From aad4326b06f9d054311590cb254ec4436642e548 2017-07-02 19:57:50 From: Tom Bannink Date: 2017-07-02 19:57:50 Subject: [PATCH] Add attempt at proving circle version of lemma --- diff --git a/diagram_circle_lemma.pdf b/diagram_circle_lemma.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0cdb2f0fac20dbeed6c81e62ba5b047d1d593ce6 GIT binary patch literal 50923 zcma&MLy#^&yJlOqZEKfp+vYCYwr$(CYnN@?w)K_G{?CbfB5p_bpmX%(AR}}5tal~3 zqNq3>6FnOY`TX+8It(WfBayw46$~#g41=txowR5Wk1wx*|x8-^^Vwny)}cacSt%)ludiZg9~`4Vv0^5BUUR0S%;19@l{wgwp=8sCLl6Uo)0 z%Jk5a&cCJjd-z|koAJ#S*_+Gk=z!}0Np>T&fWNt!Tx%kgPGWv}imxmMa z4g#DRdWAtF1T^L&z%UNxgxB03!dm>s+cO_LNJx^vDTo466g7~##38k@Cg}Z36HnDL z*Akm~E@GQvop`ocLD_yUVSDBsz%Ww!gqdUzOJ5GQR$NJQ7RiXuI{*Hp$0!&@Ncjgx58gE$K3Y^Cv01Cwh1XN=m2#zEUc#LPi0E)x| z1X62X5{|@ZZ`Q^c*(Ni_S=~Eonl^lPI7c+_Og5pB_s$5&V+2U~5IiqGkkT>1iSI56 z$YTdc+0!u@;DijwuyI~R&TV)s;Z|t^1y(2`QZMbV)-XV)moz15FhcRD1u`;(=?(zb zI|cZ*rly-a%wqiP)a{H~kDKK^faN`gt7pO~E2;`R-l_->`wF z$DlX{U@$k#KnDlm{!xO4T=OJe$j5J_O_u11TtyA8!$4dHsZ9hN~YpPa^txR z|Kl&Ko1sSG=Bk|v1u8Ne6}hbu4eJocsL_tYHsChrG(<1(C_x#|dt^iBdeTObSLQWE zqG9(UwNp5O&dp}e5OyQ_6Jna$WQQ0c0#yM2QmM@%pSMhDG<)In1L4I@bnLMo6ha-8 zC!XLWQl}h{=jVVyFOhRgp?OG>6-%%+n?TzH8^@yT&Byy$c>PbAjvF%IoMpSM9W*MA zdUKcw~nk{eF2a zB`*Yre{5Kc9a>Ok?CNM+cxDfB9rgibp{rhN!Wz8|A>+0WVTxX|mPmr7=H>=I<*4xN zbp*vi!RwU|Y8s&ouv8y7JKj2(gqU7Lr3pO6sY;y)5=7?fLc0&Fx`0mnp@)e9b4g~1 zb^|1ws~g)8iDAjABGBo(t^sKa1j)T$-qAo%U`Y>0C&_Dii{&_-=Fd(L6pgB%PO7aBewF}=V(ih$i(~X8y z1ge{mD6zpyp9|t*A!n>Fc}Vl~R0;AG)bnG+5JbSCMw60dEGwQK2erEBmX(aZhvbq4 zbN+VC++V4a<4)Ro7IyMZ)-r&MJKf%KDUsJPb_;-6f`r=1AuBF9Hnra}m5S(ozji)2 z1e77y8!EYXdUi9#5+rkxfk4S0HY+G1$9x$<3NI&O9Qcsi#cejMUUwR98cnkjDQwt8 zFX|-dn7^;NY)zq<@ZZ&jB~^T5q|OjrNI{V@O)Y_CTElG(K7OZU9PQYhJiP%BNW%OC zO)XyJs;k>-okH?BoBzGy6EJB$9^<_xZY3SZw4vDXGj;e`bR4O0TA{g2?&bpqXd*u) z-C4-=`q~`JrAWBUbiq6Ps{nth#*T`Jr2{BOj%;fK_dwozLz0Y|O6wFDeymWcHzJ@Rh+^7L25f>zg_Gc0Gu zGIpb(R(5!6HNZ1S@ruziw*aEY5Os+&R~9|b1MYBbKPo=`2I@M$IcVS`cEc-_<_GrU z7P%5{yDY{AKvw_j&%Ersji_vjO7<3JqgBJiz_T4@5j!@(vCr-Myrzhybh$y>2BmVZ zzqQh$$T?WX1=w{u7*)m*()FHoykBBkQ~rwB=i|+&5VCwk?(a_ThUy|<)811fSOq~> zOyUdKr(*4Q*HHL;nbnrC?@#Sg;5El3Y|)?~Q2`Acm}qA}>+fFY2+<8wg5r%g92iVR z$j5iEGY5_P5Z~E#b}S?@dCZLIh^r(aYvC>?b+IjUHlL|UJoHMlMncUH$#{0dQ#}DO z2erJRn`v`omn@ickJL5;o^hEGDfjd*_s(1B&KmAvJ9qN}^a>R%&7o2%Sj|+H(v zo*0vuwJur4N#NJw=*o@&`w^$jsG@YWN2;c!`=u-iGSoALqksSwuIUWp4f|-nQpjQ6 z`--=A^vTRd1@yc8H24LCNxGAqK7KBFm+9j)89Q|>aoVxXg~ku#Uqs`w-;2j*Pi%ku z1}Q-w7-CD~MY^fHkgvCci|~A;-lC}8x04}(v_Vx*{;yV_uN(Wl-;aZ{@;*O{A_@w$ zr`|ag3>e$ew_Grwa%y4erBj8=h{kSDt&!||jX|81xSs~Iy1CE+ghrB4ZNoQHU2yi|%fS!8A1F~A2CmVEc{UQ!(BNU7;&1^q)C;V^Ky84>gcJtQ1=7S{XVV=^h4^M^1sEG+_Udp_jG`PrQRt zBlKv@9EnvH7PUMPg_&o(4hp_0qr&=DRs8xJGj5L^Wr>hh2YXsi-Io$ocw3rE4@>2} zh7YM0TDx>|;Cz>_#i4RWOZ20{qcMF{h`On{srxhbw0g)RXByI*sgn!eJX=EQh-H4` z_*Cu|-t0`eEKIpo*=sGEMOo#TNMcmBRg_v8{y6#^W{eS9`W}tuHTR-EyxV06LN{7d z2bdoGKFKA52aZPz)Tm4)-CVt&k4_((O2K06r-%9@y6|>#@CEYitXa5(=25zm%EJRq z)1^BRuZ=+i`N}>@mu_zDKC7bw1e?J^=|1}OlkbNhaQ5QGE_uTOCkYXq6|;XI4WHRx z(`72@DLJhn1I)}Ad4owjFog(mb{S-UFm+`Z++Gru*3U_`@XtSF_d(Ef-JZ^O(z_Cz zr|+wstoaq!pR@yc!tpBg%8S4(InykGwO!sj06&rmLSS4c}t|rEh*u}KvE!y zj8p;@^+eI*4(9ii^XPxF+j66;bv2WhX>K-m?P*5I09zc#-&O;^jZ+2YF612S5nu%5 zDJQ3fgaick@CX#>14c^=!Pt!YM92pmjf@rC8z{O``{2liDG6R-@9{`PaXh2 zhyWBB1>^kf2szFzlE?_P4qgB)z9o`t?;aA63-sXt1sC=56^~?t@?ya%9FxAbvy%&X z9dj5+3Mg6c0J>pp<`YDW3L@DlU=P?M2W{@3o$r%5A~pcVS|3vM$9_6!wQy%dfoWIc z93WD}T&|sR#>AyUap>X^)K>#7hz;oc0$+WD?FWDRbO0hEnEws?-1>AU%9tWFX9NRr z7Z1?oHKca~(#}9dE2u3Q2!0WC2JWY4?+Y}bMEhDLyoLzl3aqsU_E&}kO)+r=-rs`x ztwRJC8bq={Lxeef>Js1GGqj7>S{M;0xVeUN6)J}EJ1Ykg5hiXsyP5cOcgEMru}52nhY=*r7xC@+-MPz00fqu=2o#zJvaZ0N z4&B#ZuwPN!>O0>0AvGibQrV~b76jVAo%^kuw#PODb$5LD!u(Q*_B2J8ds0*7{H1^Q zk9Bf#Kn&pE5FiZ5RzQLR0tOca8f=5z`;93E4g1a=>wB;2TptAP3;v#ceJS{v*Z|n0 zG<@?BUkCicm4Ww_(xdgiFqCBjg9j#TL7x9+-}>wP*t7oPpZrcd{mx2WbR;hk@@(k` z{0iZQ#I~2-I6ncfEC%hv2TuA5N9X&@G*GQTy>Gtu)YVwu-0)0vukHTKz@zK~ z!+e#77p52xDB|K{;TZj5+Wsu!{t$3dX27aK#sz)%8rk&(^#3;Q$YT%Rzu*Y%I6X3; zV!84-oeRW%(TUiHUkW@bx&-#y`Y90R7a#$}hz9XN+hzOHK!bD(wHaH1KEDmw0WrkG z;`v^IwAXH68o)}ReAzWK3Iu38w_V1Afxv)x82(bgfO_En%Ao<@aevEnlf%_d5#9?Z z2k}af*gO9Wm2Ua`CjM?CC{{qC3E%mGARr^URlwWvRHv1?p6OvH^>4`Wc~6eBoY=t$1bP39>37e9TIN=^(Gipb19(|Wr-%zJml74E9+hhz&`&cqFH+Z~=B&w3QS z{@FQ-^q#@0k5xG)h(sstVsbezmgKoFwY~0q*1CD;tpt3eozF^Z6_M3oES9b8YUw6b z3W-r#HMdlR+x@#?(M0Ag@=Izu|6;cIvq^iHs{Uc@8u{WJ1ole3_`dtoYuOn8bXyFv zybYr9k|DBS5_Rb70t4b+9aHuw>34$GWm}f)txcmcZnNowPsu@b)xsc8aHKa=sJbX zcMyZwZrYqW$!gDxNR5>aXZ!LuB_=^ZVxX!yzB_g}KajShfb(d%`f9mp zmjDdlqFvat*jZo4;sper!rg8=VcE)C9ga`s+WO}26|fxBz+erpuUJ$%ycFMEK#r$s z?bLCR2`q8L8O>0Z)YK;P>$PvkAHRkX6iDUE7w}rj?ijn|Tmh?zNF&1PLi{}nE1E>O z{v%|%@5~O*VS9@Ab8cqhE z3FNiyV%%0#44}J{5ON`o+T?)rg^z=*=-mS!$R1NT)v~HM(4{F#a{djh(IqnOae}Sv zG-AvuLZ0wLLcjM#rt~S2O>Sb}IqL#&L|ING5`#bCUCly)BK*D^EH1#CCwQ48xLwZn zQ|*Om^@PB4s_oOX(fwAh372iTs+C=gri>hlyq>oSb!(UxV^f||?cK61z>*5&xsE7W zFJ9ca0Tq0c6XVK;tlYT>4VET0Pnd1U<5L_|-LJZeG1eD@u}RQHKS{=3!Z}OUZ$CTb z3u?!D!*KzS;GM?KH#^D(YXsJX=!!9yVp9p+f2<=|pQhH9JSOhox6;HT`5>S5{~1rm z&B#+W(WXhvPM?-o7RhWA20LjGa0anK=<(+8E3jd$0(<&H7kEfsB;g`|vT0jRZf)-G z7-ZAgogcc{g}R4~o>N$Rm{SV!{>W_`UlPw z)F6R9pQa143;GqY%I4Mix~!U)zBNPEJJ4ZxzEWB4DO4LD`i-cwF!0!l*_(S@O)7Q0 zQb@A?;bp9l%tx3YV^Vbaj+fLzw}}z-S4xXJQV++hC2ZPTCU)%ZG)*IQl4s~lJwSx^ zq_-YffMEUWskIy>A&M-%+xnnRv6&Kdg|CtLJ)#9TY$GE^b9%~!Rq}{9!r<%Br3Kh& zhpzEyerighM`BD=lQf=YnfR(Y4Gc^|xsU&wIUrsV!(m>D)gPz*j`r?%Hl(26RGtN| z2o&>u?ZX`U@`3LNvv}632$0OEL~sw0yCWE*YFYI!WSurKD}Ih-o|ygTM~KC~nO)?4 znwar!FtYz3_L%A;f`rhZa)1? zuFzs3_%dfqc)!()8Kq*vtJ5AYP4Q2Dy%&)wHV-c9^GmJbkt;8$ReYE% z0W&TXdh3N8DAkw)n4EI_nH9M`&T%Ib&5hWwi{8r05p$Z4Km16FwK6v2NVM?{dY`nU zeyim1#4x(%YRsn5B;%jxP6Y0HWPGCd!NU>T^0wA3yj5@=GVE5>vF5y=;Lep~<;U+_ z4yUaAqlp}aNiEWM8RttD6P`nB_HV#!+s-%?;&Z#2_RzdD?D$~(gf0b1%6RTI&-JE0 zn>&@rYr7sVogBM&YR^)mD1t207lVSKb#taqn=D79%MImcrRVH$N86{sUNMuV$lS4F7e|C9_yv&U?Cukx&Dt7knaO=4#(w0hU z6T+{2==pjng=)L(l*pu1F& z(tYIxSQ}Cg=pV>78%-qZc!s5^;&3nC!}ok6HXqYyJhLhE4!ZTBC`gefCp^RDht(t1 zUOzg^ax-GPoBpwhs1ai#jCsd1VPXFxHv5by&aN+@lfNj5*5)H*P@G*BA*W{f%Bxy4HbXq zAnexdJ-ZZ1=OA??KgIZAOAoqQXNdPcyNcsSDVNzaO$E7Pngd*Fv5h!_F)eIt%X3qi z68XCDTBWv%T)j+YfW9pee?Q#~Q^%k15EsV>qwc~ucs~mwh8F}83}=-T%UUN6&KJAl z+x;bhn63wSmt`xu#o!#@s0qiCQup#b%fDpJP<*u3W~;TeZ~vLJcRfk-j&%@%z(%pe z!S0*jJb0Hf9E{bIkMv<d=_Cc{W=*kzRH3*v?Pl?2K$U!HPOFOqZ0%U9qhGtzdcj{L`i zYbGAYovg3~pJ~nQ;nI&Kw%Z9g(!XdmJ?C;g-kJXp=(&qWxFomHx%{X{U406GVALIw zfykJ*((}~U!A@&*(2i8Z_UX1x)z+E7iO#Ul_pb@`eCfF8Nj zFjc0zuE=%|tM}o(|L7Y=8UNG;B4!5BoTeC_sGCBQWwO~$o%cmu>Q$LT3&7lw8y5_|mcVBCW|pW>9E^JmE^;C=*7Rj9oei0cZ6L3!fSm5 z$Ll%8NQF(H*qj#~_kH$S;wZddfg?KIOR+swdr+8W@f29t%i5#6k$UN+`fJ{IJ8Ll# z!#D?3^!%-m)J|_+G2ezB^v{;N^NeYhRHOG>NebnXBL*-!seVhcb&gqBpNI4!eyXMi zG_^S7ya$s$B!Xb8c_3)qM`A?!t;s7RAUct1Fv0ymozp5GwI08C^DjgyI$%s|t>hLm z6zJj4DUD@qQd63VXfc+`ff+MSq(QL8sh7oIt+yV&ngW4Sw*VU&;)yz*XFEy~l)tq> z59bnbyUxn;{T05xa+m5xT4y+ozsA8ArHl)cuh^*9HfBEZkyM(4!B4?)l~K2>nl;KC zef|hS)9#wWkDa7uiRFN9bN;gQh&ULd%=Ql;(H{Y8koZJ~J0u^ZmqbH9r6p9a6A5trUgv+LY`MJja{Q>YWQ zpPujCkk$$#mZCLIf+S*_&+eVY`Z_Y2#Gq5h&nWWr&0z_Sy+>x!29ivQS#BbXrGoRF zhJH7eAvzqX*Uewi$0cgGbf#&eDziCpdor>HIW3uAZ!7P$gC75nRl!E&W2cv*icDSX zuE-zR4ZjY0A{~ZiYoxU#&%X%0%blL`&!mr!d!Ku97FTew6xGjL(Hw#2Z3YumKJZ`q zUUtDl0jup96{92fDj#@NSzWBbF{+GIU^)(rQIGyA(%9X0lo3RBjZ|{A1bHIAO>~PL z$5+<;Y{sb^@%s+$TOAqmM;{th`)*F;SAKqW3uGuVc);}gU85$Ay?c%eLEDa_^H1?i z_n5bux7fsP!Qq|oBbYN;KT+XQdb_Ix+np0nuzgrrQe-dd2WO(lHM^9n`LGQhOZL*|O-h3wcLH)x&j0dLD}Q6E%*DIC&fUd6-=RNP~gS9 zyt)n?@u1L3TIcjaU35JPwWO+&S4rrWU_O8nxn8`@&DoXS(*do-3yrKlE`P z$iO+sVyZtFvQ||f+?4o;za9>4dFQ<+T&gbFjJDa0t4MVkCAYFi9M1C%+PpL6;WuV@lyOOu5Iu4ebT7H6`j0>%!q#-{*hTxKw3{; zV7JQY+(2DB-dumi?3MN9VLXs%NpVIcBrfneC=r|ReJVKumyMCWm`-E4JfrwmC}_u@ z$aJln=%~to@*SWj{C;@d`H-&H_!`NHC3gWaZC#Z`37e;cbwQ#E(w+6hzaob(%ax=q zjOgva=yv2CQ6QZg~7-;}s zYZ~XktM^0O6tbq6f&;$#g!r5A5Rg>ahjHng>1CqZ@(-{$ZxWiH1WWz~j-7qrSID9E!55>fx{3iTJV?cvlK?(B@ z^*=Tt_P9}~19T0A9;L`#30Q)VB#G@zyMy(`Um0jS6`OgEIcp(>Y?F1-;g|bhX9rqkwDJ@lPxIk1d|u0)U1lsNh24MAjIMQX~>? zEgk~PxY@(tu-cRqkq$Dh+?pXx={)$eZ7r;!pnB#iJP#vy_gEmzDSD;(dg{z&nI1dA z5bPZ1cT%AL5$%5Zw-(QbuFn%#P}BTc*|l%JI|d5NA7@E!qljascd6CV=w>>2`aUet zGj`i_5s)?JK;b?dubC@M3G-Rm)$+8#K44eQ|l5 zFAvtJ4M^`@TmjhXn18yysuC$BaCgiWaayo*CMY}C!4SyDQ&vnHl7DL{NHv9c^iKVL zTovpnUqcivxmh7BfnVVi>o!2T%$0Q)ko#$WA5nBvLIj#O(fIY~n3FoEPorkt`4zf~ zAI*3M6}H2Ue8%!s(RB%%>!epf+-43L)rIr2^o9e-c9o=r`v0oi#}K`2oG#1P2%PHC zxXwhh_z+ zu7f$G&*`iO+gAUl&TCeZE{+O~1p_D~^J8%u(m3O4S=Q#^!Ec;$*sAO)6`a^7B!xWN z%lGX^_3Ct|mQ)wFhvYqXj&7F~I=5JiP9=)RJo7hdVe^Uj zRGDqtz1A)`?cs>hEC*Z<26aZSH9x@{6{;hYXOVpw%w6S47M?SH&uvkl+}9y(m2!KJ zcxZV{mqpz}hQc1v&J)c}B{o|x2v>bVOF=hovxYR_MdGgCWkSpN-?$=ml3q-^_pK2} zo=$Ruw8F+;1=2Zt(h#xD07Ni594{#h)pVB2gjJzFUDzaJ^v?JslQ2%F7jXNY_Vf4S z94Wf)WvhnK=exge7D?HYV+4acq&W3l*wr!Gg#Y=WLQO_E#^Hv)T3K}RCqw@-d|{C`Eg`X9dD-<0AJsHH$i(6%IT!}j8y@A4HFY|H;c^0n;UmvetDgrp{EGW1Fd~3GVBPl zsGtrtZXeiR3v)cKjRJ=_gN+v7fY|hK4ljSaGXIyibN<=8CefRJSrE1Iu=Xk1H*VdV z{1co~HEGl{GUMw;)g@PbKApWXjV%Zd_ikcoaU5@`=;Ozd+eI_vq1_V=z#rI z@$M~8#Pb8TH0pGA^*WwiG;Mt=lJhgy044C*&85nyqrcixe;Km2KZ@Adsgc=?%+;i$ zr67kh$KG&CJ@>%2ingCXrw!c}k$BF%A5A1@lPSXVRm)hb{;vE4({afBRhyqX!Rm&y z$0U*EM}76sn{h(WpVdoHbjLvt4J^9ZQfd#}Kg2)7K2uSix#-A`2W18NbOERFf;6EK z!$mBsd44I+K*_hi?(qg|riI_&=uly|b3a{OMIcG>F4I}PNBu@$VrTJ`n^XG(pxy*a zzF4ktp?b;oVE-k^l>H2xTaJmDF>1V{b{v>J)@7EY9jqI%Rm(y;b`w9`+MxCiX^2 zhno0^z@KtdLkvv+@lO33vKs_9JB?29;9&U6v^FjH#WmQ*)Qy7+Q2^XpKKXac%>-Vt zCB)o`f8qNnhW^Jjdlu@4085@CAeMppR`Bj9S9Fbq0~u&F4(pR7Zt5^I;>l194g4F! z=i|9s^FeF)`LW-G-mDcPdHiJUo_)kTY!0f>ljNN#-L%0+EEciT69f-qh71Ib#B`5P zmBwDjK)f%k24fTh&2}d|^)ORmn0j>!Hq|CZ8Pey+bQ=5dAdV%Bybaw3yxbJ(^Cu-4 z!D7EQawh4qiJz$(o!2R2LtOCIv%yh4OK8zy*+#*~9BCmAg+ST0l9MLnb2q|*DPlRV z;#V$Z17H`n1q*p#s}1b+U$KmcAAqot z04I4Au^ct=1*3BKOnk25ukYEw+k5WSu&w999(JSd^3r zmUIse2|-Z?31X!pCI(4bypR3Z{oH;1`L*`l%WCm+erYXG-y1bkm1VW0)hXgwu0tI>AmxNr`XC#;hI=zH&7moOr2*{y<3lcSaKq8}- z-+=~JbaV#_cn3BhW@>1_f&$wI4HW%FKR>31NC@}nuLZi~1wy9;n}$cEIy=4wz_k4n zD{6gTL+OVwg$xiA3+?bL2Q|SntaG3O16B+tvNepWz%N9C45&jzLlE)$Ee6XDp+mW# z9i6(mxR?NSbUAW#YDqLX1MwJ4Yz4y4z@1(JH3NTRV-y5Eg!*K`MkGKgFo$*j48G#l zBIP5lV+4Y62oN!Z;!q9a8H7275wt~^S5pVI;2uKq4_f&Fa{&I~#s$iN{JM4WH~pms zllYPk6D*XEn?(xYz%vAG2;=Do%8Ih$m1H2p1PW^SK!7d zaTMBCG!psg9o1)tBQN)VDGUhU*7{z=|1a5ocnNbqDx%9bi30W(qy*9uCL)<(T%N|N zp{Xj^rUh?a8`<@MooRrYfcWZj z`aseM{2utPCvXl<@QBY~^)^a8U>dSFEndnbbzP2PJy`~Iwdn({?H>rKRTv(J5yv*{ za98kk>r{QCDStToA~Z6WxweJBN8e9z;SH6C(n-dTTLcQro4mIV+i&mFR5(v)xHN!h z6DNEZMmXSy8bG=O3#NI^`FI1@6VI=YCvq~a2*3LguHNbQRD^F-IKr2vOA1kqNpS+q z#K%67Z%sKHU9@D=n!kOnr6kGr;jAb=p|{~rjSnOZ5jzDhGQ{s7F?n2FJ=cmJwapYy zZhL}`zA`wwU|H68=3tbmIhkyIsS4HTM`gN6xU5kn5A}tT6=q@43Gjh*sUW>wns{IR1333a;Q&eiJS}c1`3XanwDMGeU0UWM(@ZBWPo0R!X0>) zKjv?Fm}|KBk0Fn?q2xdV)eXccdRJ*7V)52WGU-AV@BvtwifzuC$ucb#2d*(ev$;n1 z>4XnfZ+i3BqmXIWP=0ptV5Fy}H*HdmD2X^!9>8rUt5F88hBxAc=1d@E3=%5I87hx?Pn+03x2YIa zYkpv3NcbF+*(7h4ks1tJrA<3XUIW-}P>@cfXIPP&XT*+@2HRtIMl5@9L!+x3oW46T zKb=HETkGS^KdK(#IVCHaX4>UDW9Mf1A#YSecmrB?*jh5ZgK@Lm_Y(5Tf)$ECca5S+-|70@ zpEH^AuU-*nqdM)aYh>=2WIx~bB(3Ns!zsz9njE?Gt_-?mh;AAw~50HG6C}LW-Gxx6*j`A($p{dw*m~6Z(uDvMwMFYp-92od-N=-KK?Vgt)?mkmS7I?xRxJE z+1ET-`}9mgU>DG|JLR>GekB>A_#Om5fwNqH%r6f`1J}nBkDG=v^>`yp#(=$>5SKdp^}+zn9=X3BKELnTg#73bYnIk8$%zNl0{Ka=(AVY2o#{GqT<|yElJe! z@4|Z^FQQRwFR^v-Z0wlLbkOCtFC_u-^p1CCt|OzbA}P3vxU0?t8o;HGdQuuyEOP-9 zQ5l;nGUUTNk>@gt<3<)fuwG;NI8eVdb0VH(3>Ln?@PDR}NEl|{ahD6Aw#jV)(>oz- zlBg9Pt1r#0dRI={i=;jKb{COJ*OnI&uAjr&{Wx~EOJXjO$>POItJf|v!LC`J@Zwfg zGiQ9n!f`E|lY6{%&X2H5Hji8iO)Y%2K0@|v4HK;b23V1)*Dz7yB`)h+upBmjHWL4h z4~ru7&p=B6_9c!x)bL9xR0!g7rD%EdScn<~bHH-HJ@w2(u(?q z^CexFarIsby>1>KxX;2!w88&-h*gABEbHY5Hc>%`%YnMuGE@>1Nr21dK-67Jn;ENZQ3;uM6_Jt)D7qj>&U zEpQlR6>)I9=v-}cy5}}E_<2V)+k7U+tuofMAX4+1ALYPe+$9N6xWbP6H z@iOAJN9G?~vCe0OfIE|!O6rDJ{%`W5pd^&%eG~bkW_|zZ@8^|J1Seh(p54^Sl^0>= zPHzKQA?08%ThiY^J?BG;cPrNS9Vtto6m~OP+$R(Bw2N3%@##?KT$)sP|B4-v*yiw> zff5wm!%8o%$LgL5);T?=cwBVSdeCxo?7VQ-qM0M99sF62gJS>NxrtiW_@9Tf)ws=| zio?;N*No~_i2NgYSRU|QL;@I~7?3U$s|{R!0*AwO2V?j-C4{*ZM^Uc2wk>3ID)w9Q|7h+Qo6+0xQVidYL%V^n@dD?biFDuuf%`eYdwQJS_ zL|NM5w0h>6QZ2$v-m>X&#}S2x~+R`4|!vo$F-CG>kbqZ*AzDD9YndjN@5Pm_SXf#DkZL zmcY)~C%tFbjZv#zq6?Of(q621H*)`G!cn}X{x%JhGv%72HbaF~Be$H7x{rFUO< zq&EtYvyyLay?6#qEjai~&105ddI=u-4T)|hm`C3y^~`*0f{-5Q}^&@N_7ydqL6mv-4jC7(`Xk2+MHEWMWr@rZ8=x$Ti z(?f}vSdwWblQjGBK7A=Yt24z%#^phRe?fuIU1xE(pH&y&IL(SkGk53b`-hgkYZaP4 zA$u)w9kC;JC&>>K&|U&1tG6)3(dOG_q8;vsWbL1Uv9QvcA^FN}q*zxu3+&Jh7ij!% zy(x(MWiflv?e6H+iZ6yV_gYLV%ja#Em_u<_up>LIGKq6 zvol(HflivTVJDl-4&L9`T2yy|=sA(20dL3=2Ca{$8anG;KU@(Q3vfy@=PyT+AJt#;m)ma`&>T{j)= zb(8<7gxcm7uq}q~*I-K^@B>fxTdm6BZs;HIwVoqJD?+gr7yJF_qnR@+?=$uloJTJE zC@Om7^%N7$5&WtYO4&Ugn%VgBnXX|;5@^Egm4C+dmnZ^A*cOCdgQ}}k##4H)xUWCc1o`&#_IEVA9FQ)NwlPW?RQ_Ulj10u(GUPNz-x5a)ko6FuY5=8o;bK2Q>) zEOV>zyCFNJh0~<@$5BsvoE7Psy^Pq}#xG@8bGZyhF^1+cgD+0Nl|MSLyO%wBc=;EC z>S8^@X5^G;Z7jQIVqp(ekdzv70$49TEWeLa{)o!T z$}i6;musMlj@KvaDD4fQRaf@IL#N&O5;nSP?3g%7bvf$+)=mB^Nwj~Zv;6v=@kJtH zg+c6iF5G*k7U>tujT~4nUN5$kKs-UouX-z)m-ERxE;r#@k@m}U^3cxSS9l# z5KZ?m-!YVItqHvLG{I4~?RNHi-a4|L{l~Up0rV~N40U(;^{!FIz+NCW!8{35jJn7H z8c*P*7<~}CdIz3z?!$E0Ld4wh%#LAJT9CSSXL=S9!IfHU>cElw$prKo&pg^!Z%CA> zCoyeYM#219cHL{%up}qO)uvwFk%toZHyGJQW9`rN1KG&N4tBX}x8b>ovRD#06f?at z5jVzOlxNzX@LYur=_Pm)rO5XK*ZaBQ{zeM+FV(ECx$I16jt55c%q{;Ah0So@w95M< zS|MZO`B`@w3l4Fk|X`g8I7UjGtbmPud0+M0@p=Y2cA4g>C-=r24A zCy(5`g8G=>OPW?nt+@I8wia|Ir6-)3i^833(&k7{n_W{V2PNMXA5r;hRgQXv7Ea z2{zBNlF4<)kuwCu3?wv{AcWAOZIw=w0uxc_i;xG=~{%r9&8;H5exfZ|6?C)WPg{#qYrDc6qMV4HZTi zlydpqR3d8$?n^`5gM{2Ak6WKI>zkM4DP*<;>5qdB#D!P=q_yF~`tD7NO7Rl6Y4EKL z;OF2wc~VUd);EhFz4~1F{+DfQaUEacSh*Bq?c%mNBM9SEaoA|RIJ`XsP%V@h`{I2> zduSc;y`{I1kG39}b(T^XV?)e35%>ojKtrjjwK1BXvsx@I1b0Ilf-EfwK9Iw=87D8h zisoA_LW1y;dwk^FO$N~SESeR~QEb86DLGBF$&kyqEz(i$M6$_u0ao{6T>aPahUpD z(<`FRkOLk(*u~uLnw(rM)I#jQMd&)c!!!O6OIlmfx7S`%iW-w!ySc zZAaY_Jlv2U&U@Wjod@7ZsyjbtL|?Zt9Lgg$H?JwCKMfa`LYKbn>*U3^4gp3#_D+LV zqgZ<$;+yL{gJZ_&Md5|0%PRZ1o-*V{e4uNI>fwX@<2twrbF0MX5(&$xC@k=Cxki1{ zB7^VZ6{UvfQ(#7YA7~z`Yd%&i(2qvf+r{y%@e92}qO&%Pcy>zHrA8L&kJRqrM{&f{ z8+uOT#9Xw1MV(pUoJFv#-U};Fnn7Hd60C)l_0s-lXJ?%Q%VFP z{siqgU=ZcT^|1e4Wx*uBW{ylgW*UFBf9bk$w^%jGD68!+|K{d{zDdjnJM}5r??X2X zTa}QlXcSspz-UCf>^6DCu@vR%nh($Ybf|V6sT}jyAAgRm!Ll7GD~RkcymbksE1jv_ z>Nrbbk(m&KZC@H4vFLiXcs`6>E|Vh88*!~)ot5UJE_~B+co6v|9e*-hEvUh%!e|P~ zp9yby35Pe)HS{f7MO-m2ADm|xC+a9)AGA5S!2~Jn$J(=GDSR9vQqwZ_f?WrD#uEfZ z)p-CLa_<74w(=v6JS~l#oI$3K*?@Hx_vYT_4}qo5{?wCu0~f!XDRZJevZ5$4#k=6i zmnC8KWe!}H?(vbhw$(-I0Gg-tytgvG5ko~&GJI-nhK|L9cZ9vR>_Umb5UUxfwSp3v z_~ttdwHMXpRUM@e!%|)ox-9FU3lgrKNvk1dt5~DNqK=h-!sjEri#u1lac7b6)-H3J zT-Q4Yw56pl>$fs#FqO2+EY3Rc$yCwN2;pTLdt&677Sbr$K~fSb<=BcR<@L+oe)=KQ z66iBsr#Mp#GN&p(j7q@H5;AaFj!yOvw##en<{ocl4qoFv=&@T#fOKL`KbrNT@bsg6v6W-VMe4ZY@Kj%eI`{!0}w zwllh8e-F1~Yed&-m#WilYR`Z$JNzFpKQ)7kU8X5Wooy(K;mZlC_kQ0A8#f|Bm_%Wua`KLgg;! zweU{>MY!QOvG9aMeE~c6Y+ngLow?(!;nC?xwj*cmtQ55w%I+M1q4->Xst@ao$Zf!( zte>-(q>&aaqWd&78WVa|@6p=km^Tzhfu&SV>lQ6cO#C^JZ{OHl?5sLx^x6bxSs~Tz zVsopv#T=8ai7SMN&)Uvr{6*fN$d(a~_%K~C067q@=5kEjWlCv2_~A;z%%@ztuc?;1 zYiLlDi!AM2>Bg&sgb3kg-di1a*^FVzBp8h2Db26Dn6q32Q6ydefY&^=x1h6r{@~+5 z#UiBkWTd9#*b}+Q0jyEL`*8d-=T$Yt-u+mx6QGxXp>v~|f3${!eGtx!``onGFb!2; z=u+5-CBw`eZdom8Zp(YaiuMya!Vc69d}@H$X{LrBlo0%8Hwa zG=e4m#V>YL{@)Pm|7-2~hgdUn{3qZ2hgdT*aQxS_mw}b(|DX0kvMkwtvX8SRTuJRq zy*Q8b7_+~4&vIc%M`MnxK@t)n7FTIRA`&&EvL>;9(s^nSTc-Rr@ytH{^4@yReCBYT za=$j~u6ld(v9pK9m)Au=Z$QF{6xIzOAlBjJmDmCS0_)YaL155;9h|TU>BZmmS|GH7 z;v7c*DLnc~4a5S28$58nV4&HN!;c1X@`?ix#_c2NnjS zk7v)93>nZK*yIM{mfu6EiFyS-#2?rV6lotu%>9tqOP<_o)*W0gcPOAdsv$rK{@tzH&J0En`O4oRZ2vFTg0otdx4$cY+RO5JdG;u( z*w55_FCbvHFnzB93w+r;v4dZPK)+G75Gl}Cb%omA+#)gx5p?+Mte<=7+)&T>x!5_- z;y?oX7a9%#wxIn!xjQ3&ByG|5Ab+icz7!A1As`=wJ%MW82=@pf)Hr_NQI23AgMrnf z;CV>Se6fEh{Rt5IK=-3Sv-WHtP|o<*?=9Lc@iBY7U{5&*r)TzRz3oJ$$na8R8X(sK_I340QDIq7y#ZunasZkk8;hU^ul_rziQR3 zlc=%cz87hKGJclpZ*>9gzDdFP?`E~Y%y=yJ0d{@=*g&U%a3Au={o0oO!oBwreyOGW zqL2NHc+23C-Z!USH-CK7^zk5!`gFm0%)IN|fX)ifehz7Bv?8Cb<3hHHV|K=UX?tGz7#zCNLL0m__Jy!t?ye)sF!phN}ZoZWc zV#AN`l5ym4{?e5DJBW6E6{MvVp#Ti(=>ZaaCEO>FAb|8c&V%0kVh{P1K|nNPU^Bm0A3i;kIe`+!;%_cSuj78-~1TwCj2Y-n>f{+TxT=AC%4_QP)>x z0?e@Xt;p~G7RWgDKE2|K1rZ>;ygvXHU_Twg!-EBv?PWrKv@tj_=ouf9Y!X}K?L#i= z$S4@(z*1Icc-EilRn7PN<}99Dq}-<_sx&w^hDhX4cD8J{eSPB|Gk8HJNO-*f-xxl6 zD6rrHhbblW*60?(p>5InV#{)V&aS+08tf5iysR!=B{SmhnHT4L(7bOqINrT;2L20K&Rqt5-^NIIofmw_nSFmPxU*?w}S#?vXQ+ z4bxYE5^(LU#B2O>^4PzZ$uv|Qw7tqk+8Ef&4?|hL!RE%!F3cP8R9GL$5gW(E;p*}0 z*QGzBp}>P;ipY+uH$g!39jZJ)M9eE zU}mmXb-nHPSFFLEC=-qJR6)S5-wVM z8Etn~UAZp zfWz@p*OH^@H&v>0*@SS|=qA;YlLw?z9S2a(3W0foaJj?Wvt8uR%1=r2K4SHL<#?xk zfe1dmkl6MYM}9&q@>fshh#hEZc4%QVs)w3KuA)2)$Buht+&CB@I%r~X4)v1n^IAm} zAJ$~rxx;SGcuW@UJC>-b%JYYFha!4o;X;km1Qu|xkSkmLbfRk~(~l?=Qk?$EHuhSN z5N!Y?m85(5(nbbCR1N%5<=ZTaBP)=BFqTj;T1%0E9)l6TU%2(rRSYwB>vh$tYMQn7 zc0*8p5Phgj$DqH9_RVEAyD|9_eoFSCdS7GJF!u+S*vD*je{Bb{S74DzI$rD=UJDR9 z!)$lQ0>^N=Ee-yvJHGb8BC8%eQ=7`y~z(K2*bo|SDr4IW3T_wHYg%B7Uj&$RyZ+;gSfe)LW2 z*3&g4yyAl3P;YB~p91lzl`pKsRP43}Z5XR%i-ku|a+p`Kk0fZC;jeWy<1UUTExuGb zEs@%7jUe?fzBA>2dFRieTOe@Ol2vvf9M8c9#|8fx#GwtfJO7->?W^%gWEw8kDi?GT zR~D6dr*FZ7# zdlRMFU$|Xr{R#T}G=~GQSbkzO4j1Xa#MYA%IWlO@+m_S9Y!2AwVaH?j8fABx6&K-j z)O#3gq~`V(8dUC$;OXM;3wd%fC|;|VBUNDv@(Coe4IBZcAn{BBWQJ!X7bk4YqCFzu zlh@F)RHpeI{bt@!;2ZJ%r725|rmJ|YieNMswK-D3sHUl#?09{38dpXWA#u$(^U^?P zt}mmu!@z4brF0xh=a;cBmMyzH*)4<*Zq7RBYp(!Ul7g~HpvcTj3W(h#v)t@kgrwQXu4u951cJtHC>7#s~jG0;aIwD*ziW zhJ>@cs|gCynF3ZzkVjkyBevUoSGVu3(3`<&1Bgrjn|k7$5*h*By1{Ygs~oR6mYPhS zt|9d-{pA5cADYs9a+5nIQknAnF#Q!A!7>5S61N~)gYIXYq|Ha{7WR`Q&V0wwjya)5 zoYLG`yzWK_->@U0j$}4dSq^Wy@Z6I!jbYeQ7~h6(2M>@$h7g9Hn;u}4$r(=#l%=jK zI)H?(Yg^kRk4_$GF>N@nC23L3gw_Olb_W@H>bUhA!e=%(A;iCNk&$<-^1kS$wUo4z z#nqu4hZ3roj-4N%!AVdAQ7eRUepss46lV$a)tJvm*nG+g`X6VW^qfv7%uK1K#TRoP z=1mE%U8#Kz1-_I4rIhG6HPBg%i>2X2V~B^j-ecJf>Y7V`m~gHAsWd?cXNMsL0ExKS z)_!=i&R~`vhB|1rjK~&ARF=1mMd&9H)lx8IbOF9DqBaGEcd5g*Xny&QKiN$HvwlNj8bCa z$7|#FPY2u29BPZaTj$U9)l2}|AawLtv2Es-A%9JM(mjilGe?txTOL* zq39mlBG##~3h>5_o4fMf!7VPP=Jaq+C#6@t;uYV-Hls)ADI4E9&KFu#ElVyDj9RN_ zMLs+S3StKQY=X@T*>}`JHi92NX7s?45)bB&dXb2>LN_cdQ;kQdM-aW1~QZ% zO^jFl5vm#pMdy@5+b}0Kw-?@#p*KRUU_qu>X2B_>={f0`Wp!z`U&JPV6@;D>jkbOS zUP8>*Y8w)r0kppl0VZBoSbxsE-)Wc+>WYCTwMx27gPbJh3>*c$%-#v~#Eq?7PnOUk zrZZk^)RYyw50iMCMgw{wgWttHPe(%08vF$LevD{j48tPyvuMWT*WY zU<7#WYs=y(xva0gPfV$l2i!s#h}!r3s^|u+yFG|2y~T+4olV-jiWbPOaT3H#&y%wqmG_%2%XsLgg6+VL_z5ng0SUIE^T zpb{C#9GU1=-Z*qrzv&u&Oo}4Q6!bY~%W)-Nzv(WUk~*H!t#%J3)2Fqa*6e-M)E5jj zQ4&#rD7ME#|wpdwCe$nRBD+9*JZ< z(&K#iscNyB90eckzh5z&K@=rFuL z%4I>-6&&@Dy+015I_#&e9*dNW^r{@brXSM3Luj!OO;>&)i#8Yv^6n7`Q8@nHI6d|8&J?>d?f6viG`$pvyD7qoTxh40UU3s*#`MM?~J>d_W-(_c@ z^U_Ufr*+irdHZ2iaO%wt<8T&e9|@p0vE_8ZzK=WGdgT4;6&)HZgC;V=kO-wz2&v30 z++DpN95WuJqEU;evTM2JcEn6~!=6lRe(xoJGLR;kxkA<%J|*?!dFOYN6VN)@Y@(uz zw!SgZ^)nn#{I0VeB39as`nE!KQfJ<^be>hkaAJX->{eWpJK{|wQC$ANC7+#clTJN_ z7-ZYs;N&tp5D*rcF>id7qDhC7vBzC9+4jq&QeQTQ-^en{us18okBw4vTaLoP9pECNE7g>fQbO`im1WiyYnA&K6n&w>tyT{%j;0;!VN8qcMCWH7BUcU&Z1P?# z?fFjR4>O(85%yCDk+$t6XM|M_${33TwP|fiEhZJhg{1SOW}m~w%ahxQaZ>5@Bl9}7 zWc>lBa;p59`qv(!`IMmK4=-*=5{!V0)Kzj^u}_5+fmSly;lOxwFAXW&OSk6u_0oLD zV;reua!$`cQCj@fK)llgvn1Zlj`>T-zs;o(_>Q{FT{1^$8r|tVd+u&t<-#N6dJ)i# ziA>#}L*PeP2Up8f6ze#4p@&WzHXM&{7y64I#*X<&1FmaUKgK~>E3z-#gJmreJ&NFM z`fqBgWy77!wLVB-!)kn$j@APi4ve}zt)QYtBE^>5O~Y>q>4dKd&pTlonF%hjwJI;RlPumOxGakj@RcKk^gfFE zBKR;0S<}O=v$laJBsZx-lJbmj{L839JQ5(O+V(aLH(FJr!Ph}96MsL%Vhmq$Em@sk zk4YmX9=^F&F6ikj47*yCp@y|pZe}!eehJf>q-^q zb~|_J9iCTaI&}Y~w)TF%#*BOARvDYKcUxSB6BNr8I)zr}d$k9=c>`sZ4bMN=4-Ms1p~>i2IAL zTkAymI~A!7We)#bs=3qPB_*BF5ft&_^9B=z)Q02K4pZ*bNMXl*GcK0FF`R*#4(lSP z-wl{E)0Hdyw<9LP&(N6AHPTS>=H%&<&NfRi`?mB0{-uTd6*S>YOg5U|hmyt4AsQ*m z%8ea4J@|wq7T?!gb_ao@?(${28ocF=VZIb8gEy{XPSVmDhe12gbJ=%aJ#SyIbn~Y@ zC?PpbTdF`fUsY4i=HpB@!`rbp6%R){{(j2K@O7p*FrcW&$9&lN2mZaYT(fMb3feiC zlxiVMH*o1TogyqkU+#I6+Xu&;z8m7x{jcigg;;{6j=~7Q{FnRgxtzepJ-RB^CB0IO zQ-s&&!2Htg1Z5txIhA+2`9MZ>)YS&n{Ytfjq-xIWAyp+;DoA9i;m+zx;=J_S)|wRK zn8rcELp6gjV`E8=uXjSZ^1nhSd^|SxAXlRxKb!jQ5YFqvJ)1!2NlTvO+DTC z#pyYpK~y`j5~=+ebOXOpu7iuaZRClHQLYoD(nI0{P%0ti+f%_`Qsow@+rO}Fuh2^< zdhf;3digbj{YLT`kI|6T#O_g1VQX^f8o^wpf%LWxs9&?Oi zHMfo5wevYf)A_;lw72FGeEihsLy;bq92_Rdm)u49mI;|>v+0tmIB6<=vDNn7^owa* zQtwup*!BCI5qfpdc~!FQ(Dqk$2|IC%RXtm*s2~F05oVU1S)CD$d7pOb$EDU%0t@B+ zv^9raZ9SNfIW~-B{gY-6xE2EQ4&`BiD8j&c5;iKc*|jOe!{@1yoicsgnj%)9 z6+;bXhEiOnI-|7z7^?Q|a{H7?|1+*J?&?YNe!#CbZS}$nFUi{TICYFF{ z7AoejS+e3HEM`f$gbBZ$s8~jSG6$=jskzpTXw`WC?olxH;0{^?(Ao}SP%oL*bPj`e zJ*DkkcrWx;n=f(>My=rH{;Q0)aD5>53oD$FN!e9y3YsbA`%S8K+su%E-V=6wx_zrd zNB%FH2>zk$#j=1x-27g22}+HMEYLZUxgHtC+6(2rfM!yDLz4VBt;3QS&*h zTE@Z%+E{S=K?pCFPD@#sEoqchreaw0ww@1*!Q)7=2hF=M`Jl@MEb!m)W?Zn-XULMe zMB)m)1i3ZWRsvxB1r=2W)B2}k8Q@~msco(vTEUL1{-7i7WCd?I8WSXB=0CHt3(>^X z)+Ht=Q<>FAHQHU=jO9l(ZL_>;vOW6p{KXbjl%BT?1@Ex>Jamzb&PB9`(VG?Pt0Cti zo~1;SE=&m$Jy3vjd;a+2m4hliA8X*_& zWBoNS{MHm*n2s{G!d{6?(H=mZyFvQ=Fk1$r?B$Go*#fS>f&FoX5u@H^B_+O8m7tK~ z=TEcON8~Cw^J%bCEo)XnzesSwP}jS6Cw4$2rq{(O{7ey+r1%nWS^{eVs*_sXM_+WB z>^O6XGq)6WMT0Pjov?i#B<%2wT@+BFvamqQn4evmd!}+$(fxaI8=-Cwl6LdULwQ%DYZ;XYwmXBs>4dy~AFj@vbrX%_h@dc*E(mVq{&b zJ0V4999-bJkd^?7R;zjqLy%yBjP&rwK=GWfw@8VT>DYAmNei_G|MU5p$J)-?K4k#x z^H>V_*q|dj2v9?V|M9)^l&vXX@ZVdN$DGnjDl#lL3yYk0p-}fT{v0|-@(5{2BKY-jPhSJimdpL70yV)^|^@gx!YTB{hXCKUJ?b0{00|7E~ z_r7=KBGu!_s$^`<`_=qO>C3-rFx(V>YMIT6h(Op;KT8_L$1=kTlreOk>aXruc&cKW zJ!gu+CQ*<@1Ghp~AfSK1^ZMHQ{x9&!%au z)sd}M`GJrM|2dF?7B-3)Vu8u zMbQS11SSIPoTpCCaq%FEv4mB*7w8p@!fS`^~9-iY)aX%uce?BJ_PR9(q z4gBuGCNDmOO@0%2H3HK2CMHy%zPycm6F$;4gfM~j z{vTcyYr-meRR9PP?k}dG4i5vm@BAJ4BI>1|g+12;`Min%Km-qsUEO8(fqzawbvZ)0 zqRUz+f!+ZvHVx40ydZ)6J+mtEJvE;Rad?iahD)EkT|;L9L$A3uep3*nYri*@fWTDj zF_37d*8oe3Ut{}nps#AjAVdFu^JaBHB$%L1;Qby3^10f*5g(hoU4o<352oKgS=b}6 zVQn;kcwxeQ^u7Qba%Jo|Vb6B~;0Hg*Z>1o9fBfLYG0@E5U4;t(eZ_i}2N8XbCdWMq zy!~JMQilQo`g{3)`80GLCIJIoog@<- zcR^Z#BK`GKb>n`Sr+quWe%HRKCw>M0X*9)_CzqeuCmx}HeB~*i&M#ikea6-3{~dH^ zhE#07A6q5nw_N9Ce{Owm?24XvYtsWPXoCUFeet z;lXFdoH1Q^D0u`RuwQhSb`^%yNwLAlgxFrfx8sGcM@du`n8Q2E&YzMtygui|g!vtI9E|GeJeNuWRK_*1@pNPps1ZSV0x`@4APg!Fxp zZyGjy$9s3-IJVH?fTd;h^jz^d)Og*zmGk=!h})Pbm?ydoUXs)h*pKf=F3CylnQkkm z=G@o9?$A+W)Sjj|}8k zOKVA|L|K7=H{FWa?4ztIv0(BYL_nWBj8=6)6$EixPS|XJAvshB~LI{nV-?tOzTqc9p=Ns+s{|<}~ecr11BaU90 zpx|Z7oBPb3yVZIp@v3D5Nm9Kr5nR;Ge;XOJy(ca>iHimolPN-WZ$ere#cxn*X^3Y; zOhFRud@ijKE!(mg)9j`3TZG(%Z!0V%kTSqdkg6qx0s~E;827P2Yb^G{W=@_%*s&->TH%6vLfTH4MxfB)!BMByzgKk;Lw7O(F~ zm-6GVBmEJ@TcfpQ`jXWP z_3pDLbJ>WW{N|gJ?vlHDNnxXdl1&if#mM)yMDC_+S`_J>@!LqPzvvYiFW?(d0kHo| z-feVswQ%~K#DQS2thidao#WWTwawF05`4OGjY&1ZqJJyP07*Cb8kbORo^t3!}~)HUT;FGPNy z%C@5ipEOO|u5%D)6V|-iLo8*Rc#` zD_^mYZ$kT8=ceM;LLth@N2vXUt7Gx3Tst5c7WGl&1Jc*u84bN@^Q)yN;W5#i)c}-( z|Ib1|B2@30s64 zX>0EEL)JW_Cd%G3SAIXTiFU~4y(qMUn@-cH+oJZ7>D%A(l za(387PF^{8_C&kpoLZw51=ajlp~S!4W?I%uSx%noKh~N(^d!!$kdn}y!)iwP@K_iQ&kZxF8(Jh;v{rJ#|#cqW&?BR z#ncl^M@+pFLb0bpsq<%4Kr5Ujbfjj}IU3q^iv5u$6)yl8FgF$_=A_)NKWhW!4v!9)*J~FN%G<$4yN6Nw`;GVAD#X$@A>oY9HeTRkx!=9lkZ` zK9u-!RGE$`48a?MpBp{+b`S>~tzxa1s*&m-x4&Ir97&pn&8t$`?XStHCSu>mmIsWW zj$h`<($1zB@p?J_gY%_2UUP^2U% za1L_Ebk2^TR3U!q6jcb=)H)e;*yXkq%?)ZV3)wkC9-o?pce!Elwzc>6ZdW+PORXbY z%^i)@C#X%04ssFMx-yi)7QK?mxk9N}3_KG!U6}gsD8oq=e#z`uP-|Q`6Qa|1lgddM zF7xHc>BndAzG&}gb^*DEA(Ft3Ea**X9O6~W-d6e--KG{f&6zaHVcw6D&)QOXqi)%{ zg(C8@KCr|F9WT>&9Iwqgb;Q0efuR;$xv&8vMh8!yuF0WV_bIQ<=26=oG&>#Ubhru3 zA{zrIFgX&}Q-_@7)-w=9`m zX|W@aQ!|7XHb3VJ3ppi0W>;LlIT{MY*($R*-#yVhJ*pST>?DBY0=^SgB@?SdcjeD< zQJH4n%KGp-Gf)gU7y0=B6K9t_nv8vNjAd09RJKKSY#QO7-MkY=s%kC3%N)zrt(@!6lz zBEM-gN^3^VyTUVvQj0xrU7cy0pqQ(anLANOY6xf+OWbH$6Ms9h2N@DdfaY$@Hpvl` zT^|Ou#Yq9eM!N7NSTn?b zHB;=LIReB@l$t2$WKD}!gS>re3{$G!-Ma2ndaH^x9-sb0fgH?ZJISd>UMLd_A5w8X zrk96hvqnD|v`%8~41VIfOxa#$2}w4cmNqsycW|>|uNk2HT#r zEGyPrjFl6+x5R?%nb{pvmS(cU<16xRS>zgf=ky=TJQ0Nvi$3uap?|PbK?kALVCTKV zLtFQx9k>szrDFu1)QMW|AXdaIeB)3eBOnK#79p_moo-?-9FFq#ZB5y+o3*6hHmRKN5EV>g{xY^4+JW|~ z1^Jfjj-PlosgL+F>aH5!*G_a*bVy6RIa*#aQB^*@t?|~@pLK~Hp*PZ-%N_L$P3CJm zaWXD%DKypVm+$*b9wMef_3WCvStgTW^uRm;UZ06NR6c0xz6qE9DD;}KoR$@B3V~Yw z+c2Q*D6CK7ap+K^jC142qf9)Z`!qld*QEsqA#4r2ZtBs%AvJanb>;Mz*f+p=GEAsE z(>hsIN19TSejE(#nD z!Ak4$CimSR=ws81{x=ZO9U|FUN4KAkE7JpKzH=T66a`lT==tI3zA~|e{uI_{gbDOd zvGElo=W?mD=TOt}%IYMhr2~b~%(^`6EEjr6MlKgvtlU)mN=gyt{tkZ2AuN|W0<)T> zIH9s5v_y8J_i);#FpVUdb9{$XzWbt6=TC+w!x`7xJ3;MOf%Ut>Mc`}Xud=i-%#7g# zBu>AIzEq+2^pBvY99gw~5gT2}4$a0{9ZTEh`^_jNM@1siJWgN(zhuOcokt4Kbdjlf z@vY|&19dAClqa0Ji+tOC9kp4}Re!zq6kbXLAo?jL1XmyFE0Jzw=f)JGdIan)gM8Y) z@`vLs?keAds{I%(h~%%jC$MQy$r_{9M}ZFmvB7n-%3)b5q0FJSXh@EN)5;j_$)xr^ z!^35yVYbz=vfM$;NMft%4xhC4MD-3Hif~rK}4s3@ORw z%H-vPUb+5>TCPX!u&BS6i!M9EVvXih6X(&K%#)Qz<2nqgz!zPK_JbTB13|2Zr*Tyq zr+pgh+Of}dgHcJ>XK9@kgX~LW5j{J6mimbm7j)yUZ7t5*JO&WBdG{ZE9(f$QtGG=g z5QO0z)(^sH7Y^&4#X(+nm8op?`~Z$Gq$y>UHaECxyEpGn!vTapDz|li6nt%bR-aw- zaYuq%Z&_a3N%>#FJjl@f?{1`A#t|s95G50$wsYhgT0sz{&~n7`ZSC8gQU%UVc*G1= z69&+|+{ta{+U8dUMr}2%TkGMuo3+fh@Wojl^2pP=L-GkopZ|S#Y)<&3TG24g4UQRV z7*Y$J5x$3~RYa4tH15${@3%hVhr?TCR6y%-JhEC+rQz)j;Ji+s z^*@gf6$!`oX>0@SI+kQ{mZFe53u3NN0Ue*H z!#v~aA|Q%j)XRE_m4-BFtJ!Op()*?cP{eLk_diizs9(_X$(Bdm5^*{Kv!R_1l2D#6 z&8y6!6x&JcE7($iO@9O%q!`B54GOld9gDA%Ol80t8RlPSt5G-l+k+i-dw)CvKq2r z8QJtQll1D>;n>vZ_tC2NR?`*8`#!9gfkF*8@S_v&K=3+4ylyjmU||)f_#5woNh9*? zzlPgNUxaoO`?H77kLT{(V7bN`9QPBJoh>whN<|eD5lt6>o~ggJN$=%YnuK7w{aQP`q3`L`W13YR3zlS{fEofslb?RB4s z%ozO;SjrHIME~*oTa5@@kT#i3R$* zh~6)N@We+a<7x?)0BBZV!VlK^Z!4JKtkNF?vY>gfCYkwNC%WCHnu#<3-lZVBC2yMv zHwDd1r=-|Tx2=oANssfX>#UF?f@4B&)Q}s!#U@uwDcr8`y}1g%FWWGdsE>xH*LUpA zY|{)u)Qc;UlO=*1>_dElXjCOt!qX~5J8fqdX>$%KTBGD0&=bdno)aBI#gFO5Rd+be zXMG}C*~(KQ&@qHTwX$uJfAVu>Cy6QD)Lp!UiK^Kas-_xKP(1!J2Z%6V?fmWCkcAX~ zfYXJ_LL;)&757h#?*w5YxNkExl7@|sWIv#-IN$L_OSymd8^)M0}|7Ti=JU?Q!p<7 zTnSvNi4JUjTz5ypcgeL3~b8f0A$)hooS_%MmOxVyXMQcP4)3*2Kz*HslVAxI) zFn6nSDj~(&AOdMKEr%AqGLweZBAf!;*vC1tta{|>^f41FvFE-Vl8w%YG_9wT9<1Up z=RPZ!_Uy2bG%?S0D2>}VH?=Zn;{*V3;~psk2^CG!-^`uz72QE+dO^oUUA1~t^ZRxc zrKg1i6rT@a$s zA3DmJqMmKztFW+nOONI0Fd6pu?Vglint`OJ#w$lP4bvwox!8Tz2 zNcY^Dk$b9kg?w&Xb!Z+fa&&uz6pN%@m4nt4ya7pV>$vbN$qca)zc?xj5*;IUaOUDt zItH&@;c#?V4w_*`>bk|z*33yWd3vyfM=%a!K)Phiy(%!TeUu#f_SAXSPDpU}E@b#& z4@IW3rBky}-UWd_*25c8HFJrXK)GvOQ3?8}t`r79P!+|b4D@ZQX`kBL%b|e@4}lV& zX-$qCVcgux1Z#{QB*yBV*Sz~sb0Ix++y4<76DOaRq({_gpY$UdA_KoCxu< zs|$%Ui9CcTH^0`Q+ zzoBjcusjzeko$FcgrQg9=2H)QmBy*Vk8L!Jhu_pujBci){{+wBU0goe3Guf5RTux{ z*SQ~S-FzAZ(d1B-Va5qd-6@Zgh4RW0xm@iqYfKK6)oc24oM+ejGb`q4*ziT36yL6% zMMnq~i*Dr(pPj~BxDkln+d$>9CFSN3v)XoRmGaY2eJSoOYrE2&^$vfjEHQgqG%^Mw zCwx{AjZ!vuB<@I29GRJSmo#UJxV-+F)iPbzDQ5@vRL>Zmpu|APTBmhab69 z`WwoBCO605`vkm3#;s|?uqrb{z!L&GgQwn7Y^A?v$URCJ(z3H&Tl@`@#;P(V@%vof z%(e!3K*lSJF`bR;^_4J8Ewan+KyWY(kH+b%E_`4T-6#!Y^&2uH$W zE)cd9#U;0ij*AbRk!1SG3i+0ILyV404MgNiT!vH{!Hb{ijCZ`pOj^cabvYBE&Ckvp zsPZ5myKSkr*Srv*OTib(;ril6WQyORv+r$9SE{6X_O=XTMWlj?@cx?c1fKU$%yxMsKu^xA0)Vl;{_PiO6 zce2@wvVJSymvX+;j^x@18SFgsMd6Mt?)d|HKTdnN?GzCWiD$WJvRpaJx$~j3j2&=a zqmqJTBIwIS=YW;_MW0fW#z~X;KXwSbV0To@sb*$nDZiWx?c>%qg1)-rzvs&b#RLYjwq_c4)o`38 zq;x4}$ymco8(Nni?yvxhwHKlU`*#!R_kn7jna(g9`J4)L^C4P_aTiN~2s?#!fl9o^ zVC!39b=L`COgY`sQ48;k-$U$FfX&&;e7f#JU>)GkT^y)QBrPuYUate%!%wjZ=r>_Z zQa@oH`1VjNc7Sf1Dp05rqT@Zhd~|%c@a`NXY5FCl(}Xk8W(-1qHEqj2lfB;qNXMnx zN35k5h#;NCrFltN<9u0DJ_mmQ^85*t{sT8>`cK@PnVs{0i)EPzSUDN~kC67ib8|Kh z7LNb1bk-eIS@~*%4G4&6d)nU_j$nIxn}~C&ua6;UYK|ahfi#8y7id9(f>LJz3P~!= z-yh;Vrl;p5`{naTbGOyBhWDfW=A!nby_UC;jXNrUvG+GC0kAsMzvwjt#9s-(M8`;) zkO1H}i3I>CAq?30N^wS?-B~oA7e+rCK(L=;*RS|rqJvn$RtX4bcXv1F<#s#} z1UxyXEuTL-$`K3$5OF`AuD*C*T>_xzv+Y0rBFKM@z%tndW4$;d5D*iaJHRjk*!l_x z7_lN@?gO~}cmU2W{`IAB0=u{{KhV_==>G7xW=;S?KtexKFXZoPL=>NHe|aH-?VVb~ zK7{tpacdw0{K8gSe1tl376$k)7kiU<>Okgim`fEp!AGcrD zJTu69C-D{_!Xw5%0Sb`l?|~Qb4X*$|3V3<@{2xEA@Ad(7Bp?nUK?;B1HAtw@A2-(~ zOqe&T7(PGva`--|@X|a4fY*zMI&LeLK^#Q;^qbzj-Z*uI>4AYInV^^1u^&e&N%U!VpU`~K>(5;OZ0Aezt+pAFko#zkOA0N zGdf<7%yxZ%)!vA21PUmS0(zkzvyNXG?N9xG$!Jqff3_1c$zh&1XP?mj{oHMK3L5nI zA($U~7S2)4iRL#0{EBWlU+K@z191y-ZGWzoMb?X!^C8@Xm^1ZGdjM{~^;{{GV9$YF zS^NtHX8h`p={wKpyZiI){{jL*emw;XiUS7x90^Ck97g&^63j1sUI7)cKF)Sj+9M3y zrL6^v4g);n;XU`fo;Hfri2!~JnX7OG`uHYc5>Nn$=g$!WJVplqKnwu?to8!|2HGmL zk>97G{@`Z_=HlU*XMB|cbX@LU>}G-m`ScNd4s(J?aBUr#xR09T5OuTvlrd&6H`k_< zOn#dyqKtkr`? zpu_oa`y7CnI*QD3%7*jDtP}7`u^E%JG!C@v%dqBk#vU(cNswnsC^J;;TvXMobyhlsw+9Nh<*ZM>_9x6!}P^x`EUx5sOO0eV1~> zEDBtg2rkN#mHn73e0|Q3XY*_=5I!@0{8F9Gux5Ll?u|@RY#|%ra9rgzQ97i15$?eE z8@5XgM14A3Y6$d?#hLbJDMv6lMNj=)pN@5Oj}H#H{*6b{$pSquhla#DhYmUk*;Uc-LO*g+9~t;oa(xBg#N#SqwxbRs?d)Jk|%68 zSg)S#7D>lj4m6aJ#(BC^&wtixrK1%g9eKYi`qCXcstwt2oM?GklnU<#SjN)2->(LM zB;Q^Gt2K?KML^KQp{L+~)0?i(@?kemilTGi!H_VR^yIH!Tpdg8-ppBaa*XhkxM8ZK zVH-lz`YJP#MK@FUVH8HvHDT8-oU@zvJszWlGG}`bAvAO(c6VVMTGwqCxE+6V-y|eK z1;nP>5eO+86pOH@2+!hV5p2+kg?{(rM9=v)P%68_(fRMeSd8ojk6Y0zqF!m#^IF7^ z+O#RT@$=o7ybA2-&JvWfo`QGuQ9-ICy4}QCeeE4|(^@@h7BY$hS?scVjJUu^krWqUYG^KVHQU;&;H`no7<8mQ=)iw zD6w*H3s?$TjqAMuXtUeEDWU;)LS&Zq4lnR2*Z_0}4={vTDV7U^7}*M#(cZ_kXN}rF z{xNlWu)tTMd=)8!e-}FCNWK47ojF96DNYw%?qT;Pfc~;a*i3#>`7K z7wb=@_lvhANo!k7iKf~C1lm+u>SFoZUu`{huRBFX@3MrmaO-pgq!S+e<}Gg+i3_#P zj>pD2XYK!L6{i+Wct!*PH%ep04FwPK(!E0>r;ggiG;aI^DG>i#%vVlyY2 zWE|t$5&@t^9LU@oR;pM?^O;i^JCZ8!uo<6xBeLigq#8Lzj}+s*mz=LB)cRdqW)q=` zgRdW1qsC!gN^@;8UjsL~`;(_WBR(a}o-*H)jz(Q0t~cX@P(RiXNrL8Y=!D-8rr5_@ zs_WyEkD8Chc@Vw|lvAmH^8E`#4?GyeGCB29o(|)#$sIz(0q;PAreXhjZ<{5i-de6l z`0v#5*G6yv@#XAsC{r0IHMea&(E0!Z-j6uFsmFFOXWH<-f_^Rc7R9ON-QJ4KdQ;a4PmwwcRNjzRx^MZ;`qb zYWq{!rlpCK4`anfFTuvAg_eI(+~31~-=8YD#FQ&3ob+cY?)ki)c93xc(ZOWlAr>4}{cnLBkfp~Nn{{!bzTM)R61e(P;+W}gj7;d4K$j$Q&>!qCt~I_;;aEE+xha@sL#YEU-g z%QiNem;G8STzlGSxitv-zexa$sC9PF+XH$ALovuVeL!0S2+Vdb)YpqxzLsUxko1q! z#PAZb+V_TJ-1|P+%G!NPX{8$=QtmH~h}5EP`gXd1kFygj;hPZxTxtMkVzpTmY`5+f zxVH>d(4k2vw#j*hXvbP@k*Ji5EvVEAas&;o_k>ORs7OxgdY-De#!L(=2-tbQjQyo}eua`1l;!UGc93 zzn-plhL5h(lt%?$Ev&pN#NMf$r)#3JEp6^T6Y1|bA7!l2EX8%~`-RXM3EqL1!N>Uv|B;E_gvldzv1EC##b_ z6_aJ`~9|?DO)5OO@vniDI z>DajJy3Fcw1x9I(@76c-R?}>QQ5ZfPgsfa6eE?UKI*Zj%Tj*i5@?iv>v##rAKakTy z9UKxU(+him&cd4Fq8I95>c$eJ@ zD6g-ZEqPRq^e)txx?_1B&2UEd_HR3fix<@q#u;)o-+iDf>U6XmgK*cCc#NpIq7kO* z-6wR%QQ$sM&@FVu{3g`;%DK0aq$4Tix?MJSItkG>c+)3Mr?Y*d;N`?oG1J;^`rv2D zui2(Dor@EmO-X4lqU(?>u5fcM*UAMZ?W(2XO5&M3ZnQ$Y)O!@F)5UZ^5A}_=2H(gm z(R#o$m}0hlj1iFB9e8mGKA7pSzx+zYmHGSf(ao?kXx~brDGBW#{!qd55!WSk8ZzkN z?qaV4q){0tDjgLl+-27_f>4TMHp|MRV@L#oxPmNCoSlVji1~|RQWR}9KdZ|Cs)E@u z1Tf>;uqO!KGT$C&XMKzpsG<}q7S#Ik5+9ab-PP_p$4jht5FW|3A|-E20u3>6ZglUjz!F>k}ZJR~=?usc1YPF9VraN&1nbKnu0>iTpu zx%5J^r&3mOSlla|h5!a1Z|9u<>ES&12C8q2iQR;m<%yiEPUP>pYxt0un^*qf&3&)t z9y<}rxiSOvaUZ`97SK1?eBGXJN_9McLDsP}(y++yc)vVU(8$bWM?gE*n&KhONde*r z)Y{?jVH88U(uRtPl5hBe5Uk08>h{FRw3UD0=SgeS_qa%f`(~FRww6GdM{*%riI zpWPmo+F?OZl(lb^*)!6UYxOCXNj;MbFXoQ2K{w4X7+qMEol)w>c^WBd-pp_mN&h-B zs?({)9{=|Xx*Rn_eD9;E!226K(-2#^aQ_8I-+jlGZ^4-B{6zfb9q3Rd0Xl*){ z3LoWuv^TVOG-&MZR+S9#feC6t*I7#HQW|BMQDjGF`=XC?`<>6gFjxDeTE22pgKZkr zWGl^mJ*qCAF@q&7iR)PYL0W6!H$|2F)3N7u7W34VY zEKXjI_2QW4w|7|3;bDwu?Fj7~m{^b(`V5Jb?=0NTBAkF+-pwrP8(Bh$kPbtQJ?T@JCwieFNq$bauUVO?N_R z&IcxEHf7gE83E8CNj*w=!35!IhH1+1@53pWCBx%_$yM^9G&MJ$-eec=;ogp<7P-it zl`r)ay9L7c4O_-%uh)NhE8-n_>wDA z%2qyVxqS->)*km3(~AC6g5-^+Zz-d>-eqp#Ps;ty-_#ezH&9_b%*py*pW-gBx zWbNIGG8+^y>PyRpIw3DG@Z>`K8T47`R?TKZE4a;TF>vYw+$@E+8X4hh z>+OV$afzr-xONja{I9BNDweY0b%)01fN8q9Q#d#Wifg60!F&A$7hVH3iG(l8fj`pP zB=q6Nt#?CKHmx5t5?L;ZuS@Z)+#$_~yep}s=h&JB1*q4aaQHc*J zH8F?l5Dp3>=bqNWYZ2#VYLDgg!47PC&uOK4c{`r&uJsBGL;1*o?dYoJ@e_oxLWoAW zUX>q`R@~cqTse_uiMDr#rrQ?rJ)U;ix$_6hoQ~dKDsK)!-mR)@)cq)|s*HaExbnBq z;EpkGyle(D>)8U0CJ#BgtO_n}1WqyJL{w00tFYQRgQ=B;@ ztiMZ$0^6}#NJ_g*Us#FVt%3b9Hh+$4j|AMUkstJ_Xd?r4m?&I5$CJA#yTDgk1D!&vPhW>!BRnMsfQx?WKsnAaf~)z(LQ;wLDjUj~MIfCdpONvKn%b@mZJ`y-70ojLDx;joF$3UWuIrf*#)o8*T zlO4#Jxi+t~ksN=?UqDN=kCuHJ6$BQc8w^S`tv-Np4y$gs+Bq?$k9PVd=eodQFsW^a6P~1GoCKwpjphVBz0dIxr%>bv;WV8 zb42p>MJsQpz4-`IY<>s#RVeYtk}sfPC}Ae*IfB%bB-Ui)4c^o)nEP#A4oy1MGEXqK zEt4nEv~c#~@o`pKb>|8i%K*{q>kw;3AtFs+yc<&yfXcoP6_CQZ+snWKqYXVp0>!g$vM zd6KJTX|L^LW_qS-n`%?XAfsZkJ{nNhI?)G3Sl%ROS;W?drABox4$3FQ&~@8*dM*W5 zNZDdB5M56F@KU&gO)xB`Vh34=nRUV-|8&+tS8ZB;;5gASKDsXw4PgAxMDHO3f~vCp z+{~*$+Qitx4!}y|;G}V|nDaBRpOAV2Ge`XdXbMD{X#(sTG`PHt`sh4v$?Hs>)XdWsdZxC8}JT-6iA1q?zi(;ygpf z#6v(G?XWbm_{!4630#+!-6L@gx_&bsKId?kJlhbBE>)6ql{xd-!9q_&;GsPyrMs4} z)O3TGXSHh(-pw#%h0mrz1EDI)U1bk*TW#I}tIp&cjrU{}!0yd?lH|q2GKBpdaL8cy zL$;PGr*983Wux_)e!;oPs>Xa-tXSlStM(Tn)VE|qtIzaNM_dv6QEj4Ct2Zj&%a1j) zPUpbKmsEk8uqVP{o&(K6)ggf{pwZr~q&0s}jRNA+H3e?mZasom`=XQ63!&I7#Wb}j=)~_$U?adye zIQH{Fjt4cTELok~7RGMZ#H>M1UGNbHPTq%P>#~{U`wX9QPVbkgG5g4N(UwYDz9^u0 z0r=G2dxnH3S+yGnn2GmYdt46Vp}&{b+_w3xqsg5MDRqo}61i_F6^H0!Vyj&}oW}FN z;3T-X!k(H-tK(7HF!RAQXR}e45rzh!v~MQ?B-$MOT69{lajDyPBUeC2o8B-_l#u#8 ztyOGv<;u=BZ)|<$Jqslb9ZL{Sc2K`?H1oMjqxvP5oU_7P<*KcCg~!iLWz~IK;fH54 zgLihlvc#K*(+?jzbb#jE4*!Kp8qQj^U3wasc=GF;iR*}@5{(68)?B?F-kqaAC?W7x zwUxuA(8m*caU-TRh?<+kpjc1tExMUa>;R=CNxreHlarYYvWq^(+pb!m3@Y5)p2vzF z`$w;Wv*63E<403v4T)*O^F~@bjaMQ3#~&d9-n5IcZf5h^cmY=wxY|#F1sTNn2R12& zxu$hF)D~Yym|Lrx^I`asQo33>oyzUyRJ&9U&6HKgQe44A+G?hylDwgZgtSvYJlWoRF03+m|| znI*DHg*9#=u1dnDvh=bm-%wTQKa^&T#Fp(&rYV<5KVmaZ z*8hvm*ch4qv)q({gZ+Pu&D_8h7I#-@yNjHQu~6cPND8QwieF-#0}z0M69tew!6_Gt zp%+AcKa*mK8{FdXg{ntg5rw^|qn(eh8!zS?S<}Q_4X-PoXFjuMy?@3H~=3L6gXfGp5l;r5}z@m zqd*W&_x1=O9K@iB6kQj(+yL?Hqj?43LqiL;4A}hjfPtF`X8!UgK_)r?!d&j1^0)CV zpdA1T_5-N*LBfR)&aJ_)2gAaE_SwSCsVxJTclIIpjbZsgxB>djg8g>-r*mufQZM9R z`0h9Y1%FC!5rThh))$TuZc(qY>(nbW|#oMq_1;%z}P^+`q699Pp8IM zh+^=8lbZpw*uV8(tidr^(6@*{*XLk=R9>a~g^YjtFZD6}DNcLR)5mxLrcePMf;WKu zQrK@D1HRw_{rcA+KfSf%=>W7Y5g7Q}Xkk8tkAp(n1_3aVv8QHF{J_84iGYIu+3YF6 z^kEx=h{wMF;6e{t|5Vqf#RYl+O$n%91pwyj<@xdRj!aB~McCiJ!+*Ve0l6$4q8{Ph z{baoEGIDYjllO@O&_E%y(L;gxhy2yse+dTg{h=uW4*0PFe$3NW`AvSc|B-K9GkRAW zz4im`dsxNk^7mU=5HeJO2EP3iKRhN-L_qlf{r*XPze)bd8t_wj^NaZG#a423a(u@< ze#8CYA3!i`OIpn*CpfLMWOfZ(o92kj#Y zBuL=rz^MUw30^<#V^sLyPkamFe>Y(z|BV+75*hUC*7~bKFy}Y?_dAfwa2Maan$9mw zU;X3LNlz36#o}SoOsLQh00Ij1jtJP`h=ha+>K4!l)<9v&I-e0U=;W{|~?#<`+PV z1j>4y6)G42q50*MeG~CB+NB?;-vI%@9A>P&@DJE`J@FE3$N<+(E5qNbPphxZV}^8c z7%4W^h+sbVeTBE;JFRw_sIa~?!orY}<1z;_9mGP6Yk7K~OV)5bin#U(jrHVh4ENfH zb`ERjg|ra6o6Wyv9$I2^-jPylVB+;wqSpchWjCK~({LlULB=iEh<@MBA)frYX;&R7 zm;viKB1Pl)1^ZV67j`T6zHCLSvNv|X*?^#W6&yAuKLmFif@ z8kJcA=S2y5?uxf-%}D;&n^hJCP|N^Gx#|RSw3mB##X9 zqh{sHg$`{;BE1vnfBAbnYTj-u9G}uc)r7=KOx3Y4eI^L({{7C4%O3v0}M&QYr{MY@o;&hgX zvdX3MuQPRm8EZr&^xFqZq5!6uN6!~!SsP#`Zk&ii_sRHvdy-r0$o4nvP>m^sXR}?G zrRmN^1^VDbKrSt3PR4na#*$>ewC9sEu`|}U%d1>_#@WA{JC(vwvBn`}nBnJO9 zx%H2l1e0{747Kk^kbE(_(E$|B`7M=Rio=#yrr5TwSJ)^vDeS;2a~S_n5;FcrhKkM# zh0abKAq}THv;HL=>v}wA_?~n~;*x8_AN*$;dPEg=y`F~NP6f0~wtdvyX_xKyWsmOB zg;w~LIe`KICXJ~A^XoJfwUf;aBuh9EC&-MwoVh=X@PmM?i{#lvR54uF?Ib|pp{6`C z*a&yI9*SD(ks~5gGNM88C1u-OEK`LN2Sqnsq!qs|L}io^*^P4^qi;OVBO>;z;tTB7 zV#b^*5(4?n9dgol(Kh^@)z8*lDb?iA0r4-d{=eL@dTnx@Jl9vV5tUbCb1{fg!L89H z>hGXJf035^NtyzXn>p^u=T%eH~hNDm_m(JU{zJbETYEttJ-5-MK- z-7#gjq--Eh@bx+D48f50C4Vo2NyF1KUt)yvjTvECmSM-gXa>$YMPVRfow#@K3eCjl zL5RZDP>I-QkhuPqFj}*lJOxv-^7(PaIIk_l3NLh$holMlJ&5!XT!FjdRpp6W4=Km~ z`?(9$JI!GL$@I9obuWhM1E9H1oO0gEP9DA6nNJrH3`Ij*%aq7+iIb)=Fn(Hskuq<` z^LA)$;P@N*u~gT@hv|yP5WIG-ru95tOm1(c`vhpw)Z&i)gejk<$#n_Ovsa~mLj{9kHnq)!H9mnLJu5}7h+)5NM0*gn~}t&=qW+sws7rpum!pP{mh}t zDKDi&In3yqpeNIxB(thReoAueLlg*i7E@6l*vAEpjo8J2sPvN`hK0NDpffLK48C90 z_Fopew$H0BLQ|NN%a#hW7gv~xbJH`a{kXA`2Oox!ZY1pMNv#_>l5HL@hRc~1p^v;6 zA{@bCRc@}*X`wrfsWhf4u*fj6$|0e=?A5zU(r^2o>?F^M85W~SoQ=-Sak3rfGR5;6 z-dM9Y-8yhJShFUc5RsJfXC*(roV-Tjmz!F($_6HXABP?IGoCdf6)ca{JqOFuuWy4I zHWS#v*O@=qcxyVVg=7Np5>p>d2;=!QkPl$CtKykoZboj~A{E4ErgSqMIT#C2?r`6V zOKw28B0EnnGPXz-T32<=;jp4oDB&CxMys;J6RjO51>brh8w&eBcy5r>sfix1|D2|y zoza(z#Z8FCBeRsx-8_VTxNqa?)Ob4>J#Uy!e=V`YLROz0hz=WJ(+B-whqE6s0){=t zFrS=T2YIG2<#@NyvrSp> zJh#F_rGQ`idzcH-dMEB}*_EnnHb{d3EL38JwG*G!vikHr7yTme>1=~O>Yi=^{*%*u zznCkK4sk#5@YomgaK|I~n8UPoSVUfka!tB!A1NNLP=JnjjQe;*t7Hg1LKqY+?I6{u z4Y09fU#a*hk+rKYgy(dE{^B;etVr zt?mVk$>grjdtN8D1NSY1Ybeb)0`ZFJXbZ7w>3$@;iRz8RKEpd|hgF82k%c2jS(hOy zdsBz=fsOktufT8swJG7M&<&|J7Q$>eMLRh0Q1>aU=|8-la2Etx@3q#io&oet2(OK|*_tpTwnK;kV zKhK{eoxr5K{}E4vIxXB4xQs{#cb#)N-%439KG5>gK%PU#W+UWleqI7?y#tC$okM*4 zyMe?0I^A348DraZ^Mat^Fsi*l*QAb#%=TB^cK>J#Gl|aP>hjcd_&QoXK zI;x4bq@{gmlF|07q+W$k@Z~|msAXh=M>p@JcEy(MWbBdJkel(nk1@2QRzWw(z@i== zXLBUED$WP?Bc~S$Su@QuU^=9@cGM<0wW%|B$WNcn!=1^)NEQy0tYpo7arn0MlqStP zWIif27Je130wxj>ae36c7Db2%7HbRA*TbSPG{`?M+x?%d!aKT&khbePa+J+tL~>`< za)_>jtyv7DR$R}q{1`Rmp&w-L3V_SNQ_S2w+4MsCmO-4OK6?3i`lES*?rKYiOBwZk z>SVhzn+MF*gk^Uc`{?8yjyBX8+zM>*ftomQ`el53-_AF2rn*?6Ki@mH?b5wr#macf zlQjEF4w#mmz>GcQpw13+cZs!zx7^H#;v2_3OEb_bukMuDMh zwVFn9Y)rsi6%>2i5MQq+;jq6F$fW-JMk`-{94%{jt}goc&{5HTukj*m+Vb`$P(8%n z$4v1M+uyEQkLn5iE)NeR!=a(L7lkYF&+^l>P6d@CGGq%T+|Pi%;~>3q8K~g9K#Aa!m>WCn8d_DfiI<%n zX<9`K_XlCY{OtCTB|A~d_pM3w#l0_211E<|)8CEctLYM-4R-p_jGbFsny--ai8Ni) z1-9AJFS#14qK7KKyXr+{i?>?>X4B1^qJEggqnWm&M@2xc65?GW7|sy?e1~4efEbm! z-LS}`qxB#UQ(K=%s%kZ8Vvu zoKjho^0q-oe=n-#L?D`M;JB&1op*!oGwai3GQlYS zc){2O52MWts>3JJWnFxFell76L?vlb_NvGUFNBl0C2sflcA0ZU^Ezb>5zKx6>!N#w zoJ#-6f<@I9$+Li$tbk^?_7a)ud!RWX%)#zwpQjNVZXX>pTA!hC{Ytb@h*rV=?BeD- z2Xm^|PlZpwBChDs6wo_)BvGk>+;EDlb%1n_c>;Ekv=9e)Q3!8<*F0a%bYdN2cJ^nZ z$0Y?`H7N6srq=*kce1H#OCNY0C47^Ao_g#hbTn>YYtAJvQU)xGRjSn{?ddwx;ybNp zY+Sa1I~BdwpF1{t!h$?}F5V8-0fYDU)%t^oCV(Im=sW3UKFwxUz+eC1oUmpbTUvQ| zOIME%GcK}iNW4KLr(c2G_}SMs6H7Dc^)zr1{#*G%^yDqoxZNXgBucRk5|)xJj2Bd~ zRV8#;{_$aCK7Q6cdndDMcbS(an-tlwcq~w18MnYSwda(n>eKS&JpW8-r@PTBHi&kX1k#6!9Lpw#YT|JZ3Y`3vNe z?-H2C=`dn#5Y98gpY~OgXj#}5lYmtG{`;&ii0@7f2vOHzmVSVLAU1tqH+|M zp=vw>Nj^xI4Hm1aYQ%F)HYi$R=z$V|cAxbO#v1aswPypPZ=Z~ZT%0Y0uzJHkWxUG7 zF+?jp=+LrPS%&%cVvs$Fi!BEN{l!ad-g3~Au(DT(t$#*0*BJkCe>E?LsAvtyBo@gl zi`tF3mVd48%ikC;zR|6aSw-v=ZoibBf&u&xfpkv$K|CHP zQ^4aYakMzz?nQO{vB5D*nCs57ELfhHTs&i{QFi+@*bDeUm4e9DO5ChEYU9W&%nLIm zU>@V^womS*FJ0GtF5qgq0>F74$&&or{Y5l!uKZ2((V(+9ulmA=tkT__2iEGW%TS>A z=o@bq4r*(p67RxE(sIN6ZH{F=s-PuGRJlU5AxK+Bn^-8i<*-D|M>@seo+T$}dEa$Q zMtZH$dERHWTf0Y|qulJ1mpkV2o{Jya`?C|0l_!!SvF@OB`^;%AZR6UVo5$#sCCNj! zMmfVG?sUoF*CC~?T0BjlNIqD?LoDso_L`t;5!*TBIVt>)aw*jY7TMczDr+<0sAsN?!&gemG5s~gy_}#5hE6*v{WTlqXy+3D zJi%uixCOm*vZSYnr0$Wty4Ek+@`W-}zu}osg=~2Gw3Qqi>zRbWE1C6AtfqAd>+Hqp z$2&4C6&Cp=*P=>*j1w>h+VL~1VbK9(v4C3~GQ8)@sMHX51uJ!tu?_stY6 z`;CJ(D$Oo$iMFBB1as{|SnX>eU0w!nDVEgsN9`o8F4iz3tjDs|uP1VBi6?t_IIwd8K+-Xed{8Zg%< zf9R65Wn?dOHq?cSl;v&?MgqE6-F`kP*jK}6CRYf!M>m4VER60hvPJ$dz0MQ*H4{P+ z6%|d_^&_j%XC^{JX)bZr53NRPi4_aGi+BxNiq5-&+U9AgoIC3oDK#{tC|hT?frZ%S zmuMJBs1wMf>uM@gk;V-T8Uw^sO=0*`f3Yp+H~m`#jg~cAdaS zB^7R1vuU0R^)bRd8t5A`ytJm*ND`NRdlZ8VXn7{#M@JmG@7%Tfhu1do{phQz$<`P) zI8RXQz%Q$dhb$X0A5ebIu`5!N4_b!I97FT>8i2Pl=6_Y)qkNdf9uu`OJ+Jw7wk$wd z$>mAbNvLwN#DCHd=H4#E9RGF9 zhckTrH=U!D)$C5#DD~p{O67El)?Umqsh)i|pg9c{=7@I?v04cd2Eab*6(OZ=AZgNf zsA=w9xDN$2s|`{_@fD$~QogsNodQ^LbuxZn4|~u2HndsvOh3kR7%N%pmaxy*rya0iEN~ zIW2L}(tQ(XBBUq>zFSfg4m%873SN#DC}FNTY_fk$-*U0&{QO*Fh?aiIbTf3D!P=6t zwR_1kyD_dcUW}rOMX&X2M~18u&uJ;`>p&M2+vA|#xxuoemKODCTgt76#IpHSfieU60W#Ea}RGIsh#o($A2fRuj=wqdy@C zuQFgzuRH5#PPzS&1vgymTX*(WvG2Tt!$kpg42GtRD(B5yYvifHrSB%MlQH86uRM&% z62Wsv#{l)#Pkz7Fm}67+@Wf@Sg?jd!{P)2iX|0MGz;ckV%$LGtd-|)nDJe8ToUbZ2 z)#@1i=naFt0C{1uxSKi++{s}l2c=Tw`OQO#yRybo<9x6ySX(=_`|FOw?4D z*K&Jg5S=IygY065wyoyJxb1QoaTuhi%Cj6-az5PM3}lz55X7S5%g(G~M=}cmfGTx^ z<>$O86*1NN_10hhIOz`dmUYFmI~4;g`p+1%6RY{%vU<~;>x$%*CFHH*@2^uDy2UMz zQVSGVSUHD<{tyTD?p*Z=Lz(BfC6e88L%4+|SOqHDLu+tIbuj!&N;cA{Fbu@Yib^80 zl8?`Clk@knQxDf-TU_0TXn6!tDx1C zl6XTc=4M4q)?E3ns6$n0wfpI69h`64)CUSs9p_P|_66bF)*?- zurn|*aL2+pcgZ+b~1sYS5OvJrxSCrwl*}d zwKZ`hpinlqa3c6U{#Scw2vkiRoqk_Uz)Z);!pO?W#Kua?M922O+xc7mw{J2Qt|kP( zjbfu0bTl&my|s05rzEhlH*hw0GBF@<{r{_Aqhq3DqJ-k(`>!#zGqi+a{ci)Q{5v!R z^zw$5f1GWg=;a8Q8KCIJEF7Jj378oFkKONo#eZ`CPcdtanua{K8j9~r9Yvbmy)qJkt~@ylez=Wc%=YU!g4_wsm6f>b%Sog3Q8ckO+JJdtcMk?#a}OQE@@#_ z84E~0qdqF+dZ02P`2akabip`XVF;pmVN8*}G8{}BfiU_JjX>-XFoc4MJK%4f0U%_& zcinCx2f2M&55+`|`~vVfW~89Pp3>Dihb?4uzgb75vUQG|h%ZOYlf{n0F>hXnRJEfo z?_{-bQDKg8yGZN0PcdOgpn=5~E~a4L+xHG|R;RH>t*IRi^|kJb5H&Z zZT?v@{H9t4MD6V4hX)O|iG-NY50qCADktMuNM1$&s(lF<1l^Jj z1Q0mO&Li)vuevVbCZQna*t!L(eme6r&qDQjv5vZ&W|x1kZFbMbv$@!ivd^TJp3d$6o(4i3y*J zsP()~wS33=&{Yw!X4s>PHNaC2cW=o;Z|cHn2fk{zjw|O1UpEsM>$}d+$*GD2o@A|^ zV8i*G>CqJgw=*BB(`+st7D41%g(z>7*Vmpm9eZ!+9W`psVEyoHEq=G>kRuy8f9=dO zE(o~SHFLvPc;vO;kklY1IBvss(*7cZ6D8!0ip=P<&c#SUPHwWj?uEhk!)n6s-}-Uu zP1I)DYq`6(oZNQn-nUylk(=jU%e}oLWw!B^nPG+1msdYdxVkCOU+1*wmBhxV`u3Mw z*)^sVmtt(RscevfZE%k;_ka>xChT6HgXJr7?t@5k01D|eMII6htJ zU7A3=%#_8ld#;{&J^fGFsrQQdX#($8PoEOS_;KA1?N_Gv*q0sk4vT9%Zu5EZw)qQJ zo-SNp`ud^hy5a*R{xxMm_ntg*p1ObkdiA4I1ozB;diic(?3CcDoG0~+uf6Z{VOEkR z$S1c$%Jzt$)HE)A&%Bh>3WaC`1rx(qF8$!ls#GA0OFvv8+EBqr0VEcbpI@S247&U| z2&6nMU%?nuj^=J=mgX)7j!u?tmX203e`1GeZ|F%T{k#` zbT%9liq6?^h_9B%;0Q;u&z$*>=l}lFEw=9d#Y@xkOI}4w&;1$a`uxV@hQ;%H|LlCl h@Aiy0A2UpgOA?DpDuB%gE>i] ({10*\step}:\r-1) -- ({10*\step}:\r-0.2); + \draw ({10*\step}:\r-1) +(0.3,0) node {$i_*$}; + % n-1 + \draw ({-1*\step}:\r-0.3) +(-0.4,0) node {$n-1$}; +\end{tikzpicture} +\end{document} diff --git a/main.tex b/main.tex index abd3cde6df99068eb8c35021caca298e0c8427c9..b59a0a477f53431cf926c5f24cf4caeb78da314d 100644 --- a/main.tex +++ b/main.tex @@ -580,6 +580,49 @@ The intuition of the following lemma is that the far right can only affect the z Also this claim finally ``sees'' how many empty places are between slots. These properties make it possible to use this lemma to prove the sought linear bound. We show it for the infinite chain, but with a little care it should also translate to the circle. +~ + +Here, I (Tom) tried to set up the same Lemma but for the circle instead of the infinite chain. +This time, it is no longer $I_\mathrm{max}$ but any vertex $i_* \in I$, and $I' = I \setminus \{i_*\}$. Without loss of generality, we can assume that $i_* \leq n/2$ (because if not then we can relabel the vertices and count the other way around so that $i_* \to n-i_*$). The goal is now to prove: +\begin{align*} + P_I(Z^{(0)}) = P_{I'}(Z^{(0)}) + \mathcal{O}(p^{i_* + 1 - |I|}) +\end{align*} +Note that when we refer to an interval $[a,b]$ on the circle we could be referring to two possible intervals because of the periodicity of the circle. In the following, whenever we refer to an interval $[a,b]$ we refer to the interval with vertex 0 on the \emph{inside}. + +For $a,b\in[n]$, define the event ``zeroes patch'' as the event of getting zeroes inside the interval $[a,b]$ but not on the boundary, i.e. $\mathrm{ZP}^{[a,b]} = \mathrm{NZ}^{(a)} \cap \mathrm{Z}^{(a+1)} \cap \mathrm{Z}^{(a+2)} \cap \cdots \cap \mathrm{Z}^{(b-1)} \cap \mathrm{NZ}^{(b)}$ (where we assume that $\mathrm{Z}^{(0)}$ is part of this intersection). + +Furthermore, define the `inside' and `outside' of $I$ as $I_{\mathrm{in}(a,b)} = I\cap[a,b]$ and $I_{\mathrm{out}(a,b)} = I \setminus [a,b]$. +The following diagram illustrates these definitions. +\begin{center} + \includegraphics{diagram_circle_lemma.pdf} +\end{center} +\begin{align*} + P_{I}(\mathrm{Z}^{(0)}) + &=\sum_{\substack{l,k=1\\k+l