedes)-285(Lemma)-284.9(3.1.)-428.3(If)]TJ
/F3 1 Tf
20.3743 0 TD
(x)Tj
/F1 1 Tf
0.8564 0 TD
[(is)-285(less)-284.9(than)]TJ
/F3 1 Tf
5.0381 0 TD
(p)Tj
/F1 1 Tf
0.5031 0 TD
[(,)-294.6(then)]TJ
/F3 1 Tf
2.8018 0 TD
(x)Tj
/F11 1 Tf
8.3686 0 0 8.3686 432.3906 250.3875 Tm
()Tj
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[(w,)-139.7(p)]TJ
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ET
153.93 597.025 55.571 -1.435 re
f
BT
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220.29 597.025 55.571 -1.435 re
f
BT
11.9552 0 0 11.9552 275.8167 595.6458 Tm
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ET
153.93 551.065 35.161 -1.435 re
f
BT
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(\012)Tj
ET
199.89 551.065 35.161 -1.435 re
f
BT
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(\)\))Tj
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[(Figure)-383.4(3.2)]TJ
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[(:)-444.5(Beha)27.7(viour)-333.3(of)-333.4(prexes,)-333.4(case)]TJ
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(1\))Tj
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()Tj
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(,h)Tj
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(s)Tj
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(m)Tj
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(\)\))Tj
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[(w,)-139.7(p)]TJ
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[(...)166.6(v)]TJ
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(m)Tj
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(e)Tj
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[(and)-379.2(this)-379.1(w)27.7(ord)]TJ
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[(is)-379.1(con)27.7(v)27.8(enien)27.7(t)-379.1(\()]TJ
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(e)Tj
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(j)Tj
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[(do)-27.8(es)-379.1(not)-379.2(o)-27.8(ccur)-379.2(in)-379.2(it,)]TJ
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()Tj
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(j)Tj
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[(o)-27.7(ccurs)-379.2(once,)-390.6(and)-379.2(the)-379.1(w)27.7(ord)]TJ
-26.761 -1.2 TD
[(is)-329.5(traced)-329.5(in)]TJ
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5.2189 0 TD
(S)Tj
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(\()Tj
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(w)Tj
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[(\))-329.4(since)-329.5(the)-329.5(asso)-27.8(ciated)-329.5(path)-329.5(is)-329.4(made)-329.5(of)-329.5(fragmen)27.8(ts)-329.5(of)-329.5(the)-329.5(paths)-329.5(asso)-27.8(ciated)]TJ
-7.0215 -1.2 TD
(with)Tj
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(and)Tj
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(\).)Tj
/F17 1 Tf
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0.0001 Tc
[()-5021.8()-5022()]TJ
22.7622 3.4143 TD
()Tj
-3.4143 -3.9833 TD
0 Tc
()Tj
0.569 -0.569 TD
()Tj
0.5692 -0.5692 TD
()Tj
0.569 -0.569 TD
()Tj
5.975 -0.4553 TD
0.0001 Tc
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7.6823 0.4553 TD
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()Tj
0.569 0.569 TD
()Tj
0.569 0.5692 TD
()Tj
0.5692 0.569 TD
()Tj
1.138 0.569 TD
0.0001 Tc
[()-5022()]TJ
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11.9552 0 0 11.9552 184.252 292.9768 Tm
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(1)Tj
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(j)Tj
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(1)Tj
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(j)Tj
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()Tj
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(j)Tj
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(j)Tj
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(j)Tj
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(j)Tj
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(v)Tj
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(1)Tj
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(j)Tj
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(..)Tj
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(j)Tj
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(m)Tj
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(j)Tj
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(m)Tj
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11.9552 0 0 11.9552 386.6461 292.9768 Tm
(u)Tj
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(2)Tj
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11.9552 0 0 11.9552 208.3965 326.3107 Tm
()Tj
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(\()Tj
/F3 1 Tf
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0.0269 Tc
[(w,)-139.7(p)]TJ
/F1 1 Tf
1.6904 0 TD
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(\))Tj
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()Tj
ET
165.81 318.385 55.571 -1.435 re
f
BT
11.9552 0 0 11.9552 221.3918 316.9728 Tm
( )Tj
ET
232.17 318.385 55.571 -1.435 re
f
BT
11.9552 0 0 11.9552 287.722 316.9728 Tm
(\012)Tj
-10.6465 -4.7034 TD
( )Tj
ET
165.81 262.225 92.988 -1.435 re
f
BT
11.9552 0 0 11.9552 258.8094 260.7427 Tm
(\012)Tj
ET
269.61 262.225 92.988 -1.435 re
f
BT
11.9552 0 0 11.9552 362.5574 260.7427 Tm
()Tj
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()Tj
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(\()Tj
/F3 1 Tf
0.3889 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 254.8065 250.515 Tm
()Tj
/F3 1 Tf
11.9552 0 0 11.9552 258.1599 246.1763 Tm
0.1667 Tc
(,H)Tj
/F1 1 Tf
1.357 0 TD
0 Tc
(\()Tj
/F3 1 Tf
0.3888 0 TD
(p)Tj
/F1 1 Tf
0.5032 0 TD
(\)\))Tj
/F6 1 Tf
-9.8537 -1.4913 TD
[(Figure)-383.4(3.3)]TJ
/F1 1 Tf
5.1104 0 TD
[(:)-444.5(Beha)27.8(viour)-333.4(of)-333.4(prexes,)-333.4(case)]TJ
/F3 1 Tf
12.6695 0 TD
(d)Tj
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(=)Tj
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(e)Tj
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()Tj
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(c)Tj
/F1 1 Tf
0.7661 0 TD
[(\(=)-277.8(1\))]TJ
-30.9433 -5.0383 TD
[(It)-394.2(remains)-394.2(to)-394.1(consider)-394.3(the)-394.1(case)]TJ
/F3 1 Tf
14.1371 0 TD
(x)Tj
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(=)Tj
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1.1569 0 TD
(q)Tj
/F1 1 Tf
0.4823 0 TD
[(.)-626.8(B)0(y)-394.1(construction)-394.3(the)-394.1(heir)-394.2(of)]TJ
/F3 1 Tf
13.3725 0 TD
(q)Tj
/F1 1 Tf
0.4823 0 TD
[(,)-409.3(when)-394.2(dened,)-409.4(is)]TJ
/F3 1 Tf
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(h)Tj
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(\()Tj
/F3 1 Tf
0.3889 0 TD
(s)Tj
/F4 1 Tf
8.3686 0 0 8.3686 89.1417 151.9735 Tm
(m)Tj
/F1 1 Tf
11.9552 0 0 11.9552 98.2227 153.7667 Tm
[(\).)-444.4(W)83.3(e)-333.3(read)-333.4(on)-333.4(Figure)-333.3(\(3.2\))-333.4(the)-333.4(equiv)55.6(alence)]TJ
/F3 1 Tf
9.9967 -2.5757 TD
()Tj
/F1 1 Tf
0.6059 0 TD
(\()Tj
/F3 1 Tf
0.3889 0 TD
(w)Tj
/F11 1 Tf
8.3686 0 0 8.3686 238.5086 127.9088 Tm
()Tj
/F3 1 Tf
11.9552 0 0 11.9552 241.863 122.9733 Tm
0.1666 Tc
(,h)Tj
/F1 1 Tf
1.0206 0 TD
0 Tc
(\()Tj
/F3 1 Tf
0.3888 0 TD
(s)Tj
/F4 1 Tf
8.3686 0 0 8.3686 264.3168 121.1799 Tm
(m)Tj
/F1 1 Tf
11.9552 0 0 11.9552 273.3988 122.9733 Tm
(\)\))Tj
/F10 1 Tf
1.0555 0 TD
()Tj
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()Tj
/F1 1 Tf
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(\()Tj
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0.3889 0 TD
0.0269 Tc
[(w,)-139.7(q)]TJ
/F1 1 Tf
1.6695 0 TD
0 Tc
(\))Tj
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0.7222 0 TD
(v)Tj
/F11 1 Tf
8.3686 0 0 8.3686 345.3479 127.9088 Tm
()Tj
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0.8929 0 TD
(1)Tj
/F4 1 Tf
-0.944 -0.943 TD
(m)Tj
/F3 1 Tf
11.9552 0 0 11.9552 358.1838 122.9733 Tm
()Tj
/F11 1 Tf
8.3686 0 0 8.3686 365.4436 128.087 Tm
()Tj
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0.8929 0 TD
(1)Tj
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(j)Tj
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0.5306 0 TD
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11.9552 0 0 11.9552 382.1569 122.9733 Tm
0 Tc
(,)Tj
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-25.9433 -2.7696 TD
[(and)-333.3(see)-333.4(that)]TJ
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5.7278 0 TD
(u)Tj
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0.8502 0 TD
(=)Tj
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1.0556 0 TD
(v)Tj
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8.3686 0 0 8.3686 169.4844 94.2011 Tm
()Tj
/F2 1 Tf
0.8929 0 TD
(1)Tj
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-0.9442 -0.8717 TD
(m)Tj
/F3 1 Tf
11.9552 0 0 11.9552 182.3193 89.8625 Tm
()Tj
/F11 1 Tf
8.3686 0 0 8.3686 189.5801 94.9763 Tm
()Tj
/F2 1 Tf
0.8929 0 TD
(1)Tj
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-0.9442 -1.0093 TD
(j)Tj
/F2 1 Tf
0.5306 0 TD
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(+1)Tj
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11.9552 0 0 11.9552 210.2784 89.8625 Tm
0 Tc
[(is)-333.3(con)27.7(v)27.8(enien)27.7(t.)-444.4(So)-333.4(all)-333.4(cases)-333.3(ha)27.7(v)27.8(e)-333.3(b)-27.8(een)-333.3(considered.)]TJ
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/F4 1 Tf
8.3686 0 0 8.3686 78.8314 562.3656 Tm
(k)Tj
/F13 1 Tf
11.9552 0 0 11.9552 84.7133 564.1589 Tm
[(-reduced)-333.4(for)-333.4(at)-333.4(least)-333.3(one)-333.4(in)27.7(teger)]TJ
/F3 1 Tf
14.2917 0 TD
(k)Tj
/F13 1 Tf
0.8854 0 TD
[(is)-333.4(not)-333.3(trivial.)]TJ
/F6 1 Tf
-16.2405 -2.7089 TD
[(Corollary)-474.1(3.5.)]TJ
/F13 1 Tf
7.7789 0 TD
[(Assume)-412.3(that)-412.3(the)-412.2(braid)-412.3(w)27.8(ord)]TJ
/F3 1 Tf
13.0779 0 TD
(w)Tj
/F13 1 Tf
1.1551 0 TD
[(has)-412.3(length)]TJ
/F3 1 Tf
4.9967 0 TD
()Tj
/F13 1 Tf
0.8289 0 TD
[(and)-412.2(width)]TJ
/F3 1 Tf
4.9356 0 TD
(n)Tj
/F13 1 Tf
0.6003 0 TD
[(.)-681.2(Then)-412.3(the)]TJ
-33.3733 -1.2 TD
[(n)27.8(u)0(m)27.7(b)-27.8(er)-488.5(of)-488.5(letters)]TJ
/F3 1 Tf
8.31 0 TD
()Tj
/F11 1 Tf
8.3686 0 0 8.3686 178.6082 522.541 Tm
()Tj
/F2 1 Tf
0.8929 0 TD
(1)Tj
/F4 1 Tf
-0.9442 -1.042 TD
(k)Tj
/F13 1 Tf
11.9552 0 0 11.9552 197.2832 517.4271 Tm
[(in)-488.6(an)27.8(y)]TJ
/F3 1 Tf
3.366 0 TD
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 244.3557 515.6339 Tm
(k)Tj
/F13 1 Tf
11.9552 0 0 11.9552 250.2376 517.4271 Tm
[(-reduced)-488.6(w)27.8(ord)-488.5(traced)-488.6(in)]TJ
/F3 1 Tf
11.3791 0 TD
(S)Tj
/F1 1 Tf
0.6708 0 TD
(\()Tj
/F3 1 Tf
0.3889 0 TD
(w)Tj
/F1 1 Tf
0.7428 0 TD
(\))Tj
/F13 1 Tf
0.8773 0 TD
[(is)-488.5(b)-27.8(ounded)-488.6(ab)-27.8(o)27.8(v)27.8(e)-488.6(b)27.8(y)]TJ
/F1 1 Tf
-28.9678 -1.3886 TD
(\()Tj
/F3 1 Tf
0.3889 0 TD
(n)Tj
/F10 1 Tf
0.6558 0 TD
()Tj
/F1 1 Tf
0.8333 0 TD
(1\))Tj
/F4 1 Tf
8.3686 0 0 8.3686 105.0789 505.1651 Tm
(n)Tj
/F14 1 Tf
5.9776 0 0 5.9776 110.9888 508.7726 Tm
(2)Tj
/F4 1 Tf
8.3686 0 0 8.3686 115.6543 505.1651 Tm
()Tj
/F13 1 Tf
11.9552 0 0 11.9552 120.254 500.8264 Tm
(.)Tj
/F5 1 Tf
-4.0363 -2.7089 TD
[(Pr)51.1(o)51.1(of.)]TJ
/F1 1 Tf
3.0891 0 TD
(Let)Tj
/F3 1 Tf
1.7778 0 TD
(u)Tj
/F1 1 Tf
0.8919 0 TD
[(b)-27.8(e)-319.5(an)27.8(y)]TJ
/F3 1 Tf
3.2222 0 TD
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 186.2007 466.6475 Tm
(k)Tj
/F1 1 Tf
11.9552 0 0 11.9552 192.0827 468.4408 Tm
[(-reduced)-319.5(w)27.7(ord)-319.5(traced)-319.5(in)]TJ
/F3 1 Tf
10.7028 0 TD
(s)Tj
/F1 1 Tf
0.4688 0 TD
(\()Tj
/F3 1 Tf
0.3888 0 TD
(w)Tj
/F1 1 Tf
0.7428 0 TD
[(\).)-439.8(W)83.4(rite)]TJ
/F3 1 Tf
3.8732 0 TD
(u)Tj
/F1 1 Tf
0.8919 0 TD
(as)Tj
/F3 1 Tf
1.2139 0 TD
(u)Tj
/F2 1 Tf
8.3686 0 0 8.3686 417.4934 466.6475 Tm
(0)Tj
/F3 1 Tf
11.9552 0 0 11.9552 422.8573 468.4408 Tm
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 430.1171 472.7795 Tm
(e)Tj
-0.0512 -0.923 TD
(k)Tj
/F3 1 Tf
11.9552 0 0 11.9552 435.5706 468.4408 Tm
(u)Tj
/F2 1 Tf
8.3686 0 0 8.3686 442.4139 466.6475 Tm
(1)Tj
/F3 1 Tf
11.9552 0 0 11.9552 447.7779 468.4408 Tm
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 455.0376 472.7795 Tm
(e)Tj
-0.0512 -0.923 TD
(k)Tj
/F3 1 Tf
11.9552 0 0 11.9552 460.4912 468.4408 Tm
0.1666 Tc
[(...)166.5(u)]TJ
/F4 1 Tf
8.3686 0 0 8.3686 481.2822 466.6475 Tm
0 Tc
(N)Tj
/F11 1 Tf
1.0121 0 TD
()Tj
/F2 1 Tf
0.8929 0 TD
(1)Tj
/F3 1 Tf
11.9552 0 0 11.9552 502.5873 468.4408 Tm
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 509.8481 472.7795 Tm
(e)Tj
-0.0513 -0.923 TD
(k)Tj
/F3 1 Tf
11.9552 0 0 11.9552 515.3007 468.4408 Tm
(u)Tj
/F4 1 Tf
8.3686 0 0 8.3686 522.145 466.6475 Tm
(N)Tj
/F1 1 Tf
11.9552 0 0 11.9552 72 454.0946 Tm
(\()Tj
/F3 1 Tf
0.3889 0 TD
(e)Tj
/F1 1 Tf
0.8786 0 TD
(=)Tj
/F10 1 Tf
1.1908 0 TD
()Tj
/F1 1 Tf
0.7778 0 TD
[(1\),)-434.8(where)]TJ
/F3 1 Tf
4.5743 0 TD
()Tj
/F11 1 Tf
8.3686 0 0 8.3686 172.6346 459.2094 Tm
()Tj
/F2 1 Tf
0.8929 0 TD
(1)Tj
/F4 1 Tf
-0.944 -1.0421 TD
(k)Tj
/F1 1 Tf
11.9552 0 0 11.9552 190.4249 454.0946 Tm
[(do)-27.8(es)-414.5(not)-414.5(o)-27.8(ccur)-414.5(in)-414.5(the)-414.5(w)27.7(ords)]TJ
/F3 1 Tf
12.9758 0 TD
(u)Tj
/F4 1 Tf
8.3686 0 0 8.3686 352.3975 452.3013 Tm
(t)Tj
/F1 1 Tf
11.9552 0 0 11.9552 356.6068 454.0946 Tm
[(.)-687.9(B)0(y)-414.5(Prop)-27.8(osition)-414.5(3.4,)-434.8(a)-414.5(sub)27.7(w)27.8(ord)]TJ
-23.8062 -1.2 TD
(of)Tj
/F3 1 Tf
1.1799 0 TD
(u)Tj
/F1 1 Tf
0.9467 0 TD
[(of)-374.4(the)-374.3(form)]TJ
/F3 1 Tf
5.348 0 TD
(u)Tj
/F4 1 Tf
8.3686 0 0 8.3686 168.2042 437.9551 Tm
(t)Tj
/F14 1 Tf
5.9776 0 0 5.9776 171.8157 436.7596 Tm
(1)Tj
/F3 1 Tf
11.9552 0 0 11.9552 177.0799 439.7484 Tm
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 184.3397 444.0871 Tm
(e)Tj
-0.0512 -0.923 TD
(k)Tj
/F3 1 Tf
11.9552 0 0 11.9552 189.7933 439.7484 Tm
0.1667 Tc
[(...)166.7()]TJ
/F4 1 Tf
8.3686 0 0 8.3686 211.0007 444.0871 Tm
0 Tc
(e)Tj
-0.0512 -0.923 TD
(k)Tj
/F3 1 Tf
11.9552 0 0 11.9552 216.4543 439.7484 Tm
(u)Tj
/F4 1 Tf
8.3686 0 0 8.3686 223.2986 437.9551 Tm
(t)Tj
/F14 1 Tf
5.9776 0 0 5.9776 226.9101 436.7596 Tm
(2)Tj
/F3 1 Tf
11.9552 0 0 11.9552 232.1733 439.7484 Tm
()Tj
/F4 1 Tf
8.3686 0 0 8.3686 239.4341 444.0871 Tm
(e)Tj
-0.0513 -0.923 TD
(k)Tj
/F1 1 Tf
11.9552 0 0 11.9552 249.3619 439.7484 Tm
(with)Tj
/F3 1 Tf
2.3188 0 TD
(t)Tj
/F2 1 Tf
8.3686 0 0 8.3686 281.4008 437.9551 Tm
(1)Tj
/F3 1 Tf
11.9552 0 0 11.9552 290.9011 439.7484 Tm
0.3461 Tc
(