From b09c769c096e1f2a0473f9f4c190f5efe8353c8f 2017-09-06 14:16:20 From: Tom Bannink Date: 2017-09-06 14:16:20 Subject: [PATCH] Add better explanation of independence claim --- diff --git a/diagram_paths2.pdf b/diagram_paths2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5d90e4d77f8f1ecca7a07923eaf1779c75b4de02 GIT binary patch literal 61329 zcma&MQ^0-A>hnnBvw#?;A-keQW@nehK>&I|ABK6$Xo^2HgFM9FsW%{Z6+2Wj+tc6w? zafc*vDEa}tKU29R4xc(HHIi6DN^~+e@78Ik=wa|TWb!(_8|P16LU=r6_RDqDiE82p>bf30?il-e1(xUj~o9Z(O^8j6_U; z1`T_fY{PFbMN=n#Ub`ThECENSa9&$1#?mk<$@ywdJ8zeE>}oW+Jd4A!`fh2v{O3N^ z*!MnlRVxDo;=49V?cUgA0QJd=b9RF7->Bb@5?NC?i*Ms|+@HP?tldc5=cj(7Jussc z4!Z-q93887{GZ>JswhC|UY4;4rz93Gr1TQirIwwnBVJ}G`ySUmV*K_?)3^3v?wzCM z7a?2E$VzSs`{ZGRa?X6$?wQ7u*|d(xT6d>vQqJ5d?K8!AtxVgxgbj-&qI&?G3Rt_n$!Rw74=fBXaBa#dXk@zaXsfu9RsrDgvO{{nJ)iL#Wvm{@|SHrT52v z)Mk0iZ5ap|Yxj=Mb3&$$Y#ZRpLjSp~dLa&5Wmg8Hb7eNR6#??FHLqZI&@X^d*z)kX z);wP@QF2k)+w1kg*;H^s}zj==MVrFSei>Wng_$OVEkCl;E-+!#o7F3`4*@o10E z^dy>RPCEyez(oJ~*|8Jo)SsRz76i+lT6dfP2+6tjsBr0zz{sryWS#0>uK?t1QLR)m zp&kW6(?^6o1c<2rEhN{dz;=ADtk5*Ko+Y~JS7cVNgARq!i2vZrqw(1Dbm4qvqJ}XD=fObdw4nc;1LVOo=KfXtd09(%k&+gmh(;Lo!4~c?i&*?P5mqo(6!N%SGGLzt z-IU;9Lp+Hh@P^J74}^u|AYszI)bNgI$sWE{vNEn@ZHX)7ojf4Gk$e>VnccVII~h^V zzKq1?%&j%UoyVYagAa`aWoMzhbkl}yW3&YQDDam(v)KS)EMVX| z%Wk(kp>uxY=q33S3&Tl(}^Tpg@HMPr;4WfCt^Wj z5MHZ%hv2JirUAm^&C*$v(a3Dcq2qu87R@`{nGt?M5-v#Lm>3jaH&eSI*gUO>&)48P$ywbb2#Gp^3=N{m8}T7UZxo0 zEjB&h?oe@lZl8+}H^X}61q{A!pWp1ej44Ea>gNzTjD9eH4CjZi40wT%%`p2|VBmsP zr9=E7CEc1W(nvP&hPg%)hHR@t2nfZ4REhQo#uw$uL#c#_LsIC^8|NadhHPVL;%PyboMY!LBzs+tRCncTU7dI zUR2sAnV39xv_5GNeiZpH{ve*K2xR3_1GIRPijcEp3Do0)Y+%u$mN%FR73cG8o72E` z!R)W>3G)P&U7DmEXMt#{9dJD;U#g9J7#Yi6tVmH}=!JME8ECRJ6I%S$GMtYUPScS6 zyyYBz^D2oTQd0uRWzJC2U^Z~ts}VS)WK!$_993UgP{3#m2O?1`ksy|#k0r8p63-y+ zUiZtDpyM`|{7}__fP|3@2&PBb-c(jHuxwQ~9RwIfNQN*x*jVbGAEw+o;-8`FOTV@T z_$0wfnEsTJ3{U_Iq|r7O1TM+e?2WA?h*>1Hy+i;I7K#4{E%KNa28g>Q`k1ALyhyd7 zpu|5J&=E35Z7k0L(3W#Jdo5yhad_5#_&4Naf$cWrvQtODSM0B=dBR9_QNOhQ5MU`k zzSD_pMj#XBnD$I2pbIKP|r&T{aQK*(Isl)A?XznbwPG$^W&C{9tIpIwW2h=;|EHp zi^C2ys^M1W2$ew0d~=mI=G>fzu+JBiBx~PCZF!) zJ2+5l4|QMKqdXyT#OSiT^Nxy9PiQZ2#T>13(420X1Nh}C%N&z)ZGn?QyD=0n6}}E5 zVhYRU3d&;&HJ*!0?(yTPb1I27x#3-A!knegU`Ss+CD@ZEqq-A@rgmlaQ`Lwtw{-OH z;Um%*gb#@W|3qg_$Um1BoS`SNMoP&n9F{rUkd?02G@a2DPT-Ml`#MM77NDk@Vap>e zxweQc!$NQCDESgpMo(P!9~01oJIEtOtwr2B|9ympZf5w1=qsXCpq%v>x{8}#+8`DO{DvEdAeg(-eMm#!YDF_PZgX?wp2uWLLjgkhnx+-~9Si!h&I7!=R* zrAr7OFpH_xYaB=0Q$GIf3LjaUo0qiWt9J>$@|7r;82~9E;*@Xck_0~2q=0v{dNAp! zi?*w_)b2Wkd~(Wn?P=70EZ+dD;FEDxDkQ(xz-{d^nD`q zGMHiXVzP3gZ|T!CR~Q8CJ)=3%wIRIV_EYibjh5fvGre0b%$(Xatms>9ZcEYg=x)Pk zxj3_dYkjH=a40Azuu@clouM*y=g{_gJ^C&qa;E7P{@#GmvX%t9L{9Yt;JZF(ZqU$@0A=gG!;f7B@~;}UZ!QDKyF6b~*z z)n|FGTTRCH>kff1E;MT5JM)e{<7=&?$}fMJ0OM;pFUK2*nf>v%zP4trgs+0Il`E%{sIQ3V9NW<^{tzhLZIb@+#4Nv$C5H3FNQQCt zb~#$NM-^K+-j3b?>KO(CNbpo`@|923<`G6*4A!dN7*t*uXPO36X+lmxcXQ6Q%8)j6 z6=jhnMV(6QMPJBR6%|Wa$HDr#w+YV_S(LF3inMqh6`~V(2#c>?bspf<^xbG_9^?;* zM_v|{z7O?Pm-a}>95TCM|LL5`URT~-{TgDD4TnTeB?R4%W4#s>x+gwPMb>7eIs+^O ztMq4QM6#Y@ke8q%zRy3<@M-8-LTx9NHLh3RL0uk0-0Q{kcNy2s~(%Y1)A9%SU(VSvEDclr>c}@%iL>9LOpUsv5f6F$-q0ZEf#-1jOPH*WI)TGe|taEX8$iGq?u<>`3 zVKf-aEuZ&)b}?kU+Rt=XVe`V=U9M47t~=vUowHC~ycMZKD~f{JYN+aDj=SpahkXN? zmGfXm)E~c4$FX~t|G*eoFfiox3<+~m`pe_$?5zCop*5L7kufy;t1w=Asq!k-G7-Mu zX;tb}i9V~tZL9KImszXl%v8At9|#Ey8(bwcS7_sLtkf*M7dB3hE{kiwEk{{mKp414 zUiTP^JOBE8K}MwgwG1Sy(7@T^?liQ;fqu{yd$DZ+Jssw=)rmJrm-T2?L%je8vF7qCN z+pOe~)?x$4!!f8z1qiRRwTTUp(Xb(sqtM~DM_II=d3ffc0&;HUUQERFI9<%r#wB_I zX5Yp9j{7TRc&LmRD+V;m-pq45o2-{NvZimZv)`RE!iMzUKm4ROB$d#{Hvj&A)c2qD zzY!x7=l=m683`Gg*%|)_NHP(!a>4EE*w zZiaJAl zAZiLoz~C`S@C@Xme)$#c#yboFIST;+Jv}`jhzLzU$j8L9(-3b1fEqs}R5;OhfowlL zI?xY9gtOcgurLlF6gA8bhET=@tV0+B_( zNdLn4Hn%VYFz`^0J)quLc;KaESYXJSLH_T2JOzLptR@g#;O~XPof`HQYb8;ZavVOs zJQ%pl)1_Ah0Cfq;_4X&gH{>!#;DZ?CXNPkTQFf+x!|ve)l|{6#t1Dl#!WY1SGT&=W zFAqxy4jxE0Ly7VW6@Wa(53+v)*6`%dT>?72H9G+2t`T2hTG&l4! zPCUK4a|t-V5{S+x_xG;yQLSa-f|N*=q`Xu`l6%q&g+X+m-&l zm>Ups&`}a;@BMa&VQS)?KsYn`8Q|*t7EJ27Eb0jb4kF5fl48GHnghQD_wD^Y>@OXk z#dz|~amfC-2-+XKn@X|f`z3P4hbQ$!D2xBCOez zwWHsYuh-qP+)dHXoBgYfKRdj8uF3zhF20a85hC|N8Gi0_tQBbdyh+8r(`w>$=;|UH zhkYoE_2Opvs^E?^{UDI3!m3G)yhO}Ji+vc)7*XG4`09(ljN_D@bwL}Pg+0jMcqRF@ zv*Qp;=6y;O<&ICKG^0>0BJ6We;J9)7+R-i#2XrLwqX&}lJ{E&k{2*C!f1y_y=4M1Pa*vfZ?4xw*<{xdS5~Ui=F)?1TN(=1wWzy_JPU z>*M03QU}IVknc&hjAfhSt@%6iJP&-hMaWvH>trS4l4|F%uvE(+E&_z2jD4<5h{HW=^`l({FlzhTO!9$cfm)6A*NPdP zEN&mmke#Qm=ae2#X+fK=PAUs$D*M)Sv;g#AEC&_%7yoz5DT0#%rE^$B8!hV zTpct_wsb9RS>t=EV$0O@sJCj_72{RPYTes zwQ-FcFJG@=If`Lkdv2JbFZrji40r9tQ#tJ6L71P<9UI${Rn-7YQRTEHJBR~Lr3U$c z_SW-!`h<}?O!Eu4@!1WI$MARtnK_kA(C!*nN&t&aV`lGkdvT2>Lvg>&in?9Z*&+UN zq&r>S2&!9c(J&Lji!}M)-h`tV_{h8?D9}wlvmlVlau&LIo_xYt)r4o_MEJ(XqE)pJ z(|>fx^kQ2=RW-5|Ry`iXs5V}S&jpEudTFIDq4yV|z@8(RlID>??6188GJkEfJ-_8? zZaV{#ZK3+3kA1|2WGCfbf?-M4u1?;ek((09mA{QQt}kod=Pq%I3I|_hvWoa9h0z}i zsqlm3w1%IBk!^YH%aJX39Lal4y5l|Ix+E#@-^qu}#!?b4&(rC%amYSBRS|aG_r+es zhCG)Rs{77zNywFRookCpH(?=?)KNEnd62GA=OmtsYiM+`p;$PORd*cHtM9*0_I4_C z<3**t^m_jzu4Kq(B>j1TXGUmxlm@nLgzE4K?7RfV9eH{`Iin4N(lTb(?2@J5jE#28 zsZOtnhiJepKdEsB&q56raQ;m?!d)NOfrG6HUc`sWEdF5hSofb7fLwOA4%w>T@35;x z8s2)~6{dQchDJJ-NeMh#Nw#{mr|o?BnM&TD<&0eqq-G74A21Nyz?4qwMqBK+4NN9~ ze3yHcd82h-3f_*=@6*r%J?^Sm$xrDC0 zO0r@!vS7K(d$cC42Hk$#8TLuVypdEi2tMgE74Hnf4v@$S z|G4HL@NN4{VI}8}YLF=iuM50`8MWJatYZ)3B2%VRsJ&@MReu1>IdZzgJkC6&Scy*Q zS6FCEv+5Pzu$!{)Bn@TvoV3#2-XpX&dNQwn^~6 zXD44aR}|TVInc5BVn;%XC1@;n^pVVzY>l}>+&tjaO_jWq<3aOaTl7YxS%-q(y8IbQ zjTNI#^Z@FK;%x+XI_X|nYGfr7Ux*V^jLtM7b=PT#v9ns;k>8&79HGv5mQAI0Hi{f# zFy%WRb!ur)N157y)jE3W)8W&;@Vw0V=a}kwz+`c;7jbmm&HexriJY;{8U6BxXTO>u zWaGMIuT&7WL%SM1qGD?KB%6eSdZg0;>r?VT6-~LJyvJ0o*(7zCAn}r%2c<@NASu4z zfbDEZb$Rcd%uLh!1Ki6b(iwr_N!JN=6%91(x)%Rlw1=^VEteJ$^X+7-Y6CI6#CbTI zCxemRpE5-Ku<&pAtCjHqCF3~1*m|>c)N>6BAxODbegw*QD7HCENwW$~6dBm({8V5^(K;09;-#u-l0ak zcsWr+tH_huHRX^@c=)h`Kf9ble=_=^$qwrcT`U51!}{Yz%k@5Z$j9jycVwbEy`^ba zcb+rR@uK8n+m|$He7$#g&!g7!jkn))7V_xzk?sjktky>}bc^tIE;B_9k}&}ld&TmK zTwqO7m65+>cYJefywJpP!aR@p&9L8RaVdH94DSAp&S~b7A7=0X_+8x8|BVlxRsXo+ zcmcJJ;{*byn-XP`QqK)hGI$xDFu@WrR60^W*nwkX^zw2{uLeui-^ZJOa(&JaW1HCO|lBn~<2wo-5H5 zckpDRFv3Fh@HbM+A?GqD(AcG8Bc_?ZLm;cga6TwdpNrKq@WAVH!Sk`g`?J>enk1Ql zWQ@CidrfmaJ`RVM4-eF+vKc3w!-~)gN`DrsSW@e-qT`KLHHyJ`n{?TLODY_SBStGh z!SqyzW@~AvtKdzmT@rxgK0;R_?&wat%#2B)KpS~Gfl>Ju8FOkkG_v>#=(0GEC_)Oo4m17 zi+b<@`rWTMV>qhS^{hICBoX|&3hiKu@z&UL;Eyr3p`^w7N_3j8`&4tPG5;Yw_s{~5 zQe>mpa~e#1Q*<;bkrH%B`Tds~Ili1d&8zse7CVh>=hki%)`IFC&{9`di6XvFDr$2! z=D8NREtbM`jZr|Bq-0#xQLhaiyxX$YTU5nRLi5N>4go7%Zp7)j<TBc~Bsm!BO~4CVcH14%K#dmbM#b2^EiKZg5LtYkBoq_ z{S_^oiu~@i)bmZhbP!KFrz2;AY}2LlQ$!U;Ca2iCCb?Sq*;&N&dL)LLjVd01Yw`8t z;}gPS;RxJ{mkF+i9>(@ORQt@^peI2#U=>%YJu)ZV>a9(}hJ9hlo{NLNmJf3~9{o7+ zALjg}SdO=#dm!O?5v@{I3dtnn6SAj|tDQB7z2?#<_obQygCLX`0fpFBh%^=q?>84% zG|W4hYz7kd}1CuyTjUnu!IplHvyu!i)LoNvUDy#=sSo}-BOs;(%^ z*fBc$*)$Nby^~wc`*EN@(EV3eHfIq1vrMyR#no}HaIW1dnm8(%Sc%%b)r`vXYEWy-wOL!uxS7#8t~JbEEP*iR1ZuI~x5+Pe;!IhM z?KBX5v0y2W%Cfo{C!R88swa_;%Nj#0VQPi!{25E-&j-#LAK^LnYGW-s#P+&`GfCD?g6xvDLgU09Qs>|89YsX z@KG$^zofxeI2vVr>UN~9->ruPGpf2yt44{HtXq#$MX?h}x6MS8NDMnXRE^R|H_~ncHSv6L+FG$dkBw0k zJJCBA%)m(nzifFzs%633gMvdZ{e`NKYRjS~3k7$YF_?Q&zzJO#roj zhhDOQC~|X3{8ib3Jjw1-Cn_l9?%AmPGT8hfJB~(?{~}M?c*oRkspyIcb%6}}<;~Av z>ij_2tcRnQ9mr%A*tmSNpZfQQd4HkaAZ~I`tyKD^e*9(WpXa31FSE)a99WDnSje(s zPbK2ehww7mAI^0j3RnVZjgMoy1JgqOT!CeMrO zNnT&2>$q=J*tcgpb8W1T_6OIx?8mq)^mO2fw#I;wq#~SNY&-K7UH5Gt?Bsf0c8By; z6-B~;kmlApF{1lYvpKY#nV+58_H$2J>Ti^GrqE4xR;~;h-Jla!XUzthWzzXm!^&4N zsytl_xQ8P!IYu% zxwT@ZYZ+dQmU%5hiUhr@3>_)uxdG~CWG}At^`k=#=)C@sexjSK=Ou}G_XLy%4I-B> zm!r%C3b<#dCQVxOm~?P{K5+gT3=mzb075;2stt6fh@8cj2YG^!3bIre!A+wOd2-uixbAcf4Ly^h4mZ$LfvG0+Td05GK+?AmYD2?%HfoN)#V@euKW(vNz$h73JMs;Tn;0DpF(yi?Jf*< z!}+;6Ip_!DUdjKeF=CPat=~>xH#cDdWyupH;Q5JYt3|n@=Xf*a=@cnnUrIl8PS8PN zZ=M5@IZ78b*OFmwjQI$-JaHq4L&$OPCq8=owB(3&78WTdFFTaDYl|n>yDrsVkD{+; zSmpfjb_MAcI02umJ23$*4?B4tv+1&{a`_VcBm)iqZ_3Ln`(d=39(M+PG9WXJcHhz1 zzHuqL{CtgT+HR?-!Mhs}gFbgoJ97e?bQf`lx5{Dq_I5>jxl*uxi1fX65{qFMt6GH5nj7CJfg(t!}^!9;_vw^%!o61iMC8@tk?qqRuUp79+fcpwJ zr?f%BImce^8I8)pS+n@wIxmW@l~#J5E~5fgt`8BT0ls6}(PX8c&H3(8#V>p1t(DY_ zezgXbU0y#{=v_>h6NJ0r-CAVe%1Dsm$hKdfJLRRm-d2cUdW@Yp;niU3^dacnMKu}1 z5Ar)&(a&yrv&qe8TA(0IW^J)yLKLW~NQeK22`rww42%gC*pk-CRA0PgWY)!r-Aa|8 zwwvFmk!4>NKLCw4YTKyZ-*=I@wrrhe?D7H#pY>U$rA&aQr{apQI27z)}Qgzaf3yz#fY*#9m zU5$k?d`9f^mu`4TaZBKhO+VHyN*fXjUx3z$KUf81ooY z%)}SMuq+cAF8?JiYS<@8r!3U^x?Yiv8?-&Bh`ppMfx-+QyQ;uU>PRWIvb(2~qJ4kE zMYnc;QobsN(BA0$bGMQWf0Oa?gA;>nXQ>)-=q>3+_F#5;D0&+?T+)2A3?+NwGa42J z`e|uW6^Q1g(7$qL{P8b>=4arozq;xAyHDm$brri0>}QD1x;KSQ{%x4B;)*0V!bB&$ z69;WYO`Uy)$fsf#ZrNLvd-iNbbKO9xn@sx0RlG~GAIPATM3-h+TcIP~elSjU*YByN z^DNSqrEw&WLQsi`xIxaO<-@|bbpgmoNh_fL%&q9uH6kN-#>y~4YCg#QQV+MwtmV!+ z4L)<4!foniXm`moI6Y^NdB?RLbbWp4#BZ>0qUDYCgLWkLM#c8d0*u_>A^SQ(p3MR~ zoEZkOHvQC|9x50==#Z`1BMgIoQGibytwJS$b+^M@in# zte2UYNU!6rbvw=dxpC(>Sp4ol>67~Jep+<#{#f4`%^+~4)JBYXh?w|0sV*c%jr&9T z=?|9PB|>}!=nMJq2K9ZaX|K817IVOq&n>d|iv_boZ5YI-Y7k~BBo1A!q*U`Sgj=8N%UdKF4S@B+ou#YP;{vGGqhVLQ| zpRp|~CRuXtts=9UUtd_xS4#7|Rs=(y{gZ!i?k0{$P?3C~2q+C8wfRrXq(E_8A^Z&D zO08SizpRU#x1&AIV0T0-7vIEH_N7GS=MEH5kN}rg<=f8RhHE&Q_!by76QEm&V7UEq zL;I)qwLSQ{^t&*+P%(_2c|@oK7=vjq;(c$&8GGYvZXF}+;4&>KsPZDH{5)T>6@au3 zr8C}ZK)=zd*S2m!#mppIkbh%ropSSq@gQL?8R@K%bx2Io{x|v&R`=$sjTE2sc}GbQ7%m zgob(aa8{P)LDqh{S3HCP_ZH&w1qB}|z9(wQf8lF4`ov*Uy@+7;OtAMX(!eyv32DQT zx^Xo`(Bh9tT>~-d8mN=({zM{U>Pu~C`-qt-!XY_MvGe8di-f1=gZ%kUt$yPnEVTdl+n;p-`sO zXmS>;T?_%;_-;HULtVIHIg0~}be}p+|8z~Sl)KW{ZGaA_;5|s|RM50s!P8Hv*+~c0 z^ywqlo^`^z>}YjS2ldW<_YmniG9n zmUa26%|&&Oh&kP%~IDf;AOx3!F?Qr(qhIB4XJm^( zdcWp*88ItiPm>@wxG~yfmqM;?vWoMk0x;oilNsw&wbEKzTs{HcX}7j6)YTKPq?ze| z@EZ&BNBy758Z+bnO<7~%VEZ2djfs$*nfd>!tZ^{2{J$$}tq{tIS}3eheiF9C-C%!; z66ZW`y9DU#)+8S6-{Dl6jaL zO=qS!J#(Lf61V3>1jd0)AQVE~^S}fG1kC*+a&piiA%Q?d!Tbdb3}t1NC@0_^Vsq0L zaRUT|4Dc6z{pAG(`Y5a9(4q!Fp7rs8tQpvWB*1`>Q^6%AK>dLX2<0yb2w{of2LP}efjmWLjWOzeOGSnuJ^(S^8SzpUSAvB*wuv+EDb{KK{^Kl!7e%fQRt!c0a_c} zXGgpW2{L%>-_b9_S%ZgpTjavbtN8~8WJK}h$ydY+0xeGTCtkjFml^t-K5*NF6#)V* zt&@8QG??^fJC_<2xPJ7eGk0z`^zs7u@&4|2)FH@&gX0H(upfs0-wE*ZQ-E3dPjr7r z*zdJtKpv2xfFu->BqPue3ebJCEBZ&bUpWEp0|n&A^~en38A513aLgJbp})L16Y(@L~U{q3H(jx%C&bb`lHnJwpX(6yH&^y=vPOYAf%Wctnb^lgdaQt__|&% z@7J{FSMvS0@cowhSFPmNR{TQO#>V$x%NO{E-@cA}Y5LX{ETu!2vI_P|U!BqIi(v`m zA=TxX5MG|)^*7ogzfYZ$(w)PAiVg@F4J_j4IFxufzXuUQm~vw0onhnanEjKEgS`x5 z2r(q+r^m2PLtyWBsE)ePby$}~f|mUk+CN@>JpTvx$QCC6ZQ1uV|HJ^n@vWNAfYBZZ z(;w&)$q0eKG936zP(7VX!0WdThmiCPSlB=bXe}KSI0gWAe;b&R00z1K?ebkeR0JfX zF4!6PdkpL!1T1v-3lIes3i|~N0||}!rVp&;F+A-m7+BGN#J=nCZh8_SuI2y4!Ew*= zzvNzmg4_qMq4;J5u1|`zv4}b4F3~W)LsrR+?QBhRxf9X?6f9)Iu$40#G$a|vXERsJ zm48`)6kA=+m(^Z)sMJ5|ZhKdTR<0;KNc&29&WB7=SzNuHjL4UN)821zMY(5~BVmnI z&mD#cV~GCp7!$+?1TN-5QHb_^kNm`z^J%l7T_A77Z-4N5Kcl&KG)*a)q z-$0@r&+xpXGF`#tg720v%uA0hMm*3By+*?6#g?Z@+LK2^B&mWroa?ljWxki>HVcW*Tz#wmSV zSz(Yqm8|o15()mfKJ0Nol|St?Eatv~r-7*wbnT4v*8357CIu#k)2DH(-^Rg+mta`K z@@zeMybzcovShVM0F@2)QCMCVq`clFQfH5WtalYP8(4?`aDZU{52?c1itSdhbdebG zEnkxIY4b<~(ENt~ex!PtmI#KmlyxH8eJ!XLIoJG$VTXD{YtvdBqctNe$dE6Ze`Fb~ zrv>5%IOX?VOm8Uug92hC`&v|TC>_Zu2wc#I?j<;BCOwHRYGc93&{ExR8Lf_l2SY<` zxfJ3vChJCnF4nGajqQ(0>FZsnj%=uGE|t!+2lSh+;9JcrUt6n_d);(vB+f;GOmZTc zRHrq8ce1l)C{F=b9$XJiH2uIW zyffj)F8w^UUicxm8(I#)ItxB*e0}kE-1fvh*~e4*3+NhC&deNf*gj-{=4%d9IH26w zbwmWF_V9w~))#i(sMl0a4J%}#2xpnDwllw|BPWAEWeo~|O0=5aT%FR|=MTRT{nJj;ockAFK#uix=(ZhCjm^c4?aUahh*;W18OIxN&J3GI!jYF%V6s zYo-Pu>Nd%qLa*S#m}qlMGkKdrKKK%NWX6vaQof8^`G0ivW}i>q+lU90G3-WKd>J{H zy0jU7e1wS;9R011;nU;e{+SPmSA*>4?On0be6k078C;_<^$^YA$Y8}-sQ*jbB#ye- zK0fdjk+X99<;m@?FQUL=Ve}(t?{=ya9>*)EJk)Tgp-=EU@()|iJBhR>{QfD>iJ!}2 zo#$U;-QNy>=T|8=3NllIbEU?wnK5JE_jHl`f{P9e>Knimmt8<3*B#aubM~N`uc@ZO zJI6(v)(T(6tS_}t6g=)2#9d3jd}wIm%49@09dwYLVp$mMe+1=&ekd}`(N~u9+g=>1 z?SoBp%=bzN9OJ$J*Kj6`(pr%=6~p883%il2Pcc#MO}{OQ-M>s3KwxetOUO|q78E~n ze95e>5q7+H8RIzuS9=Tls$tnE%4v(>**0&us3=!ZP!NOa#jOaG*1B*%jd;Q5szWEO zEz-Kqj!*7oR<*mH`GuDSq?^~>y+PINHM&@LHZqdWPKrN=ZM{JM1XKB`({r%vaR2tS z@=;u<7N@_Y+W|umsPlk7ztz;GIYkGujW<39HO4IrcI2O%cIwh0Se^FoxK~LHk2<^9a@*#z`M(jJ|WARlsf0cO>^grg9v)l?+)wLdJW-N;T?$^(?QIxwJZ`}Xa zIQ3DurQ)aCw3%K`{unm8l&p6HG-PFV_5p%b9@HC$0TGF6kO^652)d+UTx$);=N7_v zwrpR(T4XVq7Korz2oO?iV;W4Z@!l(aNpNa~`DY$at|ibF}tPjn*n9sQ#|Ggi8`Cl#Q2MSOyv z==1^cW5|r7sUsYGl(MhD<(<3%htgfBadtzy-sQ!`xjff-nC86jtVi5Mpnaq9bXsZi z*Ji}=N(SNzt**6v)Ie2HZ)N3GWD$@O;C)-)g{D3HV8wC@b!tudKNV;^FBCUQ2~% z;UQi{Wf+|8DsA}3Xsb*CUkJ79FqQ7cBdHR0dC{x|UYjvGxGcycmSZh4<$&&Vz?Z;`L;j7@`ehJ+9Nk^)Zp$|G2@L62>_wrOO@MCqE zRx$kqyjJ{)L-Uy&@uPQ%Gp$75;O8Lmmz5?r1of$D)Sp$lg75Hn?NI1El!mDpA9LQqq zcBqJ|sVUv7O|4|K;Wxlq9_DdfZOz4|ZgExH%HE{l_bKgdG!$^h{q7@fs26*XN;~tw z2yX=T>FEQXFa|Qsu1t@FK*F={g)Ho^19_M_an-2NwpybXk)N9P`pb2k7jqXe>Nk_o zH&wxZ)W^#Ms(88sSvY--y!L|Hxr_0JaNpJfL=7TbNkBPNTq?&Lv^kDcmO~tV++zB~ zwg8Q07}e>{NBNc;mB8`F_Ob#XXPT(v%o54JU=Q^om57&8r^4AOu6ajwsD`O z;IH)OhgmDiUSa~~jl;&fKH30n4}op^W;#EwMQ?y3P@9iJz$8@$=MuHj5-ZBKy&J&k zZ!s}@r2^)mvsn!eMBL(P^0y+H7MF;{W#x2`fLfkq+)dHJ>g{sv_z24pcvWQr; zFjom{YLbuF6~5subO7s}H5iImtgZf?ux`1v_~!M^&8wrR4qsb$p-KWbOJZ=W z+xwg2acEob4SZgWa@x9 z?}N&&lNYb($7IUxTHb_)g+pt?4urI(*IpEGO4Ouv85zxra|zT9v=#$N#zYZJy9Uc~ zB~j*M$YG=mGn4ZuK`1{@B&1X8*e3GZazSeWi@(?g@ij;yG%}yg3&|#$xs_u4woqI8 zpUMlyZX{H2DvR1mrfr`g22TE!y5>%MFq353Y4gK9-3^?2Hk5Xq)AU6i$F|OnT8BrQ z^Vn?A*fE9=%1Si?clRdX8(EL05qAdbJW%Cy>R%o+%o!2Gb_+>T+b6-NYm!IG>hP zukP3(Q>vz&np8){ca5wq+>SE4j`l}j@4w{pBh<|}NI~BO<>{IjG(jDN=exV8YVJh& z`%gtAmrrnsEW_muS*WA?)p~>6Pq^(})(7LErX(|v5yCHa#B37HQzw}2V@bn#`oY3; z^&@EVbe6^ekflzNNKGr6DQ7U%vCOqkq%!!nwUttp`B{O!&eY4j`-i5%>7H4weyA@_ zzxKuQgm3lIrO#IqBfocZO~NcuOck|v&x;P<2isT^#sT=N*~h@4#=B~oGZqem`^SvN zs#L0we&$kogc_#^)pd`z?E{gp&97q<1^11$Ef93!7tG|Ako|J30NZ=eOdZ zB+u17<91o!ad%Uqrx0Mx`b!5DLsb(fmT1QX$#2sh*zKYs3q{r=_%b<}rf{uw%Z9cs zd_h!trp$0_Lk}|fUeNFp96zFl*=fzD}b@13>jI|driS>C*e>k_SDL7TmBZX z`!wOSu)zo6fVf*+?IH-%(@Eh*)8(9fb}cu1>qtaDj8Q34)^d{s;laQZ`Ho<}KOz!7y82?GX;n{z5`B)wNq4WjyPl(iMS~8ky|sSgw`l@eD7h|XL+f&;_3u(TVlVs7`=4Wxc|U&L6>PBc zu*=t=N<<>J$Br!>I>o@9*EjnKCy%VH7P9P5zyJu`ps$Pyi?RZSipMr*ds|__>m3H8 z*z&`t&P|${S9BNzMO2xH8IUGPw;BW%fx?#j8McvbV`;LkV(u4SXbuG4Ws^Xzgv%9sj{3n zk})RgLCV-WTZ%LMPE;<%=i8Qvk^Bu;@*FYh$qO4im)u(7P^;)si0`AdB61Sf%`i396$sTAhFd0n>IJEv_wi0 z&LH{mL$u35S1WQ^bHbg8Lb49(5y!qjSVXe*B(`X}s!=Z;3Nh9QJ52$%4_||B7w#;_ ze@L}%u!5$C+z#r`bBp-}6Ya|4|BS%xlXu-{3V*Q!btho&+nzaV&{pZ}$fQb_b)dJ^ zCL}!u#VY^g;J+jTx?JVs9LY17lxfH0c;5U;mcpYMJbJM9HgSfZ!|-iJE$=%aVMXd4w`Ny#X7pN8H;ecmfJ8t!>A;-9<>Z=3ZVY zoO(QO@2tpELupdD+vdEMZ2hLGffc^T%5{a$*7N6oF?LQ{f&f~QP209@+nJTNZQHhO z+qP{~+O}=GYhHSK^|}xJ1Lp%y#NJUE-jTjZdcR?tlcq`skkUVF$DaaKj0e)$a=pUd z_h%y0?PE_FKB#mdHm~8nkyphGVs{dy{OotjSrRomG$JG}nO2#VQ~6M#60|Hg&hdFM z*oOpzik=s8Z=Mub^ z`^Cr|Fn&XuklUClkaiPwJ0T2?(38}L3NIGP^<+i$P*@#QdVptGz1Pg0#3JoXK+iKa z(CDA72a|4OGZ7`!UD4nX!|X10H*Vl&kdbcPbu7$OoY|23wI9y}3uBh8&JiDnOLZ2N zXuh8DP9AScqG(ZgCGaQ%LX^e578Q4^>RL6y)-8NTR_>eT4`b||Q;xWS=l-t2b3|+) z1rFsjR}#Nf%(a5nCDIqGR}xFr8>{0pweDF2W~$y1+lD4-o&@Jo*D@FJDkHV8;9u>@ zCsZ-#gXLhoo3bI8ti^c`kNT8dC+~x~!x!?ZPQVMuPl=LtaS^n*AMT2sa{Qg;YHk1Q4 z$qsIocaHFZ^jo+vH+~x778on}bWD8lc`%G-doqitON%0KAKTzSEM zTn#YCo_v2%x7~Ueo%Si!vH3j*7rNehX)fd_8Kf6rLI1(xLF$J zlV6@Mf=OHNJB0G(y6&-WKOEIsG+>jmO++2Z83NP0nL#fdHl?pAGCMmcKLI=AFvN%o zcI~t?KgVB=y$KHzI&!+rd5v&tsHOpI_Uf^yu=T2byNUaNl)x~q%x6Ceat%l8lX0EB ze-fM&y{%RD7|P1fo&>k*-IgQWgA{f;5`4V!hq%H&kUlJzt%10{5RZVd7>w%hddA;3 z*@~G8-H0vsVjE=IvX_9hVBUu;Q*6*(JQLMG(pC1tfSCEK$SjmI`nw$RE#jai9&<`T zDp@ggG?7>ZqO+?8FqfWgb(vrQEv`-N#kpxBAkOG{KO&`EBWqwt@#)n0{zH#&i+7F6 z>)oOk`DQ756|P$g&Vov~UD9*sr1npJz-2PRw`KIAg+Q*^reAo-8aj&;t09Mfq`x}F zs}iM{?A3jk9$8#~-01asn@1J-PEg2+ODLul-F+t73%v%||9mBf-hJgK>aF|aeQ<$F z;22*_1^Px_P`bHHB3vFMq8PjPD-A9qc}-A z?w~zW+f9xry^2bzzw}9P?q*8Wtqtd~Gf)UbWxME3opoM>){$uL3w`m`u)gVc(&8CA zF}1>T45B#VIHxkz_S)&L)}XM_Zf`%9=WhztB8fyUsdJm+u0J6EDHPf} zXMiN?=WL&Rqe_Ib8!CQIh<{+7kE}tLiW* z-Y)Cq^K+>`89@xT7rmcEB7UM2;iUhm0 zgI^?SS{5X?T z`Q-vxaw)f6YFK#WFHH%=SQVY!I09iVHu?siyhC7T@42RwA2lm-3^8P@!#z?H2TDJ$ ztK(q8?fpSG)gB9u!7>Caj33uzx80<}L~}nG3Ng{gNIz*D0SF4EqCpQ|TfBWQ_F3#F<&I{-DadzA}#+ zhbopD{#%ixpu`H2rh!pqY%lE)l)F_aDTVsoH)0h#)Zg2tvK#KnqnsK)JoFg}Uwn(8 z&G*prn_9#!3+QDv+Qd_yv2URLE{?k3vki$OR|f4&VNZiX-f5jEbCZ{I70y4mx9xINwCX$z(9A#K_bgP9T?h#VKQ)AQ3b?`2xo;UrCwI*NU4z|t` zA@hC|C1<~+AJ$=z!|vj-!{yib@bQTqrYtD(k@fe>!^WSq!;xjl^wJL>Gzr?`J8icr z?xuy_{zJO{+!aK5Z4zCq>D=_gF`M!vuCAgveE2NP{GNI{dAsz`r#5t30^dhIC)Lj1 zd?%P;7<5s98X{cLX=2fwyRG{(QS*xk6Em0ncH@+cL>%FG-3`fI$p?YDwm>mD=;^fE zF>A^nIwd1FI~l7 z(SkZ$N$%K)yQ_E(iEZq7u2*~3Kv#q5=V!ZM06PnArtx*Fy(SQRPeSdu1bkOX>v1w| z;7aUBzU^fR^VG>axs;Y&WqdaBj%M$+3oauka%FBQw=$VZCWEMX-AY*B>p$%szf@#b zFBKXNrfeY{h;SFZoH?kBEoJE`SLs&OE{Kl93$`!s$c11p+pwu)fLfP~ZW87(7xV z92~_C9|`ji+LMQ~lI^*jBKS9E(*Xox-qqc271-LNCv)d6!|;5 zh7!u(i1d9Z1$PbPr(3=5|2Aoefh<7pzor>Li>Q2n3=2892N zUg=Z#Gl)wo%S!_QIDz(e7rgwti2c^y@5?>Vr#}S(=u0P#?oVq9p$@Qx7Wz*7;1uRQ z2tdIOukOF=$Ms`@7%%{U&4B`1AG$t(DB`C!*3YoX&u02=Yj4+Ygb?bb4{)}6yK8rX z;kizLD9h&;_Pay>OYC8!L~C-xkNmqyP9OIK_I4i#*bOvD@Q*i#0t+|P1^UtW0}k|C z1wFT0Q3;_%0s!-|NPCj`v)pj12eS7jg_G^?hqfqWs16Nm|0Uzd9M1p|>EZu7tNfeu z@N4u|NA-(6{@X=p0#9DHYa6@!^E(FX5X$}jh3ZdO0}e$DAVVSw7XAyz8v2b`LpB9= zv;VcLtAYYT48j?V8#>jkPskTPhmUhp3-cPfe%7xb|H+^H9)$gqn2C%7xG2~;;~==jXej)xdq|HRny|?KDK8|Dz}DwXPCx_X-`*kU9nMfS zralP3&mfx5#&Ekk1$OsH7*8P$1;FeC(2sU5`eZddcmUuV?Vdx}x3>#{K?1WE_8UeC zwCDC_^K+l~Cqi?h*kq$AhZD~k-+ENIk}(@zX1kpaq`_C;XK2%5GwP<@pp){By;sb4EhX7HMdr&@zWs z)B371f{CU4bRfDZ0d1>|?udN&Mc+X*`=eRDo>DlKK&#>I)5RLEKKh@z?Sj% zi*7R(GHE($tVFJ`>nf2?EhZ05q-4#8x09*9AhiN^uP`FH&;Cn!3q0BPL%{$6i)$F0 zFs-U0!gdC|%iuk`FhWmT8=POMF5o_OvDZeI^)?~KgJFD{Jq`L;GL+XjZ95geB5%#t zbZYB>%jr$xpPNSDi|Zjmg*nV9ual5L*Y}+> zvvBeUXf4AI(aEa0m@cvB>xQbR1s7!;v!M)v2ZoCZ+SwR%H#~boRZu@96SrIxXAp$;eQzE(MRRV1n!UVb0 z>iMH0*GLdhG_C@y_cCJ}t&lCILOv(%q_*^qTo=IJ|G3EIc`e=!OBl#zCpoi(w@T|y z$=dl9RR8oC$uHn!r~6xYvQ>rpH9v%ypTkZ*+aC5caB*WTZTA zWd-QkW8AlU$w!U&*PUUr;S+B2>@Wgq1 zCUfm(Op5N2V1S$Wg%3YRPYB= zWAt=}Q%zF80;YUT6veWXFTw3A%?D_;lSN{i%(->Y*fZ@kW?Wd(j00<=ti6>?*yk>D zOOoYXO9BP$AdM4h9(k#*Ya zLesnO*B8VCl!VqetA(sn1PL-F(3Cu{rZ=uD>5KH5!@8LJG<7)z6UDPJVow zvP?-Rdk&f4*?qHeD4e<{M|!c1@f+^V zOtXH(KLzpr-oY%8vU2xta|M^!dp}7g6%UTL2830BoW`r-u}UVNQoPM8G3U}3Ub`2a z%5)tST>jl55(bpFu`x_z+v0^%j?o>~cL=2A(c=#zteRMV!sPGNz_y-NXxpeWI}X4h z4lW*8hB5FpB21m2+4d(TSJYX*n-kq>KetArBoE-(n0xB%6=Of8QQV2%HO3DyFkf*j zH0|OdsiEVXfzOXrcl}_0kaqAWG?%u1T?z{}LX1_|%o=ELTzsOuh*~Mw8m}Zi3C9~l z5bnA0E|w5a5K@&vlyolR0tJE9mf1j6Yb_a(=B3(@M+e9@z^D*2<%_)6M**gx&El8P zhDzB4a9byILFnPbcS=-@5SHd;S7s8?MHEea*H~Izdrr{E?th{Wjl@|G>1f3o@~6X$ zbapW)XN`*_)|V@5`C_t$l&jcXEK|M0Ue&#$sB{tun?btN(!!UjC^>|H&*Ujv%%iw( zq6?2Da#D+Tli*i8806pV&4t9_oOFGF&3dKp|%_8=&4Hc9!E9z$VuO9e8wEoE7Iu`P4_}H}D zA0;--wfWA4CyA$YDc{9J*Y0hn5W|^ldjGL63f|PK9!W-k zON(U{$35UcGLySZc)6Ci>X!NdQ*d&hL*Pn`P$h8s5jRWXD=rk~P`HKhQ8j;hRCM#; zS4WD~foeNFckK$AFwfA6XF@pK!jPpKs^xJ_^W{xka?t()k;Lz51odo=G4(LZLGw9L zympm#+QLoNDEM!kY+5fQEw7qVh4A^hi34F z*gIM77;*nGGSV`V)%Ljl0pB9kDw=KJf7}e_d%}kRlm^;7tm4>>y>>!gyFY-PnS+^8 zcKX9b4`xq-l{ma>j?H>4%g~H*n?`ovtC#Ppw@wyein8{?RvB$zu5%!`%g#+YeTG_X z8E3UEf?h;Gh0ZoVc2MhN+$p|%n}+TpI_~a}33L*F)2<`{4vuE@K60=q^=r9_`9pfl z(YBy8h+ga|nu`vJCKl%Ih05L#7xvX9<*wJB--CaneRGhSz&A?*k!(P&9@lF1`bUX# z8HGF~@5<((Tek$;V{$?Ltx}>^SFnSJc72JHdg)-L)fxuhgK)#AlG9f$b0&2>8;`d} z^u_ZlX)q@&O$Co-8at^|x#5rx%7GSj^oCh^a#%-2$~u8nuoBGKcw$Iy*>yudD6Dul zx9NVRUp+=Qjti*5%GPyO>6*lgs{yAe>Z)uv(wEvb5k;oDagGc<8ck-Fxh_ynEFN-D z8q%x>JWN%U_|4~LJ@XdPMlw&oOS4*&6b~e4+1W6mhS$I^3)A1qg~B~#wZVa5prI9r zE-TZ>d!b{WyWo`I)a5_{5Yb4`0#>=>Ap%Im*D7?IV%F|#a z-$>sYt$&9``oZ84F5gX+aFd-Gj;v*J{wd>um#?SG$o7Z*h)=MFL+LN1X6PXIk^>q@5k6yncE7 zojQ57YY>hPsDQ%;&c+~3H>FF(R=crlRgX+j0^E^xu(`l7J4&ix&^fXmLtFHr0PGW@ zOMN<0%WUJVZhPdh_)SQs9ya!B_wAT8E97dnyQ$Hlb$)Fdn44xMTFO;fU+JbAv?ROi z#vI7&T>68(VNb1$(dy4H>bZ|18ksn1w|}1Cpzy7=NN^(yw|=#51l)))9QQZ4SyPqv zB_M(+RnI2m)O8n%#Gl-gFmAYc9Q`rlL^;w6BAcaDj6KbuF61Scf-3d&}3y%Cj zm&>EdgTFAQ!(;?rh;^b65mLt%$)%Y>YP1EYZ=MZq3!pqLH@^0YKZ&+~|L(LO9QLv0 z2dB76^AU@?4anP6Mtj>VTihJTqWr;mM_O}S>bu}B6W<~W#LfLZm+`xJSw)Yt-^qGk zX0fAu=rn(o8u7J;gZ;*dXDgNU+b)|lb?5dgsf)V+`@JrekjVQ=^GMsqH)!siz^dTu zM;E(f#Of+@Mb&J|xP>a8MSNpPb`Pn!U>T#yvN0(WQ%b!?Izl9Yr>*3?PkS1_iF|H> z;W4DFNNGB)-bpU3VaO_s?QJtn9ZTiNs=i(K_699hlAGU-)-(jKJ_c1q)?R>$;5W0$ z@s_O+Dqn+VdTuy9!cb4!@}tSn&fZXdzBdfC-VPHxzx+90l_+}5ZQyI^7f;;J;|&8b zqu%I#+_GmY!!G}{SH_|4G(DnvBEyw|TLzI6>EG~;Sz2uxE26IEgXvjsBHfzL#)){_ zsU>*ErxwKCPJ6^J(^5E2s8cwXxDTP7a(h!C{K^!RI9754@7bCD;UEosW)64U+?&T{ z<{SSum?mV}z$Rn{ZozT*%S+6ZXT_S??AM=#J>_7qZl!=VL}75uPNU>byKCz%%OKOA zx)|+a@T2;}8?yZBNj{i+VI^s1Uqml$ z7IRr9)3TgcOD}H?bLT{uXoPRM{k%6pw4O#`+o2CP27jYtlTNU7qgKesr- z&0ZM)LJcaNEtcf9+*~p{cfGY?=5OyuLq0q~F7EYSwGJRV+(g3ql(0#0u)7Wkv^z+O zOORqWFCvrOnSyjZ_-J@Z!e-;=q?Zkq&3B3IW$(Qw&eUG)k2n+c8UK+!=e-+TXRAcM`nE3Yd=); zxVzBV8`bLaBxs>*t<{yKoOI$ZMEj^h@%q_rbUU`VB};8fa1E+xDdKm}-*tO&q!N$o zN-N>AH!ul|gEu)^=2&R7^2ZVEehIJ7%=58y&kIj=bW5gjGmWaw5IhUch4kU034_Ac z4_L;>PW}r>S5qeI2bJ_|F-PDz;`XW>seYk(!hn0KL z#1%pg*K11PCe!U36G3;LIS5&jiHY3B(AVKyJ5RW)P&(X-37S$-w|YJ>$OCPO3tp0_ zdG^V27l+S|!rTLxiUd1}BcHs0GoHZ03$}h@w|(E=hNL}P$?l?i_F!NErNn@BH$O&) zM2*S^B#FQfBh{&U1q-2Yr-FVSt8}Bt!Zp~kD|eVd3^ZyUwZWls#+Gb?tk(lqN#JPe zKL4#DKHf?=|FY;454ef|SL!D?qmVX3!!)kAG-vK(#am)MT~pxrFNr5DHx*~%x@NF2 z^$S=3XgI8|belNz5fD_G!qNR*O)a=6Iy@FBkxiU%fp_9NuF)=bk?p*&?0XTlj=#@k zfb2+J-Knjk2g9N{+!Aew|B-s+JaZ)(;sr*ApLLw8?-eiB|&(RMqUw)1|No63wweomhl9)36*+)_mgCG5(<^tH2S#6J05UbH7_JX8{Ci~_si!K^lGDTV`yU)?;CP zj`}(p=x%v+sV|3!Spr7}N0tkXsO+1ZDVb{uZ_ocW%qHS)2y|YZ1wS7-cs%GNk?{Yy z1B-N&6YUv%M3Uz{b4%gg=DE2eCBC){V#@B&gE%XzKMr#;Usb9UUIfWY?6!(88n>nl z87eC@DbJ3Sh1$iBIlGB<=Y(I7T--Zxv+V4u=5E)y*)SDilu>TVx^a1OSQ~Votok&p zQ&yYLChNJ3s9r;kMW0fO8>NS3rVQcMtzb5agQmvP#zh)XQsgz?P9Mxp_P}{5rUI5R zced$4P1VxP6yM^S0`PV8{Am2mxzsYQ#Jb^RNVEaTNSa@I;sJ;1wg zBKqkS=Vhp*^hfUlIoCjebreX(sc)J`hi2`e!>8P5{&OiD$Lxq;&9fV(n>ome4lmvf zFSZZgomzS?-LjBoMbc<<@u+nOi1E?Sf+pw@ZqT5OhR=hDiSrbW0nFFW#i z?v;VNbe+;%dKl41D32LEj;trPi~mAoAh97sia<7nYVSoLK(Q^>1W+G%9C4C4f@1Sj<3-8Zx8%VQ@uqeT6V;oNYM+FJZuVM9lxYu!4BAOOC7NUs zSC$-^r$6Uj4fQbVRA=LAd1xhvwoyy#b=l4vd{J>cHYF)8Hy)ROUfRolkRaiXm@AsK zV&b>GdVXrw6x2+#qqfRTv|g)u!3-JnX+wJ|*VSv$_6GNm*1S%)(b(C209n0-{YrFp z_oY?6siv=*a7PNxv?Dc#*K40po~bRkvg^f$PlFIf-j;NnUA9H4tLd@mw_SwvVSF@k zleYL0h;GcD78`{7BVXO~`7&kSS{r~CPyZ<3gCYsW0GKA0=}o^Y6&) zv);3;>qp&M5-!|NVGR#i`qB)`aI?=-szaHMdc7837RhLbee-qM1J} z0D=}C!2Du24jSh6Ssm|kDqe@3g|K+eQAgwzy*g$pk4K=tIBwEci(YM6*~na+qbHj2 z$!cX9x3|bCpFvYB+FQCj$`@fSU8lHbm&cg=m_mL)g3xS}ZP5Sf)SuBBsY%1g-Sx{gr%N-@dq(_U2=#y)D<~duPoUMjsT4 z0E0@4MhCtaxGxGg5uNbJNmbUyX-yPX*^Ou_jt=VDz+uI^d`-qfi6KC;0MViGqK5~L zg`;M6kJ~msnvP;)ceYB_ljL$Rnb#^?W|>S_PN==iANs2^;_6Bm_##8;J#^$RFt-Mh zsf;I)5*Z^pFtD%9p8!3)REcx}! z_mI{(ulTE|AKFYzRX_ym>7#fobQWc9DX^*QDE&%~RoSn#mMpzb5&^mBF(@$7p8lr; ze?)vIjtvN~;cY7GWY}RT4oTC%Zty&RJlH4P- zXen@4qme9y;dM^D9?>623EAqLVz#jLkv_ZuEpC z6Dk360Cyn;+WsO8|Z<7K{lFW)x8Ux^U^{=Rjb1^22sU+q+;u{dqe81cY;aL*Ms5m=K|F$1o0#LY-a$2tN$L z)O$cd!hpc_wFn#Vv;YDKGyJfG_Vgqx_y@oN1B9{aL3$a&0QHTC00U@mWjyrcC{w`K zq(`7ZzcdPOW*F9K%ZM|T;%;sfK-YgBnE6@%b;E*fy87;J$9IGYd=LZtK+rJ;*N@E* zXLj%gO(D|D*p>fLRFG&y@7PWug@BNVh=`P!i~tvq0^iwN!G59a&#(SJih;g^Yfyo} zyzzMc(5(>W3{#+y-^7qlfo}qUEECvRIB$88ev%{v2zxM2Lcp=|Y8Zh>w`b}Y7Y|ya zaQL8)z}0{1w}1hO`2BwQ)(y{(qec3^f7yOn^$e(yX>nzce|AxSwaN+upMZft|7lwU z6(A7-o|23e*M#r*hcJZn{TANf^=N~kk>cNTXj^4|C)Q5k0o{LT!13??sTaj*tO@{b zcS+h16A&B*eFc8aDt{f{e;L2bl7C~4esd7ku;G47PQFWi{gx2iFt&c<08^WH%&&mX z2pg>Y-|Up2&u&YWgSdvew|}*&LPGV?1u^%be%-JQboJBmm(~$NT^&ECQ+M0fXKDYR zuvxem{pz#{2*Ob8QIG4&l5=~A4_8-R@<&myA?tCLdKpnx{4|f6jEWFaK%OF;AJX-& zx&#meh=JQY=EZB&uzx)QP{e2_z}iC~AUa9p(W%IY34p=X;S=%~Z4|&D-Woa}U|47V zuc)Bk-0tERwfNpA9d{1-J`Q29_At_@AwsXmbwVnyDk|_6{8DREZ6`hz`DYnXAL}$}~ zQC732922Mbr$N5H(z^^eG-FeK#e(KshgrPG8t<1|`)x~yBTr9B9rlR=?q{pjThDp_ z)RSmT9ac?>)J1%*K`7|qG~PyV56gm#@uxi)Q9srGAW-JM(`SQ=uk&swFv`J(|)7 zIl~@vbfb$+8rclTvbV{oc&}|+A^oM5w#%b?W$oj5MC~`PaJWa2J$E_tPK|@hY|K%n zwl;FdtrYoO6_hDAHz^umq@QV-!gh*lXIeGAv)f>IT}}7v(pNTMV&t9a!mJNpWMkZF zoBDAu&tOK-wR}|U?m>%Ba!-$impd}eQ-||u?X%B%HMk0qvcB%lsB>Iqar!fQ27WfE zs&ZVeS#jDZ1~H*3e)8IQ$`^%7qKG4s>`STlyY5p~d1B1u)Vpcp#z$^g@Z?0H;PQOA ziW#{L0~h(JNSW1K{L;e-vLcuDQt}ORJR)dB^Vv2JJeJ9^cx3FDcz$acq`=0No1g-Gjgw#UrRp-o*$)xn(=}&3`r~NP;s0F#%~NKIS(@g8g0#LI_Wt;NbCGdbftXlfRb_)-nEvL>cr?ZC&n(s_tI1 z*v&IrPjXeGz-llQ0hbm8-a7bYjBZ#!>d#T8H*|14HS{;vlB{S(3MEgVB{xm6+C)kg zAZ@a^xi-2svogBkXs(>*lY#7xn3Yi{6OEbFj~V4>P<7)*=6#a#2Fgt-kLaqijLjuvLv#)iL2{-=i;V#`rch@F0sV};9#p6Do=ra*kvnpChVwIwVwsPv< z)3`^In_HSgVxITjWWA+}3{vMri*8a1CdifK53_oD9*TE`>>CblQOV$@!o<@ zv*QK4KoWWAN->8N2yMs9TE=&pck#q{ocF@OnUHV=V`4|~H>o@3BqA9t=`e)6LxC`+ zKFqK5_-~7T?8`ugJ)xTxFy%CI^T&42&k(Ul(K*7-N#Uqjk&3n%*weD6Y|f%EJ7h&Q zObLwRdmxa_<5TG{&0u#eqavMsd^b;C3X#Cw@0G#5turs*h-n+fd9Dz%(Qh*L*0mr=Jl3~2XC57WlLOlE;@5jLL z*)>*Nntz4tL-wbmd#i*uDRwhby{yS5zTOizGwbP7BpRbXHc;BT922l;DHb$jCoj-l zVed+i79RiF@_o^X6n&mA+nz1xuAW%kV^tQ`6NGD8*b(~N`O-@(ebXVb(1oWM?ZTj-70~Yss4K#*fcC!(@iO#o$?NzPd}bMuM+>{2_WyjU zpy%~|(fr$}UWl{^PtF|~EKkMc=deIvS)(kaB3O299^JGsT|{J&J>c~*U*y15=inrI z^pMpTO*S;tU)CL5oCN>;YNWR}LF-H9-0QylVT8?I@ie+VSQD!L4o~lxN9QL0N8aGB zCsXh5Uo|U|_=-wykxaiCCoStM+tWfu$s2HVB3SIrv`~scVGhPPY|hZ(VXRmf6Efr0 zn_;hUk;4ZLnzY75n}SkWS>xwC6k8XM7#wH53Iin~XUM)(i!nn+^l@6M8z~JKcmkb6 z9MqPzg370ysIPhxfAN>VQKE1iZ?t;mXY3JDAT(}j%jzqqPPxrJ|$_AqvY0HA(%dIKoEbRq` ziSL>QTl2*#&HLwWiM|ZA8J_<(jz)&I5p(WN7#*j>cv6<3+TEg)KgL^1(dM<^snaMk zrdR%WD_(EOm~q;EhgVyn!sOMsi<%MC*vV}6-m4c1iJ~urAiji`y=`=MG#86>Q%aJg zTyo-r=OrYoUN(VT_*eb=XkXu$aLlw3^=_4IyUx+hwvlEAprjuk^j<>nL=`+~Y|QJV z*Jw>5%l*Jf9>Np|bx`a1K6^@Te;D||JD36fs(OB39~9x$1?d=rwG=7p7yfckB6d|$ zX=xMWAScBC`J2!M3Bkom=!fJe9c{Oln zd$v4rhXeJ(R?cJcc3~HDPwxnM=4HHBku<-W{KI3(P2o@8JQ$HsH~l9pIP$zrYJ0r5 zt3_x7kXvGMhPkKOF6B!ViO85@mA)Z0sUPy@=Pps+VM$tUe*)EEo*pT3Q;G3=eN?vf z(r{oNPdi(i{YQ(s51=9LWB>8moE(obGgBDZ*`!KT6IBN$?25w$wyH9YA=NHv>C5a& zEWgEueRQ%qGG374$6{L!$y1yO10GbN;>I)8>F2WU;G3S&Zs&t9{Yw!eQhTgwJ)KtK zjnvBBMLBGe3G+#ELrU>UET_vW{b7&_^YDkWU+<=;xdKYrrfs$1m|-tQ307tQvwE0Bjd)5L;U@1t7hm z?sRkl`A*Tn4Ed|H$Ii9sZv14b-_>|1wgPSsS3yXhwNh_JINN3%nMBvOHd6SQ!5G|5;-1XQtx%|7e4Wu{xv zqb-fse7-r}+jMDsHDn8wG<$+oVHT@@}$Tp6`n%{73q%i8&}v{J#96 zMp1fO^>J{Kap8Clk)Om|{L?ek5_kZy3~DNwnI9Gh&t#_jspU8;Ng3HGUw9$zZKaTVKnMBwW=9Mcu^QFuU4SEBgD$I zvg@szSAXlZgix7I8?3D%x4)I|nGIJC-kvqe8=HQq$SrA1ZNE!d!=!8DO)+|fBfor~ z@7dStE0l{4k1H&TzX}(yC7MW#a^mKA~+>c$Nq+8KBlX&ikC=;#GGOUPcW% z^1^HOnTqc{u$XyweQrupEUFd_K+)U0TlGaT%1hLAVT=|sjC%^J5VRd=~TU-^xS z(j-f#yG&geYClSwhrXR!wl^JN?I68kqN~Q1mGaT~|IQD>i%HO<9eSm?N}!GWRsy6D z8D9C7Q-XLQJk)jQYnyCwXvju!+1sSJKFQ*cU5D;koZ(DJBqN}`x!SJiY0L|G+)$uL zbWrfC*BAPG^g-F=XjyDp;cTp9l{O1)zfBdo+EZj9@Lx}PF1I82!HKJiDrs+`4D+Dn zpJ-JE=|gp1Zl!`5SBvR+>LF!RFHWA3Fp9`GXI-EO({QSJ>vSdUVg+l_@4t^Qdboz3 ze(KF1$aGk@dkG7@l8RQXS8#FsjcLhXW)!fAn6S=&Zm{c7;qg0sui|d}=DxfJ2F1EYaRwAzrdTl_$Gnz|L)3RncY+}S#ndMuPkz(>eoq56k*aujjF7f_nmA2B^ z!S@<6Xs=M?huM@K%3fEc%u1gz2y+r}P8soh#h*m1sN!u}fL|DQ-xS9A3=qozR2jEafb!g!# zKJv-&+Pw}@sD6A`{~ee)xvJBWU^zS$<;{zH!qp*qH^a(2A#O~-8NOGdpAN?c3VqS( zYkv_`#p8Xc1v-T(nQT+g$o-gIHq?bZ$l0;%zjWm7Mp0`J7Al%h#?A&sHYV`E4c5H-6oSsy>7C zQttddTy`3Q%>c~quPAIvK%~#CYYV$+`}^grM4hCJ{uwCDvA1b3Q`tmDRzDRdm3#*9!;ndyyRZJc_;0V%yLk!CmM$G<+EaPNZ8x$vNvw}Gr zcO{y1uYqwdal8s z+=c(j{E$vVr~!6jThiHe6m*pw5*(!S-scOZFYW#Qo9Vis15c{Na?U&=mgJG?Av;Nn zxXs!E@roHd>Wp#M^tIof3w3l)d`Af>k~FuX%@87ogm5#lDI4zHzE;~S6vrZHpIart zL%)4Nd6?X{Lo~~K5OdSul#*U=qmEkTS{zGmQDfAMn;lv&bzVt?L)XQ+gl78sUI+=Y zcUiJXVNe}mO8t}qmN5%POj{_Y4(QDls5|H%K8P8+Rop73f=JqASeJq54)*SUGcKDF z7`v>5s5X7m@f^0_qaCJ9@6z0hrZ2FxR+Nu;HH#}OK4e#lEj&0kvZKGq` z?%1}SFV6Iys;NCy^H0@YXYcu12k&z~*LB;5apYHoS-+s47SoZ+u;)7;a_w_myD!z+ z>CL|%bdi`3=-EYr*o39WfAJv!4sDq%rjAyg%vF(ttt4DHOd$Qgpw8Ubr(89i&|J8h z57WI;oqdHGW4Fp(SkDhBV=iM8@$K$DOG$)OIN_p=SF(+Ik(490PI-5Pr~JK2L^{+r zUQt5q_p8&iZCeu(;j35?9bj-hN~$*Ha@!nIV+T^wZuF^$ z#Jq_p$Z92h^B6p+Vj>Z{F;4&SFxdc|632dt6Eb{8fL%IpbU>>l{$2~O+_O8+pgFCI z)|pG#9wANOI!%YD5v9b-*TY8&86j^GkNe79up(}xMb6W zm+cG$@M?)~vT?QtUPz81WA_ihj_ML2$05#$*>wlx$l;)0^IA2u4sy7oR0Ktrw7bD? z=5_9jydAjpk+Uej)n}rLKTHHkk-O6!z&UOu@XpR5Xe}jG<(1W(em>RRt#x@GdO$g& zXr=hgFDacnFf=>)RiqhTqAwx-l^^vFC0UBd`M35O5evr~^_X<|kjZPxOw22dyqCus zxQ=u!y6h$=(rdTOD%4TzjroEGNV6;2XE_gZ8Yth2ZG{zRDIQ}AeuCdmI6baNGT`3@ z`nSpJQbU6dr!X(H;JxP}1lk>l+ zO-^Puw*T|>e^Z-W%xwSjsL>r#5mkGYUQYTEfw&N6!TVNE8wyxhDj)<$2&@De zghVcZMi03(*aZp6Qn^I3URpw^>Qm{~{noz^V78MzozS$(U|#RMB7oX%hm8eG!$X2^ zXe9}zjYJ2Q1VM9&BL(t{NQoFp60Cp36hxwH=(_-jfi;5@<{!5C!`Roz2^TbKOoPP6 zA&VIaWcAh#I#49^CxlH?83zm`DnLm22U^ri22_ox*FmD74oML>A{;{{||w9W*vkkqZw0t{#Ptg~6h^ zy!p?c{QP{RVNmjrC%3dibfEWvc?Ka;ESQsPSZA>BBG{#1kA6Q?R1gPXIo-Vq!9UCk z(ASXnVZaRmym&Ao1%IFgpd>Olk-9klYN|k1+(Q2swfTlh1oU}n5nxDsw|o3O`K<;O z_nHeAD4a-;ha?Jww-4tM)HM|FZp9;P;ol5|6y)%U4&^98-0+5E0JVjG5CH$W%>`Cb zW&ssq`1Un|0R9~2U8E#T)&9X4Huw+PMASmxuZVfN3n2bW)nEUuy0^nb^|AyS;%`um z3vwRb|Ea}0fE*{^N5%E-0h3L#r!T-Hm)b}22WgLR8aoUkDp-g}iW&*&1U%$TL@&f2 zofG;l`>14+FCXs>;v>R@XsJ9=&wEj|9YE08&0AmTcF?2KfWD%`(QK+8FwXUeY%8hpW z?D9|R2ll|fyV3|ysx`3^P6-2bf`@-8UJ>5?v`zb$f9@`QRZsqi?EP%VXT^7Y*fY-k zQ&aDUdh~CDen0$vz6c(*GQfuUsb-i)zB#<#)t6%f;{}a^G6(b_BF8BQ ze+=Ql?^#q#5mtT<$oa!(@Ztfp3Ki}35ugx|XE7ioc#QKpOY2F}sAk{iri3YC;8y{#FUl^coPNEaBW+^)*fo z`B5Ak^hZ)E_=Ac^d(a1rOc(HU|BZ{33t&Y?GfN-)QRTolj0;E)u;*a7Mh@T)xxQnMVqR`B{`4 zZ6qPmzf@8?qCp$U(}#zE>tjI_Y<-e6!?6q-Pw7CeEtwe7l z37uGnn^>lDsTS$X4dJ(IygMEsyfPU$@=(zZ>W%Iw^7$jYH?xIiVHlrlWy1zF!V0)tQ zW)c2ww{Y0v&bUqNq&es2x5~GyM&vwW zjm##qp?e2wINbV@evay%lNd>4a8iD1Aij6ZLP<_-Pzi{~S{HYV^dA#eIz0G{*r}YJ z%i+2sn<_8mixz7`lL}LsnTFtINQ;i$oBs%qy2(^aj-lOIK=P88s~e!L52b|2!cz|T|Mf4*}DYH736y=e9<(Qoh zG^Afiw+T=2+-%49+~yRT@Wow?D}bm^1!3Nvev{{Fe57oN+gr15x|nNc&fS}{g^3HM zAqK|Sp{J%ne^-wruksZZi8L_bO!jQIk;w`bo8Gv<_PEr^$oKzP+HM+T?Q6Sk*!;&E zj*Gg)hAp2pKt;^^F_g`UjvXpu0X#+$UK(bT5o?T&0ymO7Ir(<}z@zn2Pj7T(knOj9 zX$$UGYd#|cs9yh6-pr|=Om(){DwUttN2~(2yLnE*m2TJDbAFngB(yHeUZH-yE)H$$ z_<0t$8YEB1pHp=DLAKto}K4Dh_{#&Vz|O~K-h(Ak$x2&z{;1bMcUyD19B zGn$jvp-fNF6TfI|?*Kay_)5k1uKM)|a9CK8x&k0K7aPn`%< zT+T1?nExX})_r1>%_KZY25Fqd2K&jd=2^9tqb*mX*`as?i-fXrV&;FTsWcbJ#&MQGO>vJKx1t&3dS0`nVb zNpBbH-2R1f9F!%SL7PJB&*#lNi6k+EM5YrHM-KYxAD#|e(J#fteY_l#%g>0Tc-99TD6aNJ}( zbNtXoKYw9z%3QA~3R{~d2m(tDALNv;ip+B$v@)+D07+3~c5PZvEVo{aW8K)Y+=)8W z4s1)cT$|KZ%0wyqlaZFE4hc=lvo6b>y%#(=lbP8oRvU2%R*Vn_m9>2yAemX)n?=Qw_AXMVo0( zI1C#bX<7g639|IhXUHxep3xUVllmj_Gna|Cy|EMU;0bO_L%1s?+)%4w`&rLUZxprA z$kVB$a_(7y)_dh~9gaUbc?0#}Gsz6{#c!3R@Ph9dNBIk1E$FXe6SlRa1K(7dq`;Qm z6iBdDE@G=mWV<_jNd2}U80K}&ug&(Y#!_I&cTQysUiYs}Ru(T2Ser3FTq zGY=Y9x~hf$v}ks7{iu9v(d{L{R%Q{%S=+h()O=a>8eZuf7o!ua!5oL6VxFq}wlNGT zu$o>RyTpKon&r45I7oN2B5$DK+g!l8m_^;}tPaTQ z;EM%3x83)9)5y2>%#PjkI45bZ;<&HA&(^gojL%;1TXOih9xI*?wOM^$d8r2ME1i5~ zE2*Co$XZ5OfFBkX)E*Y%#bf)ePqH*IXu}IX>IUzF&j%Mcck3#|!y$RA7)4hWA9-D$ zVwCN8n_;YXQ45Qmovd0BY6tau$d01fLgupiF!Zp-A7*{Jc9`gUeZp~R=H=pww9MPn zGtiqF@qoC*j<8PVxS#x4A7KSlO>DGOF-mFM!T7H1#AD&lsEzvfHsJQ>;fr*?t)vF` znOByt^85x#=DU0H6PonWikfj?E&Y~4evHYdxr(H0SbqaN675|Uz{=s$j-_8}6=dO< zOl1{kVw4Mu{k6qXH}7M`l-tW*9Ajj%Uum$w7lsk$=!4uBhv?Tg}o<6c@^m`+8U2W@aF{ zi^(x-%)Nv&zxAR7yw9#!S}HtY`P)A7RqGr=4PU(J?Vcn|-gv($eBg}m z&sA&wYmCS5MGaRTk-dbO%?X}3bI?iUgmx>#R*?GJP%$2u_EeV1cm(dU>m+PRpLkjMoB!cAa9>I2( zIjEREp4=a*k#8i$;(~x0e9@ftKQ}$7IF}sR8ES{O0Mydl87ZB8_Qryvb|nLsEJn2m z7_`l{2qXWg{Xe4T;q}22VF#jBp%*=a16nTRC$m0;>DE}hX(4u8X8hppil}0qi z$E(&7y2z>JJSyxY$bK10Y?!9Z^gNUkBkV=TYAFxd`cc|39|9IGe#Qyyff4US(U=~^N+Q)3+D=qpVrBRPw^Nj7_7p)Dw|aa3MlTbHI2#`?zI%kyY^7h1;K4&--71ez45y_ zlAE`dETklF0=NuaLn;C0F~L_T4<<+V&+jM}KI(8sl!h~#{R`EtHxRbrsIb1XB^A^) zRGW%}&(Mj8%ktB;I1eQxPhV;>_`nFu_JY1cACwH2x8vc;G;mWsrM(XcjN8{S9<57C z{@fc@9m%7ADY)&dVWKY9S6l2e8<8?Dc*PIEG5AS|v*(*4>E7Osa}tEtJs5Wvjf(u6 zTL6=t?og<{8#7cSKzWQz9kr&-)mP5hN6MPydBrFzEE#x9{scZMmwX483l&5U+`r8V zz6(mFsv2i-d}<_KufE!x3pkQ(<&;v0iAyxzDEJUjB|I?Q@5QL$l|p{iu#jb9!K=z6FLJWARrDl&PguUQJ8gU-j%&RwoAtsbI$8R!&-T6B2*^n>f4RI=Ici?$vD70mNS&%`V(*w)9%|voBCbKDN^7IZJlk z(lNLe?Dg6raD>4udwCnNsi^M&DEj7Ls1kZ?MSEgiIS6@`h53A!CbIZ(LU_JXr8rjE z-sNgr{00v6*eLn{1Xwh{DY}c^57LT}^004t)@sc}72C)Y7 z8lVY3n89;5@eG$7ob_dQ^Api;3{;6Fcnlr5?2B#Qt_bq&RG|f(G8^)GdZ#A#95*M$ zZh`0ZimJ5Zti)WQ$=|eFZwQRCXfv+qJ2%B%F~dKZ-AZ*tEYVb=X47Qm9!E@fN+77tPvuT!6v8hxhyBXYzSkCnjp>011UVQNgA32o>y^vo(_)DAfMI9-dQwH>N>8Vp)92F zaf69ID5X-$OF0O@0VCe+BrWX?_tM|wR~?67BSP{t7!iq5IME z^F>CjFG;Vd50?EnbIti4Dww|dXHXx$<14v;6XA~|8NmI%Ai(dWiIhtSg*IDXdFZCL zCPaukz6np_$`qa}J`av~CoQcKLVk*?8=o#b!4DBCCraCRzuYtWE7lpE>&eGPTvLvY z+wnK0a_g@H@&~yInXuqR%tJJE&YV$*t0Yu<&dpyKIHkJAsh^#!IIk9cw&HsZx*x@k z#(8@xfWNnTi?>9?(%|W>-+#%Jz{2{#l5d!v_`#R79%?{PG7F?&^&RQ8TYKfjf^Qdu zjCl^El`8@_ch$1)R8=Ua9S%qY=0e>Mdw2E>C#Ii|S!}tL72!L_?J{%b^!qY2wLkGK zV{X)6nvVBM3X;`mdngw32;({J9jB9V{jADV>tttEY{1z7JYBa&suFyW9PoOaBpp_-5ajM zne*tE*5X;?%|4Y1H+kfDm?}y7UAUY3ju%8AJ~5YETebv3XOD$K(;RxZZbG2OiXG98 z2IJ4M3wPME2FSIPYXy7K+ar{{ZzJCovoN2LS-#}6D37h)E2eF;M~i_>4t?kwuosk^ z+A^(ixF-1KDt5fv#_BS|V)=4Bgg0ge4MHr{BF=ftz7LLf zSCZEL`vixSPNwHk?w^vcT|-mMN`z{YB+fnqddg?hr&WLMuXsX`D=X%=cada9<93=Z zC4KqGJbuN(Fa9>vuo@=F#NQr|wW&Ma_9(?wV50I$f0Ud_JmMs&6MU|%A%eFZ)VEH} zFYzyfQi_MGz$NVJtVOa7_>k``{cEY{ zl9;n?fx99KtXGmWzm)MzNGj&0r`ijTdL@~)enU98-^u`r<8%co#?&*NL>*zH7rRxp z6q*DbFy-XhoW5ht`||m}dm8&uk1a6FtMTcaq!`y2Bderp-dyiQGFDjWRR2n+b7Oic zM>h}2>1cUuS$hasxnERN!(+8rjJCNiJr&e8y5vYfyhW~g?bm7)Ufjx5gmZ-$Bs%N^ z3R%!K+|M2ek|0AAdLbpn06KlF8G_r&kv4yPmqAxDLH%R?_>D)3s8wzio&5RO)>}p5 zWYUkQZSbeIS8zUImp{(+qBLl98mJrq=c_xv68?fLmZmUMfWl&vHmAdW-Q;TtiUof3 zwdd#4Y0eNOm3gNl)E9nGGHK$*2V?gc4NK#Rm4sygym-^s5YsJ@ z$G6?Q`R5EMAW`(g-swV+y`2n7T^_T5W_}9(m3^sshXZ(TxV2E3IJ~~XPesu4Dnjc% z?!huV`YhioGD0sNxnxx1;MDGwq<8JfCQ`Z#?IiYeQRIX*9F=mOFVH-bu}*d=$>vbc zYW`VFsf&*qnEk=t8BG0>U`L30TV4^JCXx?|IoFM}B!+RzkGI8o6Yp|1DT^4J>1BDk zlFOBV@+*CsaN?@?$ok+rxnCRsR)QX&(gr**dxk-JtL$^!dzc)yZ!hoKkBJBOal%=^ zU|*VPUQ!P1T(Nj0kmRc~*W1`T;Q3@RPx^ZO)GVo-?M%j<<07mgAxL`~$q-ojWJlh6 zI?gsRGUI*mUL(mzqEh|V0hVU+HMx|i$xMXl97;QzbZDk+Rdy;!4=VQ(cCBqwIvq2& zvm@RvKfzFx7J2xX%|#D#Uevh!7T!Ocnr?il-Tj@Snaxawxcem7kj%;rbkq%auqK&& zn=bpyN>)Y|`Hf}feO`b(X?^>(PUhS9T38-}m-o89(*@wZnzHu$1SL&Yo1+#Sh(bDN z%wOfyA&X^G=CpKE%v(~bsi%kHfK;wcW3kU3*WccNhRDSAJY&EhunvQ2}9PQW_jy`=elee}6*>tG-< zpp9dC_*W5(G`|6pu&8@=4InIs(Iz3YlQ3j^jk%l|GM1Un6oCak$v$;r)C#9qfV>Nm zqtGAAEQt3;%PcPiR_RntuMT6L+1F5NT7G59ac7mVL;-yd78Xp5yTS!YcamKDa_3q) zeO<#s5-we)N8%JXEA`naeWP;KpeD3C%FYPsBe(JVy zi7_a6T%S!r0nA~t3$&R5u zIiiwrZe^Yiqew8DKo`LWkaOxob*?S9I*%dne0Jc=XK#8S5*WCE#HWiFnjLob`dM;0 z=3^nQUy_jK{n9;%yM0wq4)t+KhV)+L@ovf#W}}v*08FJQjqRfj1Q*N|#IQbQ6f7Y& zRU~6FsYPeU%Y%5+cyd?&f=fNIAtlrdo+54i6%rYOuDjGh#gXgmXN@0olr%gO*v%tY zuP`fh)5@+ujp)v(*o4jo>1Uj)7T1|Vi?7OD7_8lV^uF29!+V@bOnf!qL6_Ka*y~xT z*Uwn^v9}3LhbOW=n1X5`ds)4qszp_k5Sh(FLaWdOVdY_my(qyzpul%Sbm?H_o*}Fj z9)C6iJq<}5OeLt&Tilw;_i@*)>M zy|uPh<*o0H19uY*6cdQFcJ*-?UC>&!E_kwF-88ypKElaPbDa1Tz37 z6ps}l^{mkm5#tUWVikTwoUD8^UlPj(M=qg`uA@K>qTWAjc2#{66@tHS8YqoQWobbIrP~kZ&2+rr$xwvV|ht z>uot|uEj>!=v=VvQem8%?8$e$@$_;yq+dQ{wx}$X9J`=XED`WBhpHj(-F6l&s8u`B zx9}Y8^~jTc_DiQqY_b}@f_<~x8T-?S#lUfLi0c;jBAvmd$Q>2Nr-u*Ru@!4Sf85sw zcFTdH0~@u82*EzY5JuzHr&zUCM$H3|-wNCA+)!X8RZ`ut_o&GJ%WUpqm5}H7ZA|8w z#b<^bUIvX18%Jufi3Rrc^MikhBiL77$XBwwN!?eyahh*34Yos2|*%4 zLLn;z=$8QEh9eUm-T*Sxrhk~fqfO_Ip6XFepcUmQ(KbU=lEE?&E_7jOhRL| z<`NVJ2#g?=#QK$dg#3O<1thf=aw{ScQBh4ZQc;PHkD7sr4G{TnLa)mb-AGb_FuV~$ z{cEp=3T>qhR7x3%dl7@EG{|*5?-5e?jQm_ycf2WeO7^^cG@E`57#b3;Ai~7FGm8{spb1 zgiHkVfCY39!3+Jl(LX*0|FHSC8FNG2KL@`L;ULro3?0BD^p`NiPcnfLEWim60QUM- zewa@yOoU{J2wn{2h9p6ibGN%bZx%6>&+heoqi7#cN|fa%6tGvn+Lu@J7p_U7-$>vm z?1!Kspe!h|t`mFCfc|HyvLfQ{pZ^(J3h0$g6dYKHND+;Qf&v8U)trDI>8YOQn^Yb2 zCICqClN!@k-aq~74HJ^<2L%dqFTcYD$>Ky9DCdXbo%9#-FQ#|+SO2n)+k_v&DSy*9 zgUEkmbfs6nVc%DdKhRG+;So&e+h?c{>ndoddLRc#E%>Xyh#awhNi}>!xL5mkn+7Pj zaP=?Xn+P*D!aLx;yc1a}J*!{bREa1PK4bZu;jDBQngh`5k61sTD!;$!{pQOR|AVsSe2oAoor-6X| zxIjYx-4|~-ATFXUXn%NhO)R2-IAM=3Y|IV7Cfz*{5s(Pich(f+`nN3x7*xCPKEe2I znEw*+i-WsF^aH2SU$O6=Z(mR$q8%ir{RKV{I`o@QlJ3h!{lJ}Qdv82<+a5}a2_GhH zu#w)or8y~BMhV-KhN4>E>+J^Fo{`D%b5DP-8-;Y#op}A@&>FJ}%pxXL2u}3)w;rghRYJjxN4-y9Qyl$WeY8`=k&P9<5h1`-j^(6@X)nof(sTMY?ZK&>#`1inAtt~-AX z%kHEa%=~jLY{|gsOK(L^f*Mkcw5yJ~!)qns+{6-{ODCVmFgBD|C12ERbQ+d#n?^)( zCx!+IX5x6HiQ8O3cbh%fNJLLeVcyjhuxwhLVtb*Hc5P)M9S>^fOw>%Oo5r{ZXWn%w z;b^bM3e7NsHz}; zJK;Jr>Gh_`Ln{7lP&%_Tykz37_m0^|G!*Pm6FK^%?|^QLqN@7k3_Nc=l&YD)g=?ay8Bn~sUQA(1%-Fy+l4L<5t8F6M zYgp0}oLDMi$beml#3v&zy#G*{bpPfQdzM|!QfSL^elQReNUJiIG5H=IX#$_!_WaSi z-6-GaEQoVUCUzdqrN=;i@S5ltw%MdLGB@*%35rd6hc!- z`e2m~iWpDI4x-WsmJUK-bLZ^F-S)>0S96XpDTubdB>x(WW7Cf9%G=dT&qYHK%N_#t zm~lwRoms6LQ&$Kff>N_o0?WO|J@0Tq<%wdUCzoJYvIy)xhOE4EX6I{v(zBEfo#$a{ z*5w*Wj%HRK|H!?~9Vq6Ne8B-&zBUuWUVD)pi`N z?YCj=CtG7|_Zpw$ja)-!-eki{?BB9j#gFTa7wHhMPD0{@QSY!cCzJf*wyA$U7t~)q zo_?lP&hhg^66DtS}d~Iwtmo z$NO1oousVL(>QI1`qRCyz0QDY;5S6vrJ71rx7%z+ts#Pk96+_=mi;8PnLV!H%;V$k z=ewF95LtJvHt>-AEjKz~w6buFHuG0$MK+D<)3Ti;I}2lWjr%eSg!)bT>P$UkI`9;6dG<` z?i4+0SLwMug&6{^f`kswP8O9w)JOe=E*DLmO7^UuJ}({J=owOcEx_`WFBzmUFQjE= zPtTy3de~XTgv$?)qX8*V?o2xXFQ`KRag5$r1a4qpiATqx{|NR=Jv6xxOK1 za4q_Vwx;o$0{jgmnBk?Ab}b~E>qA2Z&}6nYn$E{exVDI3!4V1#E7k(^a_n^~;V!t1 zXj0@iWd`1wzp+Me0LhH5y|%?yZ|paVfeqNy->$sB6F+#@>QJK8wo3 z#gAakSv~TYj)bI`>}duHpgCX1C(RbG!nylsjB_UyW9H;-#_iR-$H0MNUP4`SQ7-=0 zUgIPm*GBID(N{E$TVAi*!ec~UnUBP;f=bIe7NyAgRN>rcnk&OwHN2H87EL1V$I-cs z(AvEY+EYXM66@+Z6I-RNTo8|r+1@FnpA(~i#v~zp0J|`IMkC~;<~dFbl@qFBy+&rT zmws3y+NwJ7t;?;?O2UKHsSrH!t;Bq^6B%u{_|&Wn++nyJTXCDn-?r}%??hF-VZ2G( za9(<{2P{0cv?29i3?dZ}g8lK8f{M_-D2>xqVSUlX z(RjaKfs5lxKc%P*#aJH)Ye1{BN6;JMWj>&g1Q>qT8AYtGhNmK0Jh$;ySPez{JWzkC zpr-TRZof^VW~eHUvn7-*z{z`j7(?@ld7l~nDtO6BdoIsjGk8k&yS3~NYB9;kwOd_XEi0dX(FE3o0ku*6m2WA_e{AQo4njl6Y-V<6p8L6^!H&A#j4Ur8HVA zglkn`y>Zhj^ILOB_*hMCIlp%w&V3NMaTxLI#d0Dj5v;tsRT4rT=ElTN z6`t_fkSsHY!SZ^(J$=umWuZakkaQZ92p*bJ)+t=oH~3cVLs`J)Z997S2xSd%MutSn z3?pCfvO~#@S*K$JPv0LB3)d>npo*`y2r9Jx&A0TvC|2Hl6y7D%gn@pJ6nt#=_BG&5$U{zV5s7qj_*L3-ugA5QPYV>iq87z|>S`4}AAT0~{* z8YsVmH9^R{d zp-^8HpqWz>hznN94^V;BRhovPjeUE`-FzXj&Ir_?Cu1P&l1DFc!o*DJg54>cCb!|9 z#&j->e>5elH&d)g5)k+ZtT(BLsc&hf;meSeKJKaxdmKFp1bx5t0T4U>KL z8*Om8{WR!H_K!Tr^i9vT<2~JG3LxwG_xNq;nZymJJC%h5)LyIF35yd*aE2Q0cnFHQPAX9y%H!muizf} z7LL9*B?xr8_$LV#RQMjeWH+zCWVt8vR~@cf?!AZQ94TgYwbbKfSTfx@#=C0VAHrOD z>>~_gMYuamGSc%;+>=AJxlREt|5&%1VRZXbSdV^gu;V|Ns``P5%Nf5Xn~sV~`2tuB zK5JoTbRXy~^QZXm5sejbcKj^G07U-lf6eJqt_8>uXnq?EzIDOR}*X!V!@^o0q<*(YCP9zz^tH<=Y zv|4q73Zio)`)VH1ww3sx(G@#m2d>*fQkum8yk&oz&r(k{lFjyfl$e`rl@3b0qBJ0T zN8TeK`QQ}yu1n-pT(!*mS*oGPF3IJKx*Gm0DUgR+H8=l6YK)^?dvS0J(xy(l$D!iW z+-|LgE{}vRUkv=hN1tV0E37@MimOq99%B>vHo1E0a`^Tv+vLyi+g?=tieG2ua=S!V z1+gp7m?>Alj!`BARLDmku6<^x5GkC85-OhcjeKZbRb}pt-O!Truj&Ng9(XlcLZf@L zN^QF&0IpymIiF{KHyM)3zCuf9`7z8Mivk(cDUYY7RU6tY?AP?%Q2F{6zb>2#oSNm* zZ#EjB^J{TCrwgd$1rYH%&2)WD8%Z=|m7C3eO_-;)%%xSFn@Z@(g?W6L*J#u*noyk! zgfJXUTyIXn26v3tgjAogNWPq6oiMGnw}38Tzn?b(q+v@^uJUWmw{aZZFUx8nm3em2 zs*)BHpT&bo6Jjc!NZI3};Y%g3@$GcET69m_1s`l~juSAQ*{2;_4UW?|c2`zzh-~cO zDE;`3CU%rLy$YLK>1%@Li%bJ;o>&$h?_|ov-CSi?3@*_5_V=giYw1}0lGY2}mtU3R zOI4gnSdz*+tw*~dNyLdMHD@yY;t*TP%jf2;lAK%yl096*7j=UfsS(URJ})V$H@T~F z<6%@@FOuZu^6RpzSvP=HvN?H{!@oAV2gBfNX3Cu{q&%SW8>`g2XB0K`d6)~6op}cy zGiPnemvl^3QqSS8aS4=T2Kfzm7?l@wmz=HOmMj}>hUT$a*5eV>2L(BbUi5MuiPne{ z@@6EXyATGNeFMJL)q!UPkIp)^{O7E24TS<^aH87qo+?N(d_*M1M+&iAqV7DXC~^oz zTX$U!I(m1#F-kg&FVf5TyJ+az60i&^eT|>x7E;uslC~c8mBF-D=$qgxb@rnG%239W zGgK&V&nxUDM=+VO(nY|ZZ90id;cKU(>N;~n$|hwq#{OHqQplEG*tk~&HL*$0iIveu zJ8%sugE!F46gT4V3$*Q=Xaii@43JX`^$N^{xq5UMaZDDWhWH`q=cSF&g^DGbrvz5L z%2{!UX$s|snI+o%?V2w*rFA>lW#&(wZt?gA{%SBbDVM%jE7S=^lO$Ht28>i)!O^ax zkMb)19J&c#x+jgw;= zCdKn`9u`)zHquvPj2mJ{$N%P>KBkHY@NO1^R(gnsI!cqQTHVCqj<$oEkl1UY7PB0# zeO;nvd;h)%UEHl!v!##1G`T^=T;8|xwySsKSuB_w;#dV5HlT>!H;8kprKo{;ZrI7# ziT$f5TNdp`z8~+MrsyqeXt*xIni~2y(XICf`3F+sY*q;KPnvP!QK@P_|H5H}#Ky0; zJx}=wF-ha#El<%Sicql^P>LCgF{+%&f$gN;Nql9Pw|2hbEVh0;$g`YKTgk;;bpzHt zk^9RTRK(-)?j@|G0F{@$Y+y&|@9ejSS$tj|1u-Ut1Zl>2y!i-8EjC`+gXtPQ1vi)* z5AlM+PEzXCaA_UcL3Xy4Tp+|mE?DXg?KHoN74&3yN9G?MzjrPxxpxLN)jKLjG@D3poSqyTS6yN;38<|nIs3E%TJn0|_v6_8Nb68BlO#%5W z`@a`tvn~7vL{Tp#V6a zn7WPifN^@~ek^l*2rn$AA7YE}ahZfuh4of>O-*!%S!{%h3DUS8%d9mgvkPkgB9u&L zqVw8~5}o7EHo&hq%0CpE2p%nO!ks<73 z0U$CP?165?dH5pySR-Rf1d){r!j8(yYXR>RA{w^EUtZ6In;}-ZPCSKbs zU>T-99wmAFy#(CdpMO~t#+Rjn!wa<@L;m;c<+%T4mddX~M>@@5Q#&+_Dpon{Bo?JL zK*P6}mR$e2OKGUBX70IV-2E7;Tyu{bI%}w_hy|=N9D)*6}$%<+)LPzU;Np z{%?`D!SOuumWxdMxmnt6$!ampTmU43QhlCsn`iQQYq{e(XG>h+kOH8h3A-REhq8_x~Bx(*wGKK1S47q9I%T4I*qs9p-?+tn_}S3uD- z?LK z_OCyT#aTOWIZ)G|nJc|&syy3#Cc^`~4GJ~nnNBsenb3syH>AVV`OW?v)DO7t)Fw74 z-dzdp-hY6*IXupl`JlG>BU9o|)N7M=d(9RD(zQeRBe8-$U%H>G(~PnEqP&y3*83r_ zO?cl8H#jbOZHqWwFx8w#=l=|T&(2Fse zkeu&iVMDsfA_iWoY(|o}suR{W8|G`|q^1Eic;tVYZxN?J(O<;1DkNAqC4O6wpo00K zSHscxv{^+g@YFNTk3CQBU|(`*O(P3`IU)-Z{ce8vnO>Yel+Ln;*Ko?QUA z$pkN?$NJtPUGFDU7z%i+Of`6uon>^@OsT|R6tr@_;qb)8DQOc9g)$WrGO8EgBi}D< zR91pJI{8G7$Wb&20WU>ZQ#3g|G89-OR2*Wp9x)GkV!cNXJgqyq+A8^VStDfB6#nQ>Mtu$@1Ts zB0D$Bf2@?{;NoN^`rikJQQXqT#ng$2QQXGJ#Z=Vvx4nrejDP@)vx}3dku8kJ#{aIA zy?tTWrrKtvA|i$pi4Jj*3~t_If@2zngA;M*L|GtLrYT4Xap9DNTAU^Ykqz;sYOatBnQJAfjq!W02;SG!;S_90}FWt2Lp@b z;(9Wof;`)0LJkK;I|mWzC;!G2B7g=j{>u!`1QWX`)DLV`hX9m-1VlyyjFOCm1Ox>G z#rKMd6qo`;6|iln4g~KPfCGZyN_>zY!Szoa$I(jl#PQ<=?(nM}xBwDy_~wTJa-2sv zkqHV5;V)3JO<~*_KCvNW009U{a1r0ne2~lnMyOK~5XkoS_S--l!zWZpEE~W#fEUgb z_HSsXP@R;zvy)iv{Wbo zJ9q`_gfrb&^-Mjjh`)~&K`l1Jrpa$om;Q%26wm>UPkW@l4 zQn`f3ft`pzL$!Xu0|*RIF8pioaLzzFMZkTOkf13hu0V!t1AR}rM3Nb(lcBrdCwI{# zcN(}))U<=ON62w;id>VSXRAIiOk`+50&WU>`PnVuq+9zt{+KIF?Wxa8yqnycbkfybUw2hkI_a-# zty)VPck1Wz8r%+KgH&qp^FVF!kubRvP6GjEu-Ff=fSwPp-9iR(RM@Xjp+#UBDg?NB zN5UI(CRO)^oSzddtSG_NV0wO}!9RgpeOm=3AdB!|e(!gf_m{wLIli_rdMDjMxF7s9 zG?7?&)N54%TMm+Nq$VW?6BQQu*i!p2ya`f&1TufHB!4`J$kihw_stvh-F`^p zpTW9ay&{OjE@6dp1w(&vfcQidk&1&9zJs@hbn?A>X<)8ydpsoq0JYfXgOxPl`@@Nq3e0jNd5Y>i*ebq)jXNeMge%T3G0$pNaIur(7 ze20xBtqpuqP?5j{lOpdgkoaQF3->A8-7RV@t|uxO$bbj7A4#EdgbtQd%JLKc8|AY{ zopm)teiAE3fP2;vNgY|e4CSh6A6D?@gYoHBV1$@p5o*b?EabjPl%a*po@6IuDgV|o zS@$sL;FtGaJuDjL;ex!AzQ-#f2w>G|rDfp`QN3!s85Xbf%zATXjOdruY5~PuR<&Qk zirJ+FLVmh=W>#LcFJjb!h0aB5T==DV6ADRmur{OMqHsxtr}PesWrwrs-EMT%ykar9 z-;ewgP30~=jwu`siy?QuI~2=da-=uzgqw>Zwp1jiqVwdt5Z+TX7E&yKrd?d|=j+KC zFMnBBcsX@)A&12mc;B(t^SRD~N-#YWUw>M}6h+WOEJLpj0lG5p7xxqQ*tORY0?{B3 zP6H7GjfjH6nM?lNmxN$b>Mn3y9Y{-ZwbQm$84vNjqaso(Amthgi*k1+p_+I*e)98} z-d9FvAZ~3OLK8@Ly>PaB(;u_i;LXE@e#L?3D~$bQ0a5v~n+$iu@-$Up*H_GQ|0-8` z^;3FVn^Gph&1zA2uGcip!~Cd$sUiVQ`~+FwQeBC`Zg3tI_#?7AE-y`J9kZD2bPTGk38fL;qk4D3Rhadsc|eRgI!m;H zKFVlS=4E~KEcPxgnCX0fJ}pyo@A3jrB-n1GAf|bke_8_S zp7B;{v;9(hf6bNA<1?2CT62m{Vohq_GdNR%u;hmTEB02;MHWEJMIgr)GirEcM(SiAr#+#V?Q# zeCgQ18z4}jH^M|ynR=_Qx83j}$1~4|LVdC|^-@+eTSdE-ClJouYZYnA^EgL5L89Sg z%$S_#v}&{~avY=pt@C5=sw86pUmFhze#X)5`<R zW-q-NMxp|Z6R>9#6rP4H@&pJI9!S9;c*#~bT%tAHOdMe1_2>S&PtCcKBp9_T8~Q0| z*OWwZU(2|s$!*-%)YGSfog=iHJv-&HcC%>F^KvsxBQFzNB%I86B18Fcn$x^z$OOm9 z-~BXncBS+yd#uvrBPUzk(JJxo2C6=84-r<4Ar8b5nkRF2G=I)O$$dPS<=*~ zjwL5{ZDG{Epwix^PHOMqW=$U}qyM%ew@AuKF=%1C^VF=>nlnZ}IX8PogFNr}a#s&& z9e+~Gh#A1@M}QWQnO3M(PENkgOVK?h4;OXMT$rci5mF5S|Mr+YuYAi)D97#|O(GsM zjxygG*6Awi*H+-~J`@l9m~@km`}qWDxOF%a-xbdzbghRJs$q2@gHuqImHjGX<=X|Y zn+DBLN6YEu)eu2FW~0mPGCKz8EQdbZPcIojcdAAUd7+cPy*|?pb_hl|u5)KT#*mwy z$a^y2Wtg!{1!$SDEtTpMviunr*|Wom9eF^;g16SeII^JDa$Jn^S2>b)4}u^32;3V0 zk(ObA`XVk>?aY~=3w70y@lA+aOYp@af<>C4&C_@1RTEZX>s}EmdQ%R+n+nTX;hSPz zc`x7$y&}RJxVjOPfn%{uyqKD8zCbhHS8r-mCZq@4*R9(GmfpDCqU+{zqr^=nTz{ga17Xs4K8;1l~0J1WXol^){@W>g2oN+;Ku+|nd3;Hqt-1Dpmj2QRqU%}M0$N!* zTm|?}q|+B1vjnvY%T)LmkY{BGW~{Jctq}$4cyh;hBvExaO$;`j>NdD+Qw*f^67mf2 zTbbrn_#ydU{M~@XHb%3ZDcF!#MIIbYALyg`Fn!`rXxlBAIJZ~&a@=)_haW0cVi6N+ zzYj-WNe-!>=Zd=C?7tk<-h zo!VHf@k#nyPnJyQ_IjV&IF%`ClN*V<=K}tG%aLZm+{uGtk2bBaA`89U#jw?{!P-5i+igay6R0)f$!8))oi+* zh$Uiai#t0u_2YQw3Url);gtcJ@o#1SIqWtN)-H~_O&(UG^6uM#)EQgWVdS*|X;u0pP0trxwTm0yc z^oKj#c2N+)(mqBgkpN_D${nwcFr%D6n!c=9YK5$&{EC?POe2HK8eU7u|7$axVSmtC z)e_qT6wV#k&-9bKQxz^$(&eSyHJ0ras^6gV5e7PhJw6AL{5O2%3653^L_J=PX)zCa zAfez)Kp3zb*8N){PCRzU3V>TvxU&8Vjlx)!el zQ9vPj4x>3!0spacs|D7pxHxyJ(`oNdf^vBtY<3Uq5y3sD@rzRa`WGT^Y=K7Z3_C_m zsgBF_i8Ax}{PLh2!L*6O3w+A&2{a(QInR|Oo-Jy7`DzE|>z2eVZ=1VX+)94V@u<$m zrrsaFs0*p-0(GH_g?&sBsqJ4YZ{st3j%1PNp3+Inh zzf(Lbx5JbXsS=fFj+a2FCTcP9wMx*n|MlCGylU6@(UhAIS~4&T7CVZpKoIi;mjaq^2OIx zi9bQj%aP;A+1KYJ0>x3K!?%~E6>e0SIEu{*^|i*9g3h@1zpZ_fc53`o%NaIy8$gXr$z1{OEfkE*_G$!e7Z$+ z#Zf5X&vn#A+Lrz9G8_DpO+J_P%Y`Yw({kQ|XSiX-##VR|{kZAKJ#EQ(g|hoG@dKgr zBuJ3_C+>))n}DOlTgbjOH{E$|!zFg()8W+o?6-3nT~BHOytQMbm$!4;>`zdDtcaE9 zsm}^^JaWSX#tEhXT0cKBYj!M7G!|>MAv_)W`+@DOaT zowPSAt?Jv11$ainFtq%?=w0dVA1n_lUuS&&7<+o`5q|QD4YgIdImJyWIaRs%pEu3O z0ZM^0=c?lx%@-JGX`>-}RAOPHcj389?$J_rBtq7COw56!@+u4TKN*oo6a5S&342jX z1!HqH&@t;mY@V(PU~dlxn;%F<=^m(MBvvgr>n}YS0|$IRWKq;-+5&i)`q&yVx3M?2 zCo927OxVVP9$RfIsK4Hu?$?c25oQFBWJ+NRkI`#NGa$6)vB68~-Tnm14(@yM{OA+# zeiQsSZi8FEszS9!3wB)7_}P87~UGzVe^_tHnifmKrDru_mEr4^wHcueuRI+@HzYC zH(fZD`i)tSvc%+sU!0-@LVhfdu#Oq9aTi!A**z_23K<~SE7_dG>ECGNY-~Ui^->8g zKq>)<*Rb(cpw4;z{Gb6G*)7yAwB@R^TU9>fKI9Sdi#2K(wN;onn1+b=3AlQ@T7^yR z%L4Hca(?+4*s@n=RtzD!HA==%$jU|rZ`U-0p0~PRWCRFLTm9%T^C6)0uP^BP#tv0c1WS)+xGNytAUYgaFCpa1 z>ytdv1KD7uKW`*cj_ky5A?YTbe$;OI%>at4_Zr`99)kngKmF2&nYk{nI)Xu=y_9cK zT85&6{MQmbAb7r@r6nJ(toh*%gW)SEWdyh*a*H=&CC;e;Pv+$5?YD-aAvgf+t)-@t z;u>|fA!2MY8y$H0!k>*Q7))fz>Q~##rmBrobHa}^PB@LuX0zV9<`KUGoKgcfG^nlz*;Vri+mJNh zezF4H_jbzMGW`3XD~g|5dS5)V}=bqE#Tj z{c&J`DiXX&jtPg~BL4hfGhOAw?2k7N52^^79ge47=)gpxx6ZN5z8ZinN2t$X9o z(SItE1wxm)qwqgo3ppY_iaNd7yG-G8=Wct)oZa)67`pb|hc)|kY6!`jVQMY}9Qx$(2eXNUc$ zI2Qp90g3-aM$U!tWpU;8N!>~}EO!u^D1Rr+mpoL@I<$7i`x>pD5&1U$A&vA-6^L(49mu8!=rbfjT|?A*QCwO15Tsz_n28vt>ic@*uSd#y_4b#;VBJUYdUoS>h z_Dq;)bL*YY{T;REQW}^>vqHjl_J59MfsdF{L_k``h!e%MS|Z@@9(U)w@1zPgvsx0M z(am@i4>mQqI zo0v4CT~mAwS~_l^B{x<3nGp_+^!1}~!P{mwiQ|%4--vOqm(-8RGnKCz~q{oxmmj=#r(4D z5_V{~B%i}9G!StM2dFmwga-$);@EhYhQD9-{DHZ3C5TrunJzux=RxWNbj;rpV zb29GT)V0G2CjS24usy-$AO_}jk!Zso2Q&-iEOLK;bUe0wVs6*^AyGNLZ3deU5*ygl zfV_R8Ry}jBY;i+}QvKDP#js_CI7rG)Mpx)^T?dqmGjtnALn^4BL%5q~(G5{BDi-9R zCB9TeODe(d-wE0VQt8T4few*i?=Ojo+};;K5UX06Hfwd&CqLy;O zI*1H@I8n{u8r>$RmkyIpCugs|WkIsQn77(E3{kINoYpK0l<_bH&en1Lie@W4xal#7 zzH##sEwJl2z8VxSW0mmo*I*d?NV~V1 z&t*%-)XD>k(KDdbi-c86)Xs@~)b$JcMmgyO|Le}GztbZ}iy!nUm7!TU_M3K_V;OVI zo6ccU{_tbjphE&&y0_MDYw>CuxJvFt_FsNf-rr1(d)KrEqD=1EQjrxdI`-BE;#D22Wi+R9MLnD%|~n! zL4rKTnhh!3_!E>hS|_Vo*Z98TnhkvttZ98QBo{%JM5_J8brs@9d<2tecX@TA=Gg2v zDQTj?#S?Z)OCdYBNn%HVAue5s-+m<-{BZpXGZ5iB>O6e+Oo`8!FtM*Yim9Yt7q@+}03}>0z2z^D%fzC>APp9N`DQd5v1!)$_3VT=5>*CV)7M}^0T-4%Wh|94x5R30;xXWre|aA z#TMmXQ9cGe%}PlM9nS)A<^rQvlp8#?`u_$^`xNgbbuzVu7ZCV2eg+`r{(tZp8!6z^ zRrKFg|EcuL2H@fO--2llYCb$^egxC5SH+CUw7z?(CTzJ>97G%$=FuZLjAjR%x0c=P zQg|ZjkIxiA}wU-jkt)=KZ_8{T3g{5lZL+GHef|VG$AP)Y_q5#X;S}}qwjDcmOn9%Dyg|_ zCBY^kbP0mh?7)xjbm20XI{xiFeU*>d09x;DV91WQtv zN`vpMd&YNxQnWUV*hM1d$f#gyWIFh7^W6MAN#MA^0ZeAMI!eqT>=Ou7F%%DQcwAa@ zw_|r2HC9+s*toCMnGj82($G-m{b0$E&EW|RML#*|>2OOT-;Hm!5c!d1ts|%uuvQ~= z!%5qahG6KXQKgT(NTZqWY@i;%w1r2}%0$+;7XwwSW3Ke+5VMG%A7plz86f7Q*O6Dz z43bW@8`Rn&#t`8l)xpW7t!bkrbz!u|rzx%P!^D%?pXuK$0bWlmIMv4MAVGj=V zWBuPv&bMZ^BTx6jI(os85xycRB7&(Sa@_>&r*tx6!@6l1;9N9i01Yr;Z$z6~iV{UP z{Z4ca^5>~km$dohf`NA2tT0%-5jS64Q-(bA;|6@? z)-0Em1GkXYUrX_%e!Q!m!m>?QD|PJ6_Ue4?5$Jzt?&z{*U6J*_nbcPD?AtDDHp(?o z+8oV=Ni<%o{2Hrx!%}baD*1l?l({kAP*ybPVo5rh6IFV@za(SFBTfin*Pos81lx=esh(fuL!-;SU(zvC7FEK~ojK zamXF7@m1TsYPC=lNBR`EkT(n+WWJF%``2jZLMm0pW}Bmpc(WxEmO_-%tt|ncw@~C(E@Q#^wtzHa95T5q*E}R`VwQ?@z3fW>RT`%s zW2JykJ~iE;;T2VBXpT`-uJf@WwKS^e-kxk}Zpsy$%?AMA29UC|!Lh%{DTPmWP>A<( zSJeM8sUYN9)1^``4fr+)IkXo65)N!$C2Xil*}81X@+p^|SR1}ou|Ft>8&w24p3RV3 zN^|2}GwN$b`gJ&7(?%DQ*t`?LHleHr4FzOpS?^K9u;uXM|M5J#y8(?2=k5SQOZO z@_lWUYk0hb6S(>mx32841ivJ^cXD3OS?O%+4yH^o!3P9Zpujt`rzbjarYPpDPXLar zCYFczUGblZk%um&FEy^zbMQ~@v)8}M<_jP1y(}?1bbxB>k#64u%mKdOA$XW_^mHn_6zP8YSM;?rBRhu>C&SSNWE;&oTA z7usZMKTYGB^Dd#UE8+(&-@&z>PSli*VuvR^h_y=F~>_V|l> z;_Bd-gL?N3tND?gb4MK1Bo|w+d5K)JeSk0^NM(K;W-z`0-<*~DC-z|y0ZF^Q961LR zzQpOkN4^TN_lp5BOX$^*`+Y`wO$p}l1=q(sE&#=O9JqyUu0|o$P~|dZnQn^uGVayJNRmtkzjRn3e4Ip7gh&w_iF0 zT6ZHQolb36;iKL97K?2pA})5{}HGH1V-2Fp4c z&Lj#O3;as8r_pN$thRnE^upzknrOx$1j;~Wc5| zS7w^IAJg3YOSqO4`Wk!U>j|hbmAV8A;8p5S7-xySM8F~Kk-*fFH$N-z$JNo9KHe}S z4KebX7$TNp3-jC>s}XuWI_TI`c1riY(M}Vt46YRDMye|(k@Cwv#M84+K-V_uYmMnN_!SZm$n zIstV1&L?C!;n_X`)5h~ zlNjXwS*Kv4VWC6nW#ev5YVG4};p%E(N$PB3W@ln$LC2!v>SXR=X5mWfY;O4(#spyI zU}Xky(Xq%|_;@+Fn!C}`!T;+{HcpNbpUfjtS_wWjRsc6EFDn-Sz{<_R&cMn>!^%qY z8K>xE{y&9ixSBXSTbO^+m`v>5EZ|vGG$eJHr9ABIO-&pfKj}@h8rC*$q@UZr`ol=7 zW#Q`f*)%CTGl1h$d4Y|KlZlO)>)-wSjQ@N#1shKb($7wDv6x$U(vjLZo48xMS(uP| z{-0~PnAw=w{;B$hFlFIr{!jY&pXT%$pTk1RqHJob?(P83qD0F6=^G(s#c*{Qg-sHNk^is6bW1uI-LaL|Mzq+x-g? z)p8SIyGy%>*xX=eZ}yH#nn?(61(kf3#y8pivfD{^D#TO;ni#sc0vk)ngAhE5EVW5H zfJzP&D2t8Jx>lFEwHXo9hO&TIxglw3NE~M$GAl}RCz30vrAR!~&9w+ZCp4U>1A-S~ zaCO%vs||VJ>pij5wqtM-<)}y|IC)YkI{6n?M0UOJ;0CNCU!XK}P^<*lje>&Sst`CR1Lare^cTkJjM-p$IQw04YK;S_SGXiF>9Qo!Dy~dL# z_FCGEs9V8>7Md>l3C-}LabbIB^vVx0{GMbYvvNpi4peWfS0&QeMN?vE(aPw2ZJMNx zp#8(L(uWW@oklc z>0!+hNGxkJ5hMPaE!|b@o|a)zkCmEdYli%e^lye?*?-;%q2F zs-z5C}PH;p=Is8VhzIN|v%O{d4T?+GM zD$UGK%}eMCt;NrL>Si84c=XHN{0e%zxJ})Aqj?$iD=hmtz>%*p$wBCMt+Lgaa8A3g zA!kL~$usZu9gu&_41O5M7nBT21wP^tCY;skt|hlTHYrs;d3$kXTOyuFdwAZ0$28nj zV~I^fBmiw>BljtcCYwg%?@I~UY(aH9+Fz9tQ4+V%vi34_e{L7q{eqoWtU8u)bdAKP_)`-Ie#T22}~-+7`MD5D461SRCk3Vz$a8%W16}j>d{=5d{gItM)E+Y9 z7`)IC>}UKGAUWfc6;1J7wnvD?BJdkF~l*HgH0;ZwBI zex)wl&|;u%w*@#!1Cf**?`iDy6nYnd`{;sGkj_$Z?dqz4l9DXC>`l z^}AtHHtTggejzCwSAFGMJTjhatL6=td>r<_6rc}qP>aCJf0M!d zr^?d5KE66?7MAcVvX15!-lY1hq@1jV@GR;!z80S)couC^eE{j_#LnkAH76%`QjUM@ zp49#cw{#-q_$M9u=Rkr~UqFDO43V`$*un38iJzD0@~BV03FVkXu5v(Zrl8DOxY z#BGd{MKLcpl^6Pg>@QH*qqN0{^Dp)qQRdl~U}V&|CJA(wxTfC#88WOQP?`%W(hziP z<@_J^wtm-bEq#7#^u3!2O5yF_Pp8)ADQ*sYiMs=88%CsnrN^mjB(vJ6wEzDgySbaV Wx_f^nd+=PWyq}{;MJ1^$1^-{iv&o47 literal 0 HcmV?d00001 diff --git a/diagram_paths2.tex b/diagram_paths2.tex new file mode 100644 index 0000000000000000000000000000000000000000..9801516a32ded34b20db50ebddf6a41973d5a555 --- /dev/null +++ b/diagram_paths2.tex @@ -0,0 +1,107 @@ +\documentclass{standalone} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{parskip} +\usepackage{marvosym} %Lightning symbol +\usepackage[usenames,dvipsnames]{color} +\usepackage[hidelinks]{hyperref} +\renewcommand*{\familydefault}{\sfdefault} + +\usepackage{bbm} %For \mathbbm{1} +%\usepackage{bbold} +\usepackage{tikz} + +\begin{document} + +\begin{tikzpicture} + \def\height{4}; + \draw[step=1cm,gray,dotted] (-0.9,-0.9) grid (8.9,\height+0.9); + + % + % Red line through grid + % + \draw [line width=3.0,red] (0,0) -- (0,1) -- (1,1) -- (2,1) -- (2,2); + + % + % Arrows of the grid + % + \foreach \x in {0,...,7} { + \foreach \y in {1,...,\height} { + \draw[->] (\x,\y-1) -- (\x+0.9,\y-1); + \draw[->] (\x,\y-1) -- (\x,\y-1+0.9); + } + \draw [->] (\x,\height) -- (\x+0.9,\height); + } + \foreach \y in {1,...,\height} %somehow the loop cant go to '\height-1' + \draw [->] (8,\y-1) -- (8,\y-1+0.9); % so we fix it like this with '\y-1' + + % + % Move labels + % + \foreach \y in {1,...,\height} { + \draw (-1, \y - 0.5) node {$(z_\y',s_\y',r_\y')$}; + } + \foreach \x in {1,...,8} { + \draw (\x-0.6, -1.4) node[rotate=70] {$(z_\x,s_\x,r_\x)$}; + } + + % + % bitstring labels + % + + \draw(-0.1,-0.4) node {$b_1\land b_2$}; + \draw(8.2,-0.4) node {$\mathbf{1} \land b_2$}; + \draw (-0.2,\height+0.3) node {$b_1\land\mathbf{1}$}; + \draw (8.2,\height+0.3) node {$\mathbf{1}$}; + + + % + % -> steps of xi + % + + \draw (4,-2.5) node {$\to$ steps of $\xi_1$}; + \node[rotate=90,anchor=south,xshift=2cm,yshift=1.9cm] {$\to$ steps of $\xi_2$}; + + % + % (Red) circles + % + + \draw[fill,red] (0,0) circle (0.08); + \draw[fill ] (8,0) circle (0.05); + \draw[fill ] (0,\height) circle (0.05); + \draw[fill,red] (8,\height) circle (0.08); + + % + % Probability labels + % + + \def\x{6}; + \def\y{3}; + \draw[fill,black] (\x,\y) circle (0.07); + \draw[fill=white,draw=black] (\x+0.23,\y-0.26) rectangle +(0.5,0.5); + \draw[fill=white,draw=black] (\x-0.55,\y+0.26) rectangle +(1.1,0.5); + \draw (\x+0.5,\y) node {$p_{ij}$}; + \draw (\x,\y+0.5) node {$1-p_{ij}$}; + + \def\x{2}; + \def\y{1}; + \draw[fill,black] (\x,\y) circle (0.07); + \draw[fill=white,draw=black] (\x+0.12,\y-0.26) rectangle +(0.7,0.5); + \draw[fill=white,draw=black] (\x-0.75,\y+0.26) rectangle +(1.5,0.5); + \draw (\x+0.5,\y) node {$p_{3,2}$}; + \draw (\x,\y+0.5) node {$1-p_{3,2}$}; + + \def\x{8}; + \def\y{1}; + \draw[fill,black] (\x,\y) circle (0.07); + \draw[fill=white,draw=black] (\x-0.25,\y+0.26) rectangle +(0.5,0.5); + \draw (\x,\y+0.5) node {$1$}; + + \def\x{3}; + \def\y{\height}; + \draw[fill,black] (\x,\y) circle (0.07); + \draw[fill=white,draw=black] (\x+0.25,\y-0.25) rectangle +(0.5,0.5); + \draw (\x+0.5,\y) node {$1$}; +\end{tikzpicture} +\end{document} diff --git a/main.tex b/main.tex index 7703efa6d99c75d3f583e237c82b9d432c48512a..2c1d611a06586de104dbf85fd91e513423ea577b 100644 --- a/main.tex +++ b/main.tex @@ -474,64 +474,57 @@ The lemma says that conditioned on $j_1$ and $j_2$ not being crossed, the two ha &= \frac{1}{z_1} \P(r_1) \frac{1}{z_2} \P(r_2) \cdots \frac{1}{z_{|\xi|}} \P(r_{|\xi|}) . \end{align*} To construct $\xi_1$ and $\xi_2$, start with empty sequences $\xi_1,\xi_2$ and for each step $(z_i,s_i,r_i)$ in $\xi$ do the following: if $s_i$ is ``on the $b_1$ side of $j_1,j_2$'' then add $(z^{(1)}_i,s_i,r_i)$ to $\xi_1$ and if its ``on the $b_2$ side of $j_1,j_2$'' then add $(z^{(2)}_i,s_i,r_i)$ to $\xi_2$. Here $z^{(1)}_i$ is the number of zeroes that were on the $b_1$ side and $z^{(2)}_i$ is the number of zeroes on the $b_2$ side so we have $z_i = z^{(1)}_i + z^{(2)}_i$. - Let the resulting paths be - \begin{align*} - \xi_1 &= \left( (z^{(1)}_{a_1}, s_{a_1}, r_{a_1}), (z^{(1)}_{a_2}, s_{a_2}, r_{a_2}), ..., (z^{(1)}_{a_{|\xi_1|}}, s_{a_{|\xi_1|}}, r_{a_{|\xi_1|}}) \right) \\ - \xi_2 &= \left( (z^{(2)}_{b_1}, s_{b_1}, r_{b_1}), (z^{(2)}_{b_2}, s_{b_2}, r_{b_2}), ..., (z^{(2)}_{b_{|\xi_1|}}, s_{b_{|\xi_1|}}, r_{b_{|\xi_1|}}) \right) - \end{align*} + %Let the resulting paths be + %\begin{align*} + % \xi_1 &= \left( (z^{(1)}_{a_1}, s_{a_1}, r_{a_1}), (z^{(1)}_{a_2}, s_{a_2}, r_{a_2}), ..., (z^{(1)}_{a_{|\xi_1|}}, s_{a_{|\xi_1|}}, r_{a_{|\xi_1|}}) \right) \\ + % \xi_2 &= \left( (z^{(2)}_{b_1}, s_{b_1}, r_{b_1}), (z^{(2)}_{b_2}, s_{b_2}, r_{b_2}), ..., (z^{(2)}_{b_{|\xi_1|}}, s_{b_{|\xi_1|}}, r_{b_{|\xi_1|}}) \right) + %\end{align*} Now $\xi_1$ is a valid (terminating) path from $b_1$ to $\mathbf{1}$, because in the original path $\xi$, all zeroes ``on the $b_1$ side'' have been resampled by resamplings ``on the $b_1$ side''. Since the sites $j_1,j_2$ inbetween never become zero, there can not be any zero ``on the $b_1$ side'' that was resampled by a resampling ``on the $b_2$ side''. - The probability of $\xi_1$, started from $b_1$, is given by - \begin{align*} - \P_{b_1}[\xi_1] &= \P(\text{choose }s_{a_1}) \P(r_{a_1}) \P(\text{choose }s_{a_2}) \P(r_{a_2}) \cdots \P(\text{choose }s_{a_{|\xi|}}) \P(r_{a_{|\xi|}}) \\ - &= \frac{1}{z^{(1)}_{a_1}} \P(r_{a_1}) \frac{1}{z^{(1)}_{a_2}} \P(r_{a_2}) \cdots \frac{1}{z^{(1)}_{a_{|\xi|}}} \P(r_{a_{|\xi|}}) - \end{align*} - and similar for $\xi_2$. - Vice versa, all paths $\xi_1\in\paths{b_1}\cap \mathrm{NZ}^{(j_1,j_2)}$ and $\xi_2\in\paths{b_2}\cap\mathrm{NZ}^{(j_1,j_2)}$ also induce a path $\xi\in\paths{b} \cap \mathrm{NZ}^{(j_1,j_2)}$ by simply interleaving the resampling positions. Note that $\xi_1,\xi_2$ actually induce $\binom{|\xi_1|+|\xi_2|}{|\xi_1|}$ paths $\xi$ because of the possible orderings of interleaving the resamplings in $\xi_1$ and $\xi_2$. The following diagram illustrates all possible interleavings: - \begin{center} - \includegraphics{diagram_paths.pdf} - \end{center} - A particular interleaving is a path through the above grid. So for a fixed $\xi_1,\xi_2$ there are $\binom{|\xi_1|+|\xi_2|}{|\xi_1|}$ possible interleavings $\xi$, and vice versa there are $\binom{|\xi_1|+|\xi_2|}{|\xi_1|}$ paths $\xi$ that decompose into the same $\xi_1$ and $\xi_2$. For a fixed $\xi_1,\xi_2$ we now show the following: + Vice versa, any two paths $\xi_1\in\paths{b_1}\cap \mathrm{NZ}^{(j_1,j_2)}$ and $\xi_2\in\paths{b_2}\cap\mathrm{NZ}^{(j_1,j_2)}$ also induce a path $\xi\in\paths{b} \cap \mathrm{NZ}^{(j_1,j_2)}$ by simply interleaving the resampling positions. Note that $\xi_1,\xi_2$ actually induce $\binom{|\xi_1|+|\xi_2|}{|\xi_1|}$ paths $\xi$ because of the possible orderings of interleaving the resamplings in $\xi_1$ and $\xi_2$. + For a fixed $\xi_1,\xi_2$ we will now show the following: \begin{align*} \sum_{\substack{\xi\in\paths{b} \cap \mathrm{NZ}^{(j_1,j_2)} \text{ s.t.}\\ \xi \text{ decomposes into } \xi_1,\xi_2 }} \P_b[\xi] &= \sum_{\text{interleavings of }\xi_1,\xi_2} \P(\text{interleaving}) \cdot \P_{b_1}[\xi_1] \cdot \P_{b_2}[\xi_2] \\ &= \P_{b_1}[\xi_1] \cdot \P_{b_2}[\xi_2] \end{align*} - This is best explained by an example. We have, for an example interleaving: + where both sums are over $\binom{|\xi_1|+|\xi_2|}{|\xi_1|}$ terms. + This is best explained by an example. Lets consider the following fixed $\xi_1,\xi_2$ and an example interleaving where we choose steps from $\xi_2,\xi_1,\xi_1,\xi_2,\cdots$: + \begin{align*} + \xi_1 &= \left( (z_1, s_1, r_1), (z_2, s_2, r_2), (z_3, s_3, r_3), (z_4, s_4, r_4),\cdots \right) \\ + \xi_2 &= \left( (z_1', s_1', r_1'), (z_2', s_2', r_2'), (z_3', s_3', r_3'), (z_4', s_4', r_4'),\cdots \right) \\ + \xi &= \left( (z_1 + z_1', s_1', r_1'), (z_1+z_2', s_1, r_1), (z_2+z_2', s_2, r_2), (z_3+z_2', s_2', r_2'), \cdots \right) + \end{align*} + The probability of $\xi_1$, started from $b_1$, is given by \begin{align*} - \xi &= \left( (z_1, s_1, r_1), (z_2, s_2, r_2), (z_3, s_3, r_3), (z_4, s_4, r_4), \cdots \right) \\ - \xi_1 &= \left( (z^{(1)}_1, s_1, r_1), \phantom{(z^{(2)}_2, s_2, r_2), (z^{(1)}_3, s_3, r_3),} (z^{(1)}_4, s_4, r_4),\cdots \right) \\ - \xi_2 &= \left( \phantom{(z^{(1)}_1, s_1, r_1),} (z^{(2)}_2, s_2, r_2), (z^{(2)}_3, s_3, r_3),\phantom{(z^{(2)}_4, s_4, r_4), } \cdots \right) + \P_{b_1}[\xi_1] &= \P(\text{choose }s_1) \P(r_{a_1}) \P(\text{choose }s_2) \P(r_{a_2}) \cdots \P(\text{choose }s_{|\xi_1|}) \P(r_{|\xi_1|}) \\ + &= \frac{1}{z_1} \P(r_1) \frac{1}{z_2} \P(r_2) \cdots \frac{1}{z_{|\xi_1|}} \P(r_{|\xi_1|}) . \end{align*} - Remember $z^{(1)}_i + z^{(2)}_i = z_i$. - The probability associated to this interleaving is + and similar for $\xi_2$ but with primes. + The following diagram illustrates all possible interleavings, and the red line corresponds to the particular interleaving $\xi$ in the example above. + \begin{center} + \includegraphics{diagram_paths2.pdf} + \end{center} + For the labels shown within the grid, define $p_{ij} = \frac{z_i}{z_i + z_j'}$. + The probability of $\xi$ is given by \begin{align*} - \P_b[\xi] &= \frac{1}{z_1} \P(r_1) \frac{1}{z_2} \P(r_2) \frac{1}{z_3} \P(r_3) \frac{1}{z_4} \P(r_4) \cdots \\ + \P_b[\xi] &= \frac{1}{z_1+z_1'} \P(r_1') \frac{1}{z_1+z_2'} \P(r_1) \frac{1}{z_2+z_2'} \P(r_2) \frac{1}{z_3+z_2'} \P(r_2') \cdots \tag{by definition}\\ &= - \frac{z^{(1)}_1}{z_1} \frac{1}{z^{(1)}_1} \P(r_1) \; - \frac{z^{(2)}_2}{z_2} \frac{1}{z^{(2)}_2} \P(r_2) \; - \frac{z^{(2)}_3}{z_3} \frac{1}{z^{(2)}_3} \P(r_3) \; - \frac{z^{(1)}_4}{z_4} \frac{1}{z^{(1)}_4} \P(r_4) - \cdots \\ + \frac{z_1'}{z_1+z_1'} \frac{1}{z_1'} \P(r_1') \; + \frac{z_1 }{z_1+z_2'} \frac{1}{z_1 } \P(r_1 ) \; + \frac{z_2 }{z_2+z_2'} \frac{1}{z_2 } \P(r_2 ) \; + \frac{z_2'}{z_3+z_2'} \frac{1}{z_2'} \P(r_2') + \cdots \tag{rewrite fractions}\\ &= - \frac{z^{(1)}_1}{z_1} - \frac{z^{(2)}_2}{z_2} - \frac{z^{(2)}_3}{z_3} - \frac{z^{(1)}_4}{z_4} + \frac{z_1'}{z_1+z_1'} \; + \frac{z_1 }{z_1+z_2'} \; + \frac{z_2 }{z_2+z_2'} \; + \frac{z_2'}{z_3+z_2'} \cdots - \P_{b_1}[\xi_1] \P_{b_2}[\xi_2] - \end{align*} - \todo{write more} - \begin{align*} - \P(\text{interleaving}) = \P(\text{choose step of }\xi_1) \P(\text{choose step of }\xi_2) \P(\text{choose step of }\xi_2) \P(\text{choose step of }\xi_1) \cdots - \end{align*} - These choices depend on the number of zeroes present in the state: - \begin{align*} - \P(\text{choose step of }\xi_1) &= - \frac{z^{(1)}_1}{z^{(1)}_1 + z^{(2)}_1} \\ - \P(\text{choose step of }\xi_2) &= - \frac{z^{(2)}_1}{z^{(1)}_1 + z^{(2)}_1} + \P_{b_1}[\xi_1] \; \P_{b_2}[\xi_2] \tag{definition of $\P_{b_i}[\xi_i]$} \\ + &= (1-p_{1,1}) \; p_{1,2} \; p_{2,2} \; (1-p_{3,2}) \; \P_{b_1}[\xi_1] \; \P_{b_2}[\xi_2] \tag{definition of $p_{i,j}$} \\ + &= \P(\text{path in grid}) \; \P_{b_1}[\xi_1] \; \P_{b_2}[\xi_2] \end{align*} - These are the $p_i$ and $1-p_i$ shown in the grid diagram. In $\P_{b_1}[\xi_1]$ we have a factor $\frac{1}{z^{(1)}_1}$, so together with the probability above this gives $\frac{z^{(1)}_1}{z^{(1)}_1 + z^{(2)}_1} \frac{1}{z^{(1)}_1} = \frac{1}{z_1}$, as in the original expression for $\P_b[\xi]$. In the grid we see that at every point the probabilities sum to 1, and we always reach the end, so we know the sum of all paths in the grid is 1. This means the sum of all $\P(\text{interleaving})$ is equal to 1. + In the grid we see that at every point the probabilities sum to 1, and we always reach the end, so we know the sum of all paths in the grid is 1. This proves the required equality. We obtain \begin{align*}